text:
HomeNews Introducing our expert National Archives Committee
Introducing our expert National Archives Committee
Prev Dragon to the rescue: Michele Romanow and partner launch new venture to source PPE during pandemic Globe & Mail Aug 2, 202005 August 2020Next Національний Архівний Комітет Конґресу06 August 2020
August 6, 2020in News 0 Likes
National Archives Committee – Who we are?
In November 2019, the Ukrainian Canadian Congress passed a motion to establish the National Archives Committee. The aim of this committee is to raise awareness, in the Ukrainian Canadian community across Canada, about the importance of preserving the community's documentary heritage created by individuals, organizations, and institutions over the last one hundred plus years and those that are being created today. The committee was asked to work with, and encourage organizations to preserve their institutional documentary heritage and act as a coordinating body in providing information on preservation and archival standards, education, and funding opportunities for archives. An additional goal will be to explore a way to share information about existing Ukrainian Canadian archives and where they reside.
The co-chairs of this committee are Maryna Chernyavska and Andriy Sawchuk. To date other members are Bohdan Figol, Andrew Paranczuk and Irene Jendzjowsky. Myron Momryk – advisor.
About the National Archives Committee Members:
Maryna Chernyavska is the archivist at the Kule Folklore Centre, University of Alberta and a database administrator for the Archives Society of Alberta's provincial database Alberta On Record. Maryna has a Master of Arts degree in Ukrainian Folklore and a Master of Library and Information Studies. She is an active member of the Ukrainian community in Edmonton, and of the international archival community.
Andriy Sawchuk works as an archivist at Library and Archives Canada in the nation's capital – Ottawa. He comes from St. Catharines, Ontario and has been involved in the Ukrainian Canadian community all of his life. His educational background includes art history, archaeology, and museum studies.
Bohdan Figol worked in the computer industry for over 40 years including General Electric, Honeywell and Lava Computer MFG Inc. He has a BA and MLS from the University of Toronto and has taught and developed IT courses in Community Colleges for 15 years. Bohdan is an active member of the Ukrainian Canadian community in Toronto.
Andrew Paranczuk works in the FinTech space, currently in Big Data, as a Software Engineer/Manager. Andrew has a BSc in Computer Science from the University of Toronto and is working towards an MBA, slated to graduate in 2021. Andrew is an active member of the Ukrainian Canadian community in Toronto."
Iryna Jendzjowsky has been working in the heritage field for over 30 years and for the last 18 years, before she retired, she worked at the Provincial Archives of Alberta. She has a Masters degree in Ukrainian and a Masters in Library and Information Studies. Iryna is an active member of the Ukrainian Canadian community in Edmonton.
Myron Momryk is an archivist and author. He was raised in Oshawa, Ontario, and graduated from the University of Waterloo in 1972 with an M.A. in Canadian History. He is a retired Archivist from Library and Archives Canada, where he worked in the Multicultural Archives Program for twenty-five years. He was responsible for acquiring and organizing archival material from East European groups, particularly the Ukrainian Canadian community. Myron continues to research and write about Ukrainian Canadian history.
In the near future, the Committee will be sending out a brief survey to organizations that are members of UCC to get a sense of the range, complexity, and variety of the documentary heritage that exists in Canada, where it resides and the needs of the community in preserving the documentary heritage. Stay tuned and look for additional information in the UCC newsletters. In the meantime, if you have any questions please feel free to contact the committee through the following email address: [email protected]
December 3, 2020in News 0 Comments 0 Likes
Архіви – що варто зберігати?
Archives – what do we keep?
Що таке документальна спадщина і чому треба зберігати архіви?
What is documentary heritage and why keep archives?
August 6, 2020in News 0 Comments 0 Likes
Національний Архівний Комітет Конґресу
March 7, 2019in Holodomor 0 Comments 0 Likes
REPORT: Saving Archives in Montreal from 1983 on the 1932-33 Holodomor


text:
Dozens of Air Jordans stolen in robbery at Arkansas home
Posted: Mon 10:08 PM, Dec 31, 2018
LITTLE ROCK, Ark. (AP) — Police in Little Rock say a man armed with a pistol forced his way into a home and stole dozens of pairs of Air Jordan sneakers.
The robbery was reported Friday at a home in Little Rock. According to Little Rock TV station KATV , a man told police that he was getting out of his car when he was approached by a man who hit him with a pistol and forced him to open the door of the home.
According to police, the armed man took about 50 pairs of sneakers worth a total of $8,750, as well as two cellphones, a pair of earrings and a Rolex watch. He also stole a vehicle, but that was later found abandoned.
The police report says no arrests have been made.
Information from: KATV-TV, http://www.katv.com/

redpajama_set_name: RedPajamaCommonCrawl


text:
Place A Legal Notice/LLC
Times E-Edition
By Anthony Gannon
Marriage At The Market
Visitors of the Clinton Farmers Market last week were witness to more than just the foods and flowers of the usual vendors and stalls.
Love bloomed in the middle of the market for the marriage of Corinna Edick and Bill Simpson. The 3 o'clock ceremony began with a humorous address from the minister, a personal friend of the newlyweds, with inside jokes from their past and reminding them of the significance of their vows to each other.
The full story is in this week's edition of the newspaper.
THE WATERVILLE TIMES
© 2023 by The Waterville Times-Helen Publishing


The Paleo diet, also popularly referred to as the "caveman" diet, is a modern/Western nutritional plan based on a diet of wild plants and animal flesh our ancestors presumably ate during the Paleolithic era, some 2.5 million years ago before the Neolithic Revolution ushered in agriculture. The diet consists mainly of fish, grass-fed pasture raised meats, eggs, vegetables, fruit, fungi, roots, and nuts, and excludes grains, legumes, dairy product, potatoes, refined salt, refined sugar, and processed oils; in others words, it's basically the diet of the Mbuti and San people of Afrika. So then why isn't it called the Mbuti or San diet? Western cultural chauvinism won't let that happen, after all, the diet is a Western creation.
The diet is based on evolutionary thought in that it presupposes the early human diet was based on scarcity as humans battled against Nature for survival. And that the Paleo diet matches our level of evolution, and is suited to or even assisted in the transformation to human from prehuman. However, there is available information that challenges if not disproves such assumptions about early humanity. It seems, in fact, that ancient Afrikan hunters perceived Nature, not as hostile, but as accommodating and congenial. According to archaeological evidence, the ancient savannas teemed with animal life, suggesting the prehistoric hunter had a greater variation of wildlife than contemporary hunters.
A study of the San by Richard Lee is instructive. The San live in the desert, an environment of little resources and even less water. One would think such an environment would produce an inadequate diet and therefore a pessimistic worldview. Lee discovered the San daily caloric intake was 2,140 calories, which is more than the American Recommended Daily Allowance. Also, Lee determined the San were selective eaters. Of the 223 variety of animal species, they ate only fifty- four species, and only seventeen of them formed the core meat consumed. What is more surprising however, is their diet consisted of mainly vegetables. He found the population was healthy and showed no "clinical-level" symptoms of vitamin deficiency.
The above information refutes evolutionary biological theory, which states that natural selection had sufficient time to genetically adapt the metabolism and physiology of Paleolithic humans to the varying dietary conditions of that era. But what was so varying about the life of early Afrikan hunters? The warm climate and abundant wildlife offered choice and plenty. Drawing from Lee's study, ancient hunters had an even greater variety of meats, and through vegeculture, which is the replanting or transplanting of plants from one area to another, they were able to ensure their vegetable food sources. (Other foods that were available to Paleolithic hunters were wild fruit, berries, mushrooms, juicy green leaves, stems and roots.) There was no struggle to survive as scarcity was not an issue. Maybe in the caves of Europe there was scarcity, perhaps starvation, but not in Afrika. Therefore there was no need for species adaptation. So why are we using Europe as our reference? Oh, I forgot, the diet is a Western (European) creation.
Proponents of the Paleo diet further argue that in the 10,000 years since the invention of agriculture and its consequent major change in the human diet, natural selection has had too little time to make the optimal genetic adaptations to the new diet. Thus, even with the coming of the Neolithic Revolution (which developed during the period of Afrikan hegemony) the species has not physiologically and metabolically adapted, resulting in the contemporary human diet, being deficient, and in turn contributing to many of the so-called diseases of civilization or affluence. This too is a fallacious argument because 100 years ago, the five leading causes of death in the U.S. were Pneumonia, Influenza, Tuberculosis , Diarrhea , Heart disease and Stroke, which indicates that diet was not the main disease causing factor. If the failure to adapt to the Neolithic Revolution and the concomitant change in diet were the culprit, then surely its effects would have been exhibited 100 years ago, so the Neolithic Revolution cannot be the offending agent. If anything the Industrial Revolution is to blame. But again the Industrial Revolution has been with us for some time now, so why would its inimical effects become evident in only the last 40 years or so ?
The Neolithic and Industrial Revolutions are neither to blame. This pejorative health state is a consequence of changes is food science and lifestyles. True, scientists know more about nutrition and health and people are living longer. But people are less healthy and are dying from conditions brought about by science and technology, two of the things that were suppose to save us. They eat genetically modified foods, a gift from science; and they don't exercise because technological advances no longer require the human being to work the body to do the things it was design to do in order for living, i.e., farm, hunt, build shelter, make cloths--all these things are done by machines, robots, computers. And no longer perform the tasks that rooted them to life. Today, people have been taught to believe that a pill will cure everything. They are even willing to live (or die) from the often dangerous side effects of the pill.
Western scientific research in Biology, Biochemistry, Ophthalmology, Dermatology and many other disciplines indicate it is the modern or "evolved" diet, full of refined foods, trans fats and sugar, that is the cause of degenerative diseases such as obesity, cancer, diabetes, heart disease, Parkinson's, Alzheimer's, depression and infertility. The typical American diet is too high in saturated fat, sugar, and sodium, and too low in fruits, vegetables, whole grains, fiber, and calcium, and has contributed to four of the six leading causes of death and increases the risk of numerous diseases, including: heart disease, diabetes, obesity, hypertension, stroke, osteoporosis, and cancers (colon, prostate, mouth, throat, esophagus, lung, stomach).
If anything, the Scientific Revolution is the real villain because it was supported a particular worldview; a worldview that has helped to create a "god-complex" in Western man, whereas he no longer has moral or scientific limits; he no longer knows what should and should not be toyed with. He's messing with our food and he should know better. Don't mess with a man's food. On a more personal note, I eat lots of salads (raw food), no sugar (stevia, I've learn how to use it), and fish. Lately I've been eating more salmon (and it has all been Wild Alaskan either) and tuna, but that will come to an end now as I just found out that the FDA has cleared the way for salmon to be Genetically Modified, and I just read a report that 60% of the fish labeled tuna is fake--it's not tuna at all. Damn, even if I were a Breatharian, the air pollutants would make that diet less nutritious and my death would still be premature. As for the Paleo diet, aside from the theories and mythologies that surround it, it's very close to the San or Mbuti diet. But with all diets, be careful, pay attention to your body, and various fasts (detox) regularly to keep whatever dietary impurities you consume to a limit and let the body balance itself.
You hit the nail on the head yet again. I have severally come across this diet and had to wonder how can we start evolving backwards, after having worked up with good food all this while? Going backwards to a diet that the human has already evolved from is weird to say the least. The truth of the matter seems to be that health has taken the back-seat and our careers seem to be the most important issues around us. with a well balanced diet and regular exercise, we can lead healthy lifestyles, but exercise is rarely considered these days.
Pretty awesome to share such wonderful experiences and comments as motioned above but I like my work in my own way if you like then please visit this KEYWORD.
Most of us are easily carried away by whatever sweeps our way. There was a big fuss sometime back about the Paleo diet. It is a good thing that you have taken time to expose and breakdown what it really entails. I totally agree with this and it certainly appears to be a step in the backward direction. What people should learn is just eating healthy and also detoxifying the body often. This will be an easy lifestyle to follow with or without the Paleo diet.


text:
Social supports and interventions can decrease developmental disadvantages in children
Researchers in Determinants of Child Development Lab examine critical ingredients needed for children to succeed
Megan Fox, Advancement
This story is the first in a three-part series on how University of Calgary researchers and the United Way of Calgary and Area are making progress in the areas of raising successful kids, overcoming poverty and building strong communities. Together, we can improve local lives. Find out how you can help.
Ongoing research in the Faculty of Arts has found that increased access to community supports early in life can help mitigate childhood developmental disadvantages.
Researchers in the Determinants of Child Development Lab are examining the critical ingredients needed for children to succeed and are identifying how to put them on a trajectory to achieve developmental milestones on or even ahead of time.
"We find that social and relational support are important buffering factors that can help children thrive, and also offset risks that children may experience, such as low family income," says lab director Dr. Sheri Madigan, PhD, an assistant professor in the Department of Psychology.
Madigan's lab researches how the first 2,000 days of life can set the groundwork for later growth and development, and how access to additional positive resources in the community is important.
"When children aren't receiving the supports they need at home, but they get it elsewhere — such as from their daycare provider, teacher or coach — this relationship can serve to bolster their learning and development," says Madigan.
When addressing developmental gaps in young children, Madigan says early intervention is important, as these gaps tend to widen rather than shrink over time.
When children or their families experience hardship, we need to provide them with as many supports as possible to help tip the scale toward more positive outcomes.
Madigan hopes her research will advance understanding of how to shrink disparities in child development and inform intervention strategies for those who need them.
United Way supports child development in Calgary
The United Way of Calgary and Area supports efforts to address the same developmental gaps as the ones being looked at in Madigan's lab. Partner agencies such as Between Friends, Families Matter, Wood's Homes, YMCA, Big Brothers and Big Sisters Calgary, Junior Achievement of Southern Alberta, and Trellis Society (formerly Boys and Girls Clubs of Calgary) offer interventions and connections to help kids succeed, including learning programs, tutoring, mentoring and family counselling.
Poverty, food insecurity, mental illness, domestic violence, and lack of early supports are all risk factors that could prevent children from reaching their developmental milestones. According to the United Way, only one in five kids who need mental health services receives them, and one in five kids in Calgary does not complete high school on time.
Initiatives like the United Way's All In for Youth program have helped reduce the number of children in Calgary who do not complete high school on time by 20 per cent.
UCalgary's 2021 United Way Campaign runs from Oct. 4 until Dec. 10. Your donations help to support programs like those mentioned that ensure children in our community are given the tools needed to succeed. This year's goal is to raise $625,000 to help improve local lives.
Donate to UCalgary's United Way campaign by logging in with your UCalgary email address.
The University of Calgary is driving science and innovation to transform the health and well-being of children and families. Led by the Alberta Children's Hospital Research Institute, top scientists across the campus are partnering with Alberta Health Services, the Alberta Children's Hospital Foundation, and our community to create a better future for children through research.
Who are you thankful for?
United Way celebrates Cindy Graham for her work on UCalgary's 2020 campaign
2021 United Way Campaign: Research, philanthropy, and the power of working together
Record-setting year for 2020 UCalgary United Way campaign
caring campus
children's well-being


text:
Ingredient handling
Customers -
When tradition meets innovation at Destrooper Olivier
Destrooper-Olivier - established in 1986 by the Luc Destrooper and Christiane Olivier – is a thriving Belgian family business, with the second generation currently at the helm. Their own "Belgian Butters" brand is renowned for their wide sophisticated range of the finest pure butter biscuits, waffle crisps and chocolate "petit fours".
Authenticity and tradition are core values of the company and in order to meet the increasing national and international demand and in order to ensure the quality of the authentic recipes, the company decided in 2015 to scale up its production site to a more modern and performing facility. Given the fact that Spiromatic had also installed an important part of the initial infrastructure, it was only logic that they were called to tackle the renovation of the factory.
Since its start in 1986 the company has continued to grow and in order to meet that growing demand the originally quite modest installation had been 'expanded' with intermediate solutions. As a result the production flow had become far from lean and moreover these DIY - expansions had reached their limits.
Spiromatic was asked not only to modernize the infrastructure, but also to reconfigure the production lay out within the existing premises. Before designing the new lay out, the Spiromatic engineering team spent quite some time with the Destrooper Olivier management and their German co-owner Küchenmeister, in order to ensure that the new facility would meet all the client's demands, not only today but also tomorrow.
Given their vast experience, turning this small-scale and slightly outdated bakery into a fully automated and modern production facility was a piece of cake for the Spiromatic engineers. They developed a comprehensive configuration where nearly everything was replaced, moved and automated:
Flour and sugar are now stored in two new 50-ton composite silos. The silos were personalized by adding the Belgian Butter logo and moreover, the galvanized top was masked by a sleeve, giving the silos a very lean look.
As the dough chamber is the heart of the factory, special attention was given to this. Spiromatic replaced 2 of the 3 Tonnaer kneaders and moreover, moved all 3 of them in order to get a leaner production flow. In order to increase accuracy, Spiromatic also installed 3 new weighing bunkers in the dough chamber, each with a 350 kg content.
Dedusting filters were installed on the kneaders, for a cleaner and healthier work environment.
Spiromatic also helped designing a separate room with kneaders solely used for handling allergen ingredients.
A new spiral transport now guarantees a fast (3.5 ton/hour) flux of ingredients to the kneaders.
Moreover, a new Siemens automation system – linked to the client's ERP system - was installed, ensuring a very accurate and phased handling and dosing of ingredients into the kneaders.
The new installation is more user-friendly, faster and more reliable than before, resulting in constant quality of the traditional recipes.
The remote control access to the automation system allows Destrooper to perfectly control, monitor and trace the ingredient flux, the mixing and the production process, even from a distance.
Thanks to the automation system, there is a better recipe control and optimal traceability.
The new installation ensures the company is ready to meet further increased demand and other future challenges.


The item How to twist a dragon's tale, Cressida Cowell represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Royal Borough of Kensington and Chelsea Libraries.
Someone has stolen the Fire stone. Now the volcano on Volcano Island has become active and the tremors are hatching the eggs of the Exterminator dragons! Can Hiccup stop the Volcano from erupting, and save the Tribes from being wiped out by the terrible sword-claws of the Exterminators?


text:
Q: Swift can't access part of JSON response with enum initializer I have trouble accessing a part of a JSON response.
(part of)The Source:
{
"time":1552726518,
"result":"success",
"errors":null,
"responce":{
  "categories":{
     "1":{
        "nl":{
           "name":"New born",
           "description":"TEST",
           "children":[
              {
                 "name":"Unisex",
                 "description":"TESTe",
                 "children":[
                    {
                       "name":"Pants",
                       "description":"TEST",
                       "children":false
                    }
                 ]
              }
           ]
        }
     }
   }
 }
}

Approach
As you can see the source can have multiple categories. A single categorie will have a 'name', 'description' and may have 'children'. Children wil have a 'name', 'description' and also may have 'children'. This might go endless. If there are no children the SJON statement is 'false'
I use the website: https://app.quicktype.io to generate a junk of code to parse the JSON. I modified the result, because the website doesn't know that the children can go endless:
struct ProductCategories: Codable {
let time: Int?
let result: String?
let errors: [String]?
let responce: ProductCategoriesResponce?

init(time: Int? = nil, result: String? = nil, errors: [String]? = nil, responce: ProductCategoriesResponce? = nil) {

    self.time = time
    self.result = result
    self.errors = errors
    self.responce = responce
}
}

struct ProductCategoriesResponce: Codable {
let categories: [String: Category]?
}

struct Category: Codable {
let nl, en: Children?
}

struct Children: Codable {
let name, description: String?
let children: EnChildren?
}

enum EnChildren: Codable {
case bool(Bool)
case childArray([Children])

init(from decoder: Decoder) throws {
    let container = try decoder.singleValueContainer()
    if let x = try? container.decode(Bool.self) {
        self = .bool(x)
        return
    }
    if let x = try? container.decode([Children].self) {
        self = .childArray(x)
        return
    }
    throw DecodingError.typeMismatch(EnChildren.self, DecodingError.Context(codingPath: decoder.codingPath, debugDescription: "Wrong type for EnChildren"))
}

func encode(to encoder: Encoder) throws {
    var container = encoder.singleValueContainer()
    switch self {
    case .bool(let x):
        try container.encode(x)
    case .childArray(let x):
        try container.encode(x)
    }
}
}

And I can decode de data with:
productCategories = try JSONDecoder().decode(ProductCategories.self, from: jsonData!)

This part works fine. I can access "New Born", but can't get access his children.
I search a long time for the answer I tried so much. To much to all share here. What I expect to get access is:
 if let temp = productCategories.responce?.categories?["\(indexPath.item)"]?.nl?.children! {
 let x = temp(from: Decoder)

but this will trow me an error:
"Cannot call value of non-function type 'EnChildren'"
also code like:
  let temp1 = productCategories.responce?.categories?["\(indexPath.item)"]?.nl?.children

Won't get me anywhere.
So, any ideas? Thank you.

A: First of all: If you are responsible for the server side, send null for the leaf children or omit the key rather than sending false. It makes the decoding process so much easier.
app.quicktype.io is a great resource but I disagree with its suggestion.
My solution uses a regular struct Children with a custom initializer. This struct can be used recursively.
The key children is decoded conditionally. First decode [Children], on failure decode Bool and set children to nil or hand over the error.
And I declared struct members as much as possible as non-optional
struct Response: Decodable {
    let time: Date
    let result: String
    let errors: [String]?
    let responce: ProductCategories
}

struct ProductCategories: Decodable {
    let categories: [String: Category]
}

struct Category: Decodable {
    let nl: Children
}

struct Children: Decodable {
    let name, description: String
    let children : [Children]?

    private enum CodingKeys : String, CodingKey { case name, description, children }

    init(from decoder: Decoder) throws {
        let container = try decoder.container(keyedBy: CodingKeys.self)
        name = try container.decode(String.self, forKey: .name)
        description = try container.decode(String.self, forKey: .description)
        do {
            children = try container.decode([Children].self, forKey: .children)
        } catch DecodingError.typeMismatch {
            _ = try container.decode(Bool.self, forKey: .children)
            children = nil
        }
    }
}

In the root object the key time can be decoded to Date if you specify an appropriate date decoding strategy
let decoder = JSONDecoder()
decoder.dateDecodingStrategy = .secondsSince1970
let result = try decoder.decode(Response.self, from: jsonData!)

And you can access for example the name Unisex with
result.responce.categories["1"]?.nl.children?[0].name

redpajama_set_name: RedPajamaStackExchange


text:
Intellasia East Asia News – Shortage of COVID-19 testing kits 'not…
Home Americas Publisher Pulls Alice Sebold's Memoir As 1981 Rape Conviction Is Overturned
Publisher Pulls Alice Sebold's Memoir As 1981 Rape Conviction Is Overturned
The publishers of Alice Sebold's "Lucky" are ceasing distribution of the memoir after a man was cleared of the 1981 rape recounted in the book.
"Following the recent exoneration of Anthony Broadwater, and in consultation with the author, Scribner and Simon & Schuster will cease distribution of all formats of Alice Sebold's 1999 memoir 'Lucky' while Sebold and Scribner together consider how the work might be revised," Simon & Schuster ― which owns Scribner, Sebold's publisher ― wrote in a statement issued Tuesday, according to Publishers Weekly.
The company reiterated the statement on Twitter early Wednesday.
Following the recent exoneration of Anthony Broadwater, and in consultation with the author, Scribner and Simon & Schuster will cease distribution of all formats of Alice Sebold's 1999 memoir LUCKY.
[1/2]
— Simon & Schuster (@simonschuster) December 1, 2021
Broadwater, a Black man, was convicted of Sebold's rape in 1982 and spent 16 years in prison. According to his attorneys, he was denied parole at least five times because he wouldn't admit to a crime he didn't commit.
He was released from prison in 1999, after which he was required to register as a sex offender. Now 61, he was exonerated by a New York trial judge last week.
In 1999's "Lucky," Sebold recounted being raped when she was an 18-year-old student at Syracuse University. She explained in the book that she'd picked out another man, not Broadwater, in a police lineup following the assault. By the time Sebold got to the witness stand, however, she identified Broadwater as her attacker.
On Tuesday, the author ― who is best known for 2002's "The Lovely Bones," which also deals with sexual assault ― addressed Broadwater's exoneration in a statement published on Medium.
"I am grateful that Mr. Broadwater has finally been vindicated, but the fact remains that 40 years ago, he became another young Black man brutalized by our flawed legal system," she wrote. "I will forever be sorry for what was done to him."
Doubts about Broadwater's conviction resurfaced in 2019 when work began on a planned movie adaptation of "Lucky." Timothy Mucciante, an executive producer on the project, told The New York Times last month that he "started having some doubts" about the case after noting discrepancies in the script.
Plans for the film, which was to star "You" actor Victoria Pedretti, have since been shelved. On Wednesday, it was announced that a different production team would move forward with a documentary based on the case, to be titled "Unlucky."
var _fbPartnerID = null; if (_fbPartnerID !== null) { fbq('init', _fbPartnerID + ''); fbq('track', "PageView"); }
(function () { 'use strict'; document.addEventListener('DOMContentLoaded', function () { document.body.addEventListener('click', function(event) { fbq('track', "Click"); }); }); })();
Alice Sebold
Tulsa Megachurch Pastor Under Fire For Rubbing Spit Into Man's Face During Sermon


text:
\section{Introduction}
In this paper we investigate the estimation of an unknown deterministic signal that is observed through a deterministic data matrix under additive noise \cite{kuruoglu1,donmez,dsp1,dsp2,dsp3,dsp4,kuruoglu2,kalantarova,Ghaoui97,Kailath:book,yonina,sayedchandra1,sayedchandra2,sayedchandra3}. Although the data matrix and the output vector are not exactly known, estimates for both of them as well as uncertainty bounds on the estimates are given \cite{pilanci10,sayed02,donmez,kalantarova,yonina2,yonina4}. When the model parameters are not known exactly, a popular method to estimate the desired signal is to use the robust LS method \cite{Ghaoui97}, since the performances of the classical LS estimators degrade significantly when the perturbations on the data matrix and the output vector are relatively high \cite{Ghaoui97,sayed02,pilanci10,kuruoglu3,schubert,huber1}.

A prevalent approach to find robust solutions to such estimation problems is the robust LS method \cite{Ghaoui97, sayed02,kalantarova}, in which the uncertainties in the data matrix and the output vector are incorporated into optimization framework via a minimax residual formulation. Another well-known approach to compensate for errors in the data matrix and the output vector is the total least squares method (TLS) \cite{pilanci10}, which may yield undesirable results since it employs a conservative approach due to data de-regularization. Furthermore, the data matrix usually has a known special structure, such as Toeplitz and Hankel, in many linear regression problems \cite{Ghaoui97, pilanci10} and the performance of the estimators based on minimax approaches are shown to improve
when such a prior knowledge on data matrix structure is integrated into the problem formulation \cite{Ghaoui97, pilanci10}.

Although the robust LS methods are able to minimize the LS error for the worst-case perturbations, they usually provide unsatisfactory results on the average \cite{pilanci10,chandra1,chandra2,chandra3} due to their conservative nature. In order to counterbalance this conservative nature of the robust LS methods \cite{Ghaoui97}, we propose a novel robust LS approach that minimizes a worst case ``regret'' that is defined as the difference between the squared residual error and the smallest attainable squared residual error with an LS estimator. By this regret formulation, we seek a linear estimator whose performance is as close as possible to that of the optimal estimator for all possible perturbations on the data matrix and the output vector. Our main goal in proposing the minimax regret formulation is to provide a trade-off between the robust LS methods tuned to the worst possible data parameters (under the uncertainty bounds) and the optimal LS estimator tuned to the underlying unknown model parameters. Furthermore, after studying the data estimation problems in the presence of bounded data uncertainties, we extend the regret formulation to the regularized LS problem, where the regret is defined as the difference between the cost of the regularized LS algorithm \cite{Kailath:book, sayed02}, and the smallest attainable cost with a linear regularized LS estimator. Furthermore, we extend our discussions to scenarios involving both structured and unstructured data. Under these frameworks, we provide the solutions for the proposed regret based minimax LS and the regret based minimax regularized LS approaches in semi-definite programming (SDP) forms. We emphasize that SDP problems can be efficiently solved even for real-time applications \cite{boyd}.

Minimax regret approaches have been presented in signal processing literature to alleviate the pessimistic nature of the worst case optimization methods \cite{ElMe04, ElTaNe04, eldar_mrr, Kozat:09TSP,donmez,kalantarova}. However, we emphasize that the methods proposed in this paper extensively differ from \cite{Ghaoui97, sayed02, ElMe04, ElTaNe04,Kozat:09TSP,donmez}. Note that the optimization frameworks investigated here are different than \cite{Ghaoui97, sayed02}, where the regret terms are directly adjoined in the cost functions. Although a similar regret notion is used in \cite{ElMe04,ElTaNe04,Kozat:09TSP,donmez}, the cost function as well as the constraints on uncertainties in the data matrix and the output vector are substantially different in this paper. Moreover, unlike this paper, in \cite{donmez}, the problem is described for the channel equalization scenario, where the authors rely on the statistical assumptions. Furthermore, we note that the uncertainty is in the statistics of the transmitted signal in \cite{ElMe04}. In \cite{ElTaNe04} and \cite{Kozat:09TSP}, the uncertainty is in the transmitted signal and the channel parameters, respectively. Unlike these relevant works, in this paper, the uncertainty is both on the data matrix and the output vector. Furthermore, the solutions to the LS problems presented in this paper cannot be obtained from \cite{ElMe04, ElTaNe04, Kozat:09TSP,donmez,kalantarova}, since the cost functions are different in our optimization formulations. While in \cite{kalantarova}, the authors have considered a similar framework, the results of this paper builds upon them and provide a complete solution to the regret based robust LS estimation methods unlike \cite{kalantarova}. We emphasize that perturbation bounds on the data matrix and the output vector heavily depend on the estimation algorithms employed to obtain them. Since our methods are formulated for given perturbation bounds, different estimation algorithms can be readily incorporated into our framework with the corresponding perturbation bounds \cite{sayed02}.

In this paper, we first present a novel robust LS approach in which we seek to find the transmitted signal by minimizing the worst case regret, i.e., the worst case difference between the residual error of the LS estimator and the residual error with the optimal LS estimator. In this sense, our aim is to introduce a trade off between the performance of the robust LS methods and the tuned LS estimator (LS estimator that is tuned to the unknown data matrix and the output vector). We next propose a minimax regret method for the regularized LS problem. Finally, we introduce a structured robust LS method in which the data matrix has a special structure such as Toeplitz and Hankel. We demonstrate that the proposed robust methods can be cast as SDP problems. In our simulations, we observe that these approaches provide better performance compared to the robust methods that are optimized with respect to the worst-case residual error \cite{Ghaoui97, sayed02}, the tuned LS and the tuned regularized LS estimators (tuned to the estimates of the data matrix and the output vector), respectively.

Our main contributions in this paper are as follows. {\em i)} We introduce a novel and efficient robust LS estimation method in which we find the transmitted signal by minimizing the worst-case regret, i.e., the worst-case difference
between the residual error of the LS estimator and the residual error of the optimal LS estimator tuned to the underlying model. In this sense, we present a robust estimation method that achieves a tradeoff between the robust LS estimation methods and the direct LS estimation method tuned to the estimates of the data matrix and output vector. {\em ii)} We next propose a minimax regret formulation for the regularized LS estimation problem. {\em iii)} We then introduce a structured robust LS estimation method in which the data matrix is known to have a special structure such as Toeplitz or Hankel. {\em iv)} We demonstrate that the robust estimation methods we propose can be cast as SDP problems, hence our methods can be efficiently applied for real-time \cite{boyd}. {\em iv)} In our simulations, we observe that our approaches provide better performance compared to the robust methods that are optimized with respect to the worst-case residual error \cite{est1, est2}, and the conventional methods that directly solve the estimation problem using the perturbed data.

The organization of the paper is as follows. An overview to the problem is provided in Section \ref{sec:system}. In Section \ref{sec:urls}, we first introduce the LS estimation method based on our regret formulation, and then present the regularized LS estimation approach in Section \ref{sec:urrls}. We then consider the structured LS approach in Section \ref{sec:srls} and provide the explicit SDP formulations for all problems. The numerical examples are demonstrated in Section \ref{sec:numer}. Finally, the paper concludes with certain remarks in Section \ref{sec:conc}.

\section{System Overview} \label{sec:system}
\subsection{Notation}
In this paper, all vectors are column vectors and represented by boldface lowercase letters. Matrices are represented by boldface uppercase letters. For a matrix $\vH$, $\vH^H$ is the conjugate transpose, $\norm{\vH}$ is the spectral norm, $\vH^+$ is the pseudo-inverse, $\vH > 0$ represents a positive definite matrix and $\vH \geq 0$ represents a positive semi-definite matrix. For a square matrix $\vH$, $\tr(\vH)$ is the trace. Naturally, for a vector $\vx$, $\norm{\vx} = \sqrt{\vx^H \vx}$ is the $\ell^2$-norm. Here, $\vec{0}$ denotes a vector or matrix with all zero elements and the dimensions can be understood from the context. Similarly, $\vI$ represents the appropriate sized identity matrix. The operator $\col(\cdot)$ is the vectorization operator, i.e., it stacks the columns of a matrix of dimension $m\times n$ into a $mn \times 1$ column vector. Finally, the operator $\otimes$ is the Kronecker product \cite{graham}.

\subsection{Problem Description}
We investigate the problem of estimating an unknown deterministic vector $\vx \in \mathbb{C}^{n}$ which is observed through a deterministic data matrix. However, instead of the actual data matrix and the output vector, their estimates $\vH \in \C^{m\times n}$ and $\vy \in \C^m$ and uncertainty bounds on these estimates are provided. In this sense, our aim is to find a solution to the following data estimation problem
\[
\vy \approx \vH \vx,
\]
such that
\[
\vy+\dy = (\vH+\dH)\vx,
\]
for deterministic perturbations $\dH \in \C^{m \times n}$, $\dy \in \C^m$. Although these perturbations are unknown, a bound on each perturbation is provided, i.e.,
\[
\norm{\dH} \leq \delta_H \text{ and } \norm{\dy} \leq \delta_Y,
\]
where $\delta_H, \delta_Y \geq 0$. In this sense, we refrain from any assumptions on the data matrix and the output vector, yet consider that the estimates $\vH$ and $\vy$ are at least accurate to ``some degree'' but their actual values under these uncertainties are completely unknown to the estimator.

Even in the presence of these uncertainties, the symbol vector $\vx$ can be naively estimated by simply substituting the estimates $\vH$ and $\vy$ into the LS estimator \cite{sayedbook}. For the LS estimator we have
\[
\hat{\vx} = \vH^ + \vy,
\]
where $\vH^+$ is the pseudo-inverse of $\vH$ \cite{graham}. However, this approach yields unsatisfactory results, when the errors in the estimates of the data matrix and the output vector are relatively high \cite{est1, est2, yonina1, yonina2, yonina3}. A common approach to find a robust solution is to employ a worst-case residual minimization \cite{est1}
\[
\hat{\vx} = \argmin_{\vx \in \C^n} \max_{\norm{\dH} \leq \delta_H, \norm{\dy} \leq \delta_Y} {\norm{(\vy+\dy)-(\vH + \dH)\vx}}^2,
\]
where $\vx$ is chosen to minimize the worst-case residual error in the uncertainty region. However, since the solution is found with respect to the worst possible data matrix and output vector in the uncertainty regions, it may be highly conservative \cite{tls, yonina1, yonina3}.

Here, we propose a novel LS estimation approach that provides a tradeoff between performance and robustness in order to mitigate the conservative nature of the worst-case residual minimization approach as well as to preserve robustness \cite{yonina1, yonina3}. The regret for not using the optimal LS estimator is defined as the difference between the residual error with an estimate of the input vector and the residual error with the optimal LS estimator, i.e.,
\begin{equation}\label{eq:LSreg0}
  \R(\vx; \dH,\dy) \defi {\norm{(\vy+\dy)-(\vH + \dH)\vx}}^2 - \min_{\vw \in \C^n}{\norm{(\vy+\dy)-(\vH +\dH)\vw}}^2.
\end{equation}
By making such a regret definition, we force our estimator not to construct the symbol vector according to the worst possible scenario considering that it may be too conservative. Instead, we define the regret of any estimator by the difference in the estimation performances of that estimator and the ``smartest'' estimator knowing both data matrix and output vector in hindsight, so that we achieve a tradeoff between robustness and estimation performance.

We emphasize that the regret defined in \eqref{eq:LSreg0} is completely different than the regret formulation introduced in \cite{yonina1, yonina3}. In \eqref{eq:LSreg0}, the uncertainty is on the data matrix where the desired data vector $\vx$ is completely unknown, unlike \cite{yonina1, yonina3}. We emphasize that we use the residual error ${\norm{(\vy+\dy)-(\vH + \dH)\vx}}^2$ instead of the estimation error $\norm{\hat{\vx} - \vx}$ since the estimation error directly depends on the vector $\vx$ and cannot be used in the regret formulation since $\vx$ is assumed to be unknown in the presence of data uncertainties. Moreover, in our formulation, the estimate $\hat{\vx}$ is not constrained to be linear unlike \cite{yonina1, yonina3} since our regret formulation is well-defined without any limitations on the estimated $\hat{\vx}$.

In the next sections, the proposed approaches to the robust LS estimation problems are provided. We first introduce the regret based unstructured LS estimation method. We next present the unstructured regularized LS estimation approach in which the worst-case regret is optimized. Finally, we investigate the structured LS estimation approach.

\section{Unstructured Robust Least Squares Estimation} \label{sec:urls}
In this section, we provide a novel robust unstructured LS estimator based on a certain minimax criterion. We consider the most generic estimation problem
\begin{equation}\label{eq:prudef1}
  \min_{\vx \in \C^n} \max_{\norm{\dH} \leq \delta_H,\norm{\dy} \leq \delta_Y} \R(\vx; \dH,\dy),
\end{equation}
where $\R(\vx; \dH,\dy)$ is defined as in \eqref{eq:LSreg0}. Now considering the second term in \eqref{eq:LSreg0}, we define $\tH \defi \vH + \dH$, $\ty \defi \vy+\dy$, where $\tH$ is a full rank matrix, and denote the estimation performance of the optimal LS estimator for some given $\tH$ and $\ty$ by
\[
f(\tH, \ty) \defi \min_{\vw \in \C^n}{\norm{\ty - \tH \vw}}^2.
\]
Since we consider an unconstrained minimization over $\vw$, we have \cite{sayedbook}
\begin{align}
  \vw^* & \defi \argmin_{\vw \in \C^n}{\norm{\ty - \tH \vw}}^2 \nn\\
        & = \tH^+ \ty,
\end{align}
as the optimal data vector minimizing the residual error. Then we have
\begin{align}
  f\left(\tH, \ty \right) & = \norm{\ty - \tH \vw^*}^2 \nn\\
              & = (\ty - \tH \vw^*)^H (\ty - \tH \vw^*) \nn\\
              & = \ty^H (\ty - \tH \vw^*) \nn\\
              & = \ty^H \tP \ty, \nn
\end{align}
where the third line follows from $\tH^H \tH \vw^* = \tH^H \ty$ \cite{sayedbook} and $\tP \defi \vI-\tH\tH^+$ is the projection matrix of the space perpendicular to the range space of $\tH$. If we use the Taylor series expansion based on Wirtinger calculus \cite{graham} for $f\left(\tH, \ty \right)$ around $\tH = \vH$ and $\ty = \vy$, then
\begin{equation}\label{eq:1stTaylorApp}
  f\left(\tH, \ty \right) = f(\vH,\vy) + 2\operatorname{Re} \left\{ \tr\left( \nabla f(\tH, \ty)\big\vert_{\tH=\vH, \ty = \vy}^H \big[\dH \;\; \dy\big] \right) \right\} + O\left( \norm{\left[\dH \hspace{0.1in} \dy \right]}^2 \right).
\end{equation}
Note that the first order Taylor approximation is introduced in order to obtain a tractable solution. Clearly, the effect of using this approximation vanishes as $\norm{[\dH \;\; \dy ]}$ decreases and for distortions with larger $\norm{[\dH \;\; \dy ]}$, one can easily use higher order approximations instead. However, we observe through our simulations that even for relatively large perturbations, a satisfactory performance is obtained using this approximation.

We now introduce the following lemma in order to obtain the first order Taylor approximation in \eqref{eq:1stTaylorApp} in a closed form.

\begin{lem}\label{lem1}
Let $\tH =\vH+\dH$ be a full rank matrix and $\ty=\vy+\dy$, where $\tH \in \C^{m \times n}$ and $\ty \in \C^m$. Then defining $f\left(\tH, \ty \right) \triangleq \ty^H\tP\ty$, where $\tP \triangleq \vI-\tH\tH^+$, we have
\[
\dfrac{\partial f\left(\tH, \ty \right)}{\partial \tH}\bigg\vert_{\tH=\vH, \ty=\vy} = - \vP \vy \left( \vH^+ \vy \right)^H,
\]
and
\[
\dfrac{\partial f\left(\tH, \ty \right)}{\partial \ty}\bigg\vert_{\tH=\vH, \ty=\vy} = \vP \vy,
\]
where $\vP \triangleq \vI-\vH\vH^+$
\end{lem}

\begin{pol1}
Since $\tH$ is full rank and $m \geq n$, the pseudo-inverse of $\tH$ is found by \cite{graham}
\[
\tH^+ \defi (\tH^H\tH)^{-1}\tH^H.
\]
Hence, we have \cite{graham}
\begin{align}\label{eq:D_definition}
  \vD & = \dfrac{\partial}{\partial \tH} \left( \ty^H \ty - \ty^H \tH (\tH^H\tH)^{-1}\tH^H \ty \right) \bigg\vert_{\tH=\vH, \ty=\vy} \nn\\
      & = \vH(\vH^H\vH)^{-1}\vH^H\vy\vy^H\vH(\vH^H\vH)^{-1} \nn\\
      & \hspace{0.5cm} - \vy\vy^H \vH(\vH^H \vH)^{-1} \nn\\
      & = \vH \vH^+ \vy \left( \vH^+ \vy \right)^H - \vy \left( \vH^+ \vy \right)^H \nn\\
      & = - \vP \vy \left( \vH^+ \vy \right)^H,
\end{align}
and
\begin{align}\label{eq:b_definition}
  \vb & = \dfrac{\partial}{\partial \ty} \left( \ty^H \ty - \ty^H \tH (\tH^H\tH)^{-1}\tH^H \ty \right) \bigg\vert_{\tH=\vH, \ty=\vy} \nn\\
      & = \vP \vy,
\end{align}
where the last line of the equality follows since $\vH \vH^+$ is a symmetric matrix according to the definition of the pseudo-inverse operation. This concludes the proof of Lemma 1. \hfill $\square$
\end{pol1}

Now turning our attention back to \eqref{eq:1stTaylorApp}, we denote
\[
\vD \defi \dfrac{\partial f\left(\tH, \ty \right)}{\partial \tH}\bigg\vert_{\tH=\vH, \ty=\vy},
\]
and
\[
\vb \defi \dfrac{\partial f\left(\tH, \ty \right)}{\partial \ty}\bigg\vert_{\tH=\vH, \ty=\vy},
\]
where we emphasize that the closed form definitions of $\vD$ and $\vb$ can be obtained from Lemma 1. We then approximate \eqref{eq:1stTaylorApp} and obtain the first order Taylor approximation as follows
\begin{align}\label{eq:thmReg}
  f\left(\tH, \ty \right) & \approx f(\vH, \vy) + 2\operatorname{Re}\left\{ \mathrm{Tr} \left( \left[\vD \;\; \vb \right]^H \left[\dH \;\; \dy \right] \right) \right\} \nn\\
              & = \kappa + 2\operatorname{Re}\left\{ \left(\col(\vD)^H \col(\dH) + \vb^H \dy \right) \right\} \nn\\
              & = \kappa + \vd^H \dvh + \dvh^H \vd + \vb^H \dy + \dy^H \vb,
\end{align}
where $\kappa \defi f(\vH,\vy)$, $\vd \defi \col(\vD)$, and $\dvh \defi \col(\dH)$. Hence we can approximate the regret in \eqref{eq:LSreg0} as follows
\begin{equation}\label{eq:regret_final}
  \R(\vx; \dH,\dy) \approx {\norm{\ty - \tH \vx}}^2 - \left( \kappa + \vd^H \dvh + \dvh^H \vd + \vb^H \dy + \dy^H \vb \right).
\end{equation}

In the following theorem, we illustrate how the optimization (or equivalently estimation) problem in \eqref{eq:regret_final} can be put in an SDP form.

\begin{thm}\label{thm1}
Let $\vH \in \C^{m\times n}$ and $\vy \in \C^{m}$ be the estimates of the data matrix and the output vector, respectively, both having deterministic additive perturbations $\dH \leq \delta_H$ and $\dy \leq \delta_Y$, respectively, i.e., $\tH=\vH+\dH$ and $\ty=\vy+\dy$, where $\tH$ is the full rank data matrix, $\ty$ is the output vector, and $m \geq n$. Then the problem
\begin{equation}\label{eq:prudefthm1}
  \min_{\vx \in \C^n} \max_{\norm{\dH} \leq \delta_H,\norm{\dy} \leq \delta_Y} \R(\vx; \dH,\dy),
\end{equation}
where $\R(\vx; \dH,\dy)$ is defined as in \eqref{eq:regret_final}, is equivalent to solving the following SDP problem
\begin{gather}
  \min \gamma \nn\\
  \mbox{subject to} \nn\\
  \tau_1 \geq 0, \tau_2 \geq 0 \text{, and} \nn\\
  \begin{bmatrix}
  \gamma + \kappa - \tau_1 - \tau_2 & (\vy-\vH\vx)^H & \delta_Y \vb^H  & \delta_H\vd^H \\
  \vy-\vH\vx                        & \vI            & -\delta_Y\vI    & \delta_H\vX \\
  \delta_Y \vb                      & -\delta_Y\vI   & \tau_1 \vI      & \vec{0} \\
  \delta_H\vd                       & \delta_H\vX^H  & \vec{0}         & \tau_2 \vI
  \end{bmatrix} \geq 0,
\end{gather}
where $\vX$ is the $m\times mn$ matrix defined as $\vX \defi \vx^H \otimes \vI$.
\end{thm}

The proof of Theorem 1 is provided in \ref{app:pot1}.

\begin{rmk}
In the proof of Theorem 1, we use Proposition 1 that relies on the lossless {\em S}-procedure. However, {\em S}-procedure is lossless with two constraints when the corresponding two quadratic (Hermitian) forms on the complex linear space \cite{s1}. However, classical {\em S}-procedure for quadratic forms is, in general, lossy with two constraints in the real case \cite{s2}. Hence, Theorem 1 cannot be extended for real linear space.
\end{rmk}

Now we can consider two important corollaries of Theorem 1. First, a special case of Theorem 1 in which the uncertainty is only in the data matrix. We emphasize that the perturbation errors only in the data matrix are also common in a wide range of real life applications \cite{sayedbook}. Here, we can define the regret as follows
\begin{equation}\label{eq:regretH}
\R(\vx; \dH) \defi {\norm{\vy - \tH \vx}}^2 - \min_{\vw \in \C^n}{\norm{\vy - \tH \vw}}^2,
\end{equation}
and similar to the previous case, we calculate the optimal estimation performance under a given uncertainty bound
\begin{align}
  f\left(\tH\right) & \defi \min_{\vw \in \C^n}{\norm{\vy - \tH \vw}}^2 \nn\\
         & \approx \kappa + 2\operatorname{Re} \left\{ \tr\left( \nabla f(\tH, \vy)\big\vert_{\tH=\vH}^H \;\; \dH \right) \right\} \nn\\
         & = \kappa + 2\operatorname{Re}\left\{ \col(\vD^H) \col(\dH) \right\} \nn\\
         & = \kappa + \vd^H \dvh + \dvh^H \vd. \nn
\end{align}
Hence we approximate the regret in \eqref{eq:regretH} as follows
\begin{equation}\label{eq:regret_final_H}
  \R(\vx; \dH) \approx {\norm{\vy - \tH \vx}}^2 - \left( \kappa + \vd^H \dvh + \dvh^H \vd \right).
\end{equation}

\begin{cor}\label{cor1}
Let $\vH \in \C^{m\times n}$ and $\vy \in \C^{m}$ be the estimates of the data matrix and the output vector, respectively, where $m \geq n$. Suppose there is a bounded uncertainty on the full rank data matrix $\tH$, i.e., $\tH=\vH+\dH$, $\norm{\dH} \leq \delta_H$. Then the problem
\begin{equation}\label{prudefcor1}
\min_{\vx \in \C^n} \max_{\norm{\dH} \leq \delta_H} \R(\vx; \dH),
\end{equation}
where $\R(\vx; \dH)$ is defined as in \eqref{eq:regret_final_H}, is equivalent to solving the following SDP problem
\begin{gather}
\min \gamma \nn\\
\mbox{subject to} \nn\\
\tau \geq 0 \text{ and }
\begin{bmatrix}
\gamma + \kappa -\tau  & (\vy-\vH\vx)^H  & \delta_H\vd \\
\vy-\vH\vx             & \vI             & \delta_H\vX\\
\delta_H\vd            & \delta_H\vX^H   & \tau \vI
\end{bmatrix}\geq 0.
\end{gather}
\end{cor}

\begin{poc1}
The proof of Corollary 1 can be explicitly derived from the proof of Theorem 1 by simply setting $\delta_Y = 0$ and $\tau_1 = 0$, hence is omitted. \hfill $\square$
\end{poc1}

Second, we consider another special case of Theorem 1 in which the uncertainty is only in the output vector. We emphasize that similar to the previous case, this one is also a common case in a wide range of real-life applications \cite{sayedbook}, and studied under a similar framework in \cite{yonina1}. Here, we can define the regret as follows
\begin{equation}\label{eq:regrety}
  \R(\vx; \dy) \defi {\norm{\ty - \vH \vx}}^2 - \min_{\vw \in \C^n}{\norm{\ty - \vH \vw}}^2,
\end{equation}
and similar to the previous case, we calculate the optimal also performance under a given uncertainty bound
\begin{align}
  f(\ty) & \defi \min_{\vw \in \C^n}{\norm{\ty - \vH \vw}}^2 \nn\\
         & \approx \kappa + 2\operatorname{Re} \left\{ \tr\left( \nabla f(\vH, \ty)\big\vert_{\ty=\vy}^H \;\; \dy \right) \right\} \nn\\
         & = \kappa + 2\operatorname{Re}\left\{ \vb^H \dy \right\} \nn\\
         & = \kappa + \vb^H \dy + \dy^H \vb. \nn
\end{align}
Hence we approximate the regret in \eqref{eq:regrety} as follows
\begin{equation}\label{eq:regret_final_y}
\R(\vx; \dy) \approx {\norm{\ty - \vH \vx}}^2 - \left( \kappa + \vb^H \dy + \dy^H \vb \right).
\end{equation}

\begin{cor}\label{cor2}
Let $\vH \in \C^{m\times n}$ and $\vy \in \C^{m}$ be the estimates of the data matrix and the output vector, respectively, where $m \geq n$. Suppose there is a bounded uncertainty on the output vector $\ty$, i.e., $\ty=\vy+\dy$, $\norm{\dy} \leq \delta_Y$. Then the problem
\begin{equation}
\min_{\vx \in \C^n} \max_{\norm{\dy} \leq \delta_Y} \R(\vx; \dy),
\end{equation}
where $\R(\vx; \dy)$ is defined as in \eqref{eq:regret_final_y}, is equivalent to solving the following SDP problem
\begin{gather}
\min \gamma \nn\\
\mbox{subject to} \nn\\
\tau \geq 0 \text{ and }
\begin{bmatrix}
\gamma + \kappa -\tau  & (\vy-\vH\vx)^H  & \delta_Y\vb^H \\
\vy-\vH\vx             & \vI             & -\delta_Y\vI \\
\delta_Y\vb            & -\delta_Y\vI    & \tau \vI
\end{bmatrix} \geq 0.
\end{gather}
\end{cor}

\begin{poc2}
The proof of Corollary 2 can be explicitly derived from the proof of Theorem 1 by simply setting $\delta_H = 0$ and $\tau_2 = 0$, hence is omitted. \hfill $\square$
\end{poc2}

\begin{rmk}
Corollaries 1 and 2 follows from the proof of Theorem 1, which relies on the lossless {\em S}-procedure. Under the frameworks presented in the Corollaries 1 and 2, one can safely extend the same conclusions for the real case also, since {\em S}-procedure is lossless for quadratic forms with one constraint both in complex and real spaces \cite{huang1,huang2}.
\end{rmk}

\section{Unstructured Robust Regularized Least Squares Estimation} \label{sec:urrls}
In this section, we introduce a worst-case regret optimization approach to solve the regularized LS estimation problem in \cite{est2}. The regret for not using the optimal regularized LS estimator is defined by
\begin{equation}\label{Req:LSreg0}
  \R(\vx;\dH,\dy) \defi \left\{ {\norm{\ty - \tH \vx}}^2 + \mu \norm{\vx}^2 \right\}-  \min_{\vw \in \C^n} \left\{ \norm{\ty - \tH \vw}^2 + \mu \norm{\vw}^2 \right\},
\end{equation}
where $\mu > 0$ is the regularization parameter. We emphasize that there are different approaches to choose $\mu$, however, for the focus of this paper, we assume that it is already set before the optimization so that these methods can be readily incorporated in our framework. Hence, we solve the regularized LS estimation problem for an arbitrary $\mu > 0$ and note that we have already covered the $\mu = 0$ case in Section \ref{sec:urls}.

Similar to the previous case, we denote the estimation error of the optimal LS estimator for some estimated data matrix $\vH$ and output vector $\vy$ by
\begin{align}
  f(\vH, \vy) & \defi \min_{\vw \in \C^n}{\norm{\vy - \vH \vw}}^2 + \mu \norm{\vw}^2 \nn\\
              & = \norm{\vP^{-1} \, \vy}^2 \nn\\
              & = \vy^H \vP^{-1} \, \vy, \nn
\end{align}
where $\vP \defi \vI + \mu^{-1}\vH\vH^H$. Considering the first order Taylor series expansion based on Wirtinger calculus \cite{graham} for $f(\tH, \ty)$ around $\tH = \vH$ and $\ty = \vy$
\begin{align}
f(\tH, \ty) & \approx \kappa + 2\operatorname{Re} \left\{ \tr\left( \nabla f(\tH, \ty)\big\vert_{\tH=\vH, \ty = \vy}^H \big[\dH \;\; \dy\big] \right) \right\}, \nn\\
            & = \kappa + \vd^H \dvh + \dvh^H \vd + \vb^H \dy + \dy^H \vb, \nn
\end{align}
where $\vd \defi \col(\vD^H)$, $\dvh \defi \col(\dH)$,
\begin{align}
  \vD & \defi \dfrac{\partial f(\tH, \ty)}{\partial \tH}\bigg\vert_{\tH=\vH, \ty=\vy} \nn\\
      & = - \vP^{-1} \vy \vy^H \vP^{-1} \vH,
\end{align}
and
\begin{align}
  \vb & \defi \dfrac{\partial f(\tH, \ty)}{\partial \ty}\bigg\vert_{\tH=\vH, \ty=\vy} \nn\\
      & = \vP^{-1} \vy, \nn
\end{align}
where the last line follows since $\vP$ is symmetric. Hence we can approximate the regret in \eqref{Req:LSreg0} as follows
\begin{equation}\label{eq:regret_final_2}
  \R(\vx; \dH, \dy) \approx {\norm{\ty - \tH \vx}}^2 + \mu \norm{\vx}^2 - (\kappa + \vd^H \dvh + \dvh^H \vd + \vb^H \dy + \dy^H \vb),
\end{equation}
similar to \eqref{eq:regret_final}. In the following theorem, we illustrate how the optimization problem in \eqref{eq:regret_final_2} can be put in an SDP form.

\begin{thm}\label{thm2}
Let $\vH \in \C^{m\times n}$ and $\vy \in \C^{m}$ be the estimates of the data matrix and the output vector, respectively, both having deterministic additive perturbations $\dH \leq \delta_H$ and $\dy \leq \delta_Y$, respectively, i.e., $\tH=\vH+\dH$ and $\ty=\vy+\dy$, where $\tH$ is the full rank data matrix, $\ty$ is the output vector, and $m \geq n$. Then the problem
\begin{equation}\label{eq:prudefthm2}
  \min_{\vx \in \C^n} \max_{\norm{\dH} \leq \delta_H,\norm{\dy} \leq \delta_Y} \R(\vx; \dH,\dy),
\end{equation}
where $\R(\vx; \dH,\dy)$ is defined as in \eqref{eq:regret_final_2}, is equivalent to solving the following SDP problem
\begin{gather}
  \min \gamma \nn\\
  \mbox{subject to} \nn\\
  \tau_1 \geq 0, \tau_2 \geq 0 \text{, and} \nn\\
  \begin{bmatrix}
  \gamma + \kappa - \tau_1 - \tau_2 & (\vy-\vH\vx)^H & \vx^H   & \delta_Y \vb^H  & \delta_H\vd^H \\
  \vy-\vH\vx                        & \vI            & \vec{0} & -\delta_Y\vI    & \delta_H\vX \\
  \vx                               & \vec{0}        & \mu \vI & \vec{0}         & \vec{0} \\
  \delta_Y \vb                      & -\delta_Y\vI   & \vec{0} & \tau_1 \vI      & \vec{0} \\
  \delta_H\vd                       & \delta_H\vX^H  & \vec{0} & \vec{0}         & \tau_2 \vI
  \end{bmatrix} \hspace{-0.1cm} \geq \hspace{-0.05cm} 0.
\end{gather}
\end{thm}

\begin{pot2}
The proof of Theorem 2 follows similar lines to the proof of Theorem 1, hence is omitted here. \hfill $\square$
\end{pot2}

\begin{rmk}
Under the framework introduced in this section, one can straightforwardly obtain the corollaries similar to Corollaries 1 and 2 by considering cases in which the uncertainty is either only on the data matrix or only on the output vector, i.e., $\delta_Y = 0$ and $\delta_H = 0$ cases, respectively. The derivations follow similar lines to Corollaries 1, 2 and Theorem 2, hence is omitted. However, similar results can be readily derived from the result in Theorem 2 with suitable changes in the SDP formulations.
\end{rmk}

\section{Structured Robust Least Squares Estimation}\label{sec:srls}
There are various communication systems where the data matrix and the perturbation on it have a special structure such as Toeplitz, Hankel, or Vandermonde \cite{est1, tls}. Incorporating this prior knowledge into the estimation framework could improve the performance of the regret based minimax LS estimation approach \cite{est1, tls}. Hence, in this section, we investigate a special case of the problem in \eqref{eq:prudef1}, where the associated perturbations for the data matrix $\vH$ and the output vector $\vy$ have special structures. The structure on the perturbations is defined as follows
\begin{equation}\label{eq:dH}
  \dH = \sum_{i=1}^p\alpha_i\vH_i,
\end{equation}
and
\begin{equation}\label{eq:dy}
  \dy = \sum_{i=1}^p\beta_i\vy_i,
\end{equation}
where $\vH_i \in \C^{m \times n}$, $\vy_i \in \C^m$, and $p$ are known but $\alpha_i,\beta_i \in \C$, $i=1,\dots,p$, are unknown. However, the bounds on the norm of $\va \defi [\alpha_1,\dots,\alpha_p]^H$ and $\vbet \defi [\beta_1,\dots,\beta_p]^H$ are provided as $\norm{\va} \leq \delta_{\alpha}$ and $\norm{\vbet} \leq \delta_{\beta}$, where $\delta_{\alpha}, \delta_{\beta} \geq 0$. We emphasize that this formulation can represent a wide range of constraints on the structure of perturbations of the data matrix and the output vector such as Toeplitz and Hankel \cite{est1, sayedbook}. Our aim is to solve the following optimization problem
\[
\min_{\vx \in \C^n} \max_{\norm{\va} \leq \delta_{\alpha}, \norm{\vbet} \leq \delta_{\beta}} \R(\vx; \dH, \dy),
\]
where
\begin{gather}
  \R(\vx; \dH, \dy) \defi \norm{\ty - \tH \vx}^2 - \min_{\vw \in \C^n} \norm{\ty - \tH \vw}^2, \label{eq:regret_first_3}\\
  \tH \defi \vH + \dH = \vH + \sum_{i=1}^p \alpha_i \vH_i, \label{eq:tH}\\
  \ty \defi \vy + \dy = \vy + \sum_{i=1}^p \beta_i \vy_i. \label{eq:ty}
\end{gather}

After following similar lines to Section \ref{sec:urls}, and introducing the first order Taylor approximation to $f\left(\tH,\ty\right)$ around $\va = \vec{0}$ and $\vbet = \vec{0}$, we obtain
\begin{equation}\label{eq:taylor3}
  f\left(\tH,\ty\right) \approx \kappa + 2\operatorname{Re} \left\{ \tr\left( \nabla f(\tH, \ty)\big\vert_{\va=\vec{0}, \vbet=\vec{0}}^H \left[\va \;\; \vbet\right] \right) \right\},
\end{equation}
where $f\left(\tH,\ty\right) = \ty^H \tP \ty$ and $\tP = \vI - \tH \tH^+$. We next introduce the following lemma to calculate the first order Taylor approximation in \eqref{eq:taylor3} in a closed form.

\begin{lem}\label{lem2}
Let $\tH =\vH+\dH$ be a full rank matrix and $\ty=\vy+\dy$, where $\tH \in \C^{m \times n}$, $\ty \in \C^m$, $\dH$ and $\dy$ are defined as in \eqref{eq:dH} and \eqref{eq:dy}, respectively. Then denoting $f\left(\tH, \ty \right) \triangleq \ty^H\vP\ty$, where $\tP \triangleq \vI-\tH\tH^+$, we have
\begin{equation}
  \dfrac{\partial f(\tH, \ty)}{\partial \va} \bigg\vert_{\va = \vec{0}, \vbet = \vec{0}} = \left[ - \vy^H \vP^H \vH_1 \vH^+ \vy, \dots,  - \vy^H \vP^H \vH_p \vH^+ \vy \right]^H,
\end{equation}
and
\begin{equation}
  \dfrac{\partial f(\tH, \ty)}{\partial \vbet} \bigg\vert_{\va = \vec{0}, \vbet = \vec{0}} = \left[ \vy^H \vP \vy_1, \dots,  \vy^H \vP \vy_p \right]^H,
\end{equation}
where $\vP \triangleq \vI-\vH\vH^+$.
\end{lem}

\begin{pol2}
Note that the derivative of $f\left(\tH, \ty \right)$ is taken with respect to $[\va \;\; \vbet]$, hence we can use the Chain Rule to calculate the derivatives by using the results we have obtained in Lemma 1.

First, we consider the derivative of $f\left(\tH, \ty \right)$ with respect to $\alpha_i$, $i = 1,\dots,p$, i.e.,
\begin{align}
  d_i & \defi \dfrac{\partial f(\tH, \ty)}{\partial \alpha_i} \bigg\vert_{\va = \vec{0}, \vbet = \vec{0}} \nn\\
      & = \tr \left( \left( \dfrac{\partial f(\tH, \ty)}{\partial \tH} \right)^H  \dfrac{\partial \tH}{\partial \alpha_i} \bigg\vert_{\va = \vec{0}, \vbet = \vec{0}} \right) \nn\\
      & = \tr \left( - \vH^+ \vy \vy^H \vP^H \vH_i \right) \nn\\
      & = - \vy^H \vP^H \vH_i \vH^+ \vy, \nn
\end{align}
where the last line follows from the cyclic property of the trace operator.

Similarly, we next consider the derivative of $f\left(\tH, \ty \right)$ with respect to $\beta_i$, $i = 1,\dots,p$, i.e.,
\begin{align}
  b_i & \defi \dfrac{\partial f(\tH, \ty)}{\partial \beta_i} \bigg\vert_{\va = \vec{0}, \vbet = \vec{0}} \nn\\
      & = \tr \left( \left( \dfrac{\partial f(\tH, \ty)}{\partial \ty} \right)^H  \dfrac{\partial \ty}{\partial \beta_i} \bigg\vert_{\va = \vec{0}, \vbet = \vec{0}} \right) \nn\\
      & = \vy^H \vP \vy_i. \nn
\end{align}
This concludes the proof of Lemma 2. \hfill $\square$
\end{pol2}

Now turning our attention back to \eqref{eq:taylor3}, we denote
\[
\vd \defi \dfrac{\partial f\left(\tH, \ty \right)}{\partial \va} \bigg\vert_{\va = \vec{0}, \vbet = \vec{0}},
\]
and
\[
\vb \defi \dfrac{\partial f\left(\tH, \ty \right)}{\partial \vbet} \bigg\vert_{\va = \vec{0}, \vbet = \vec{0}},
\]
where we emphasize that the closed form definitions of $\vd$ and $\vb$ can be obtained from Lemma 2. We then approximate \eqref{eq:taylor3} and obtain the first order Taylor approximation as follows
\[
f\left(\tH, \ty \right) \approx \kappa + \vd^H\va + \va^H\vd + \vb^H\vbet + \vbet^H\vb.
\]
Therefore, we can approximate the regret in \eqref{eq:regret_first_3} as follows
\begin{equation}\label{eq:regret_final_3}
  \R(\vx; \dH,\dy) \approx {\norm{\ty - \tH \vx}}^2 - \left( \kappa + \vd^H\va + \va^H\vd + \vb^H\vbet + \vbet^H\vb \right).
\end{equation}

In the following theorem, we illustrate how the optimization problem in \eqref{eq:regret_final_3} can
be put in an SDP form.

\begin{thm}\label{thm3}
Let $\vH, \vH_1, \dots, \vH_p \in \C^{m\times n}$, $\vy, \vy_1,\dots, \vy_p \in \C^{m}$, $\delta_{H}, \delta_{Y} \geq 0$, $m\geq n$, where $\tH$ is the full rank data matrix defined as in \eqref{eq:tH}, $\ty$ is the output vector defined as in \eqref{eq:ty}, with the corresponding estimates $\vH$ and $\vy$, respectively. Then the problem
\begin{equation}\label{eq:prudefthm3}
  \min_{\vx \in \C^n} \max_{\norm{\va} \leq \delta_{\alpha},\norm{\vbet} \leq \delta_{\beta}} \R(\vx; \dH,\dy),
\end{equation}
where $\R(\vx; \dH,\dy)$ is defined as in \eqref{eq:regret_final_3}, is equivalent to solving the following SDP problem
\begin{gather}\label{eq:sdp3}
  \min \gamma \nn\\
  \mbox{subject to} \nn\\
  \tau_1 \geq 0, \tau_2 \geq 0 \text{, and} \nn\\
  \begin{bmatrix}
  \gamma + \kappa - \tau_1 - \tau_2  & (\vy-\vH\vx)^H           & \delta_{\alpha} \vd^H  & \delta_{\beta} \vb^H \\
  \vy-\vH\vx                         & \vI                      & -\delta_{\alpha} \vG   & \delta_{\beta} \vQ \\
  \delta_{\alpha} \vd                & -\delta_{\alpha} \vG^H   & \tau_1 \vI             & \vec{0} \\
  \delta_{\beta} \vb                 & \delta_{\beta} \vQ^H     & \vec{0}                & \tau_2 \vI
  \end{bmatrix} \geq 0,
\end{gather}
where $\vG \defi [\vH_1\vx,\dots,\vH_p\vx]$ and $\vQ \defi [\vy_1,\dots,\vy_p]$.
\end{thm}

\begin{pot3}
The proof of Theorem 2 follows similar lines to the proof of Theorem 1, hence is omitted here. \hfill $\square$
\end{pot3}

\begin{rmk}
Under the framework introduced in this section, one can straightforwardly obtain the corollaries similar to Corollaries 1 and 2 by considering cases in which the uncertainty is either only on the data matrix or only on the output vector, i.e., $\delta_{\beta} = 0$ and $\delta_{\alpha} = 0$ cases, respectively. The derivations follow similar lines to Corollaries 1, 2 and Theorem 3, hence is omitted. However, similar results can be readily derived from the result in Theorem 3 with suitable changes in the SDP formulations.
\end{rmk}

\begin{rmk}
The proofs of Theorem 2 and Theorem 3 follow from the results of Theorem 1, which relies on the lossless {\em S}-procedure. However, {\em S}-procedure is lossless with two constraints when the corresponding two quadratic (Hermitian) forms on the complex linear space \cite{s1}. However, classical {\em S}-procedure for quadratic forms is, in general, lossy with two constraints in the real case \cite{s2}. Hence, Theorem 2 and Theorem 3 cannot be extended for real linear space. On the other hand, under the frameworks described in Remark 3 and Remark 4, one can safely extend the same conclusions for the real case also, since {\em S}-procedure is lossless for quadratic forms with one constraint both in complex and real spaces \cite{huang1,huang2}.
\end{rmk}

\section{Simulations}\label{sec:numer}
We provide numerical examples in different scenarios in order to illustrate the merits of the proposed algorithms. In the first set of the experiments, we randomly generate a data matrix of size $m \times n$, and an output vector of size $m\times 1$, which are normalized to have unit norms. Then, we generate $1000$ random perturbations $\dH$, $\dy$, where $\norm{\dH}\leq \delta_H$, $\norm{\dy}\leq \delta_Y$, $m=5$, $n=3$, and $\delta_H=\delta_Y=1.2$. Here, we label the algorithm in Theorem 1 as ``rgrt-LS'', the robust LS algorithm of \cite{Ghaoui97} as ``rbst-LS'', the total LS algorithm \cite{Ghaoui97} as ``TLS'', and finally the LS algorithm tuned to the estimates of the data matrix and the output vector as ``LS'', where we directly use $\hat{\vx} = \vH^+ \vy$.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.75\textwidth]{ls_ustr.eps}\\
  \caption{Sorted residual errors for the rgrt-LS, rbst-LS, LS, and TLS estimators over $1000$ trials when $\delta_H = \delta_Y = 1.2$, $m=5$, and $n=3$.}\label{fig:ls_ustr}
\end{figure}

For each algorithm and for each random perturbation, we find the corresponding $\hat{\vx}$ and calculate the error $\norm{ \tH \hat{\vx} - \ty}^2$. After we calculate the errors for each algorithm and for all random perturbations, we plot the corresponding sorted errors in ascending order in Fig. 1 for 1000 perturbations. Since the rbst-LS algorithm optimizes the worst-case residual error with respect to worst possible disturbance, it usually yields the smaller worst-case residual error among all algorithms for these simulations. On the other hand, since the LS algorithm directly uses the estimates, it usually yields the smaller residual error when the perturbations on the data matrix and the output vector are significantly small.

These results can be observed in Fig. 1, where in one extreme, the largest residual errors are observed as $2.9762$ for the TLS estimator, $2.2557$ for the LS estimator, $1.9275$ for the rbst-LS estimator, and $1.9325$ for the rgrt-LS estimator. In the other extreme, i.e., when there is almost no perturbation, the smallest estimation errors are observed as $0.3035$ for the LS estimator, $0.4036$ for the TLS estimator, $0.8727$ for the rbst-LS estimator, and $0.6387$ for the rgrt-LS estimator. While the LS estimator can be preferable when there is relatively smaller perturbations and the rbst-LS estimator can be preferable when there is significantly higher perturbations, the introduced algorithm provides a tradeoff between these algorithms and achieve a significantly smaller average error performance. The average residual error of the rgrt-LS estimator is observed as $1.1928$, whereas this value is $1.2180$ for the LS estimator, $1.2708$ for the rbst-LS estimator, and $1.3826$ for the TLS estimator. Hence, the rgrt-LS estimator is not only robust but also efficient in terms of the average error performance compared to its well-known alternatives.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.75\textwidth]{ls_ustr_MSE_vs_rho.eps}\\
  \caption{Averaged residual errors for the rgrt-LS, rbst-LS, LS, and TLS estimators over $1000$ trials when $\delta_H = \delta_Y \in [0.3, 0.6]$, $m=5$, and $n=3$.}\label{fig:ls_ustr_MSE_vs_rho}
\end{figure}

For the second experiment, we generate $1000$ random perturbations $\dH$, $\dy$, where $\norm{\dH}\leq \delta_H$, $\norm{\dy}\leq \delta_Y$, $m=5$, $n=3$ for different perturbation bounds and compute the averaged error over $1000$ trials for the rgrt-LS, the LS, the rbst-LS, and the TLS algorithms. In Fig. 2, we present the averaged residual errors for these algorithms for different values of perturbation bounds, i.e., $\delta_Y=\delta_H = \delta \in [0.3, 0.6]$. We observe that the proposed rgrt-LS algorithm has the best average residual error performance over different perturbation bounds compared to the LS, the rbst-LS and the TLS algorithms.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.75\textwidth]{sysid_str.eps}\\
  \caption{Sorted residual errors for the str-rgrt-LS, str-rbst-LS, SLS-BDU, LS, and TLS estimators over $1000$ trials when $\delta_H = \delta_Y = 2$, $m=5$, and $n=3$.}\label{fig:sysid_str}
\end{figure}

As can be observed from Fig. 2, as the perturbation bounds increase, the performances of the LS and the TLS estimators significantly deteriorate, whereas the performance of the rbst-LS estimator almost does not change. The residual error of the rgrt-LS estimator, on the other hand, slightly increases as the perturbation bounds increase, yet the robustness of this estimator can be observed in Fig. 2. Furthermore, the rgrt-LS estimator significantly outperforms its competitors in terms of the average error performance by introducing a transition between the best-case performance of the LS estimator and the worst-case performance of the rbst-LS estimator.

In the next experiment, we examine a system identification problem \cite{pilanci10}, which can be formulated as $\vH_0 \vx=\vy_0$, where $\vH=\vH_0+\vW$ is the observed noisy Toeplitz matrix and $\vy=\vy_0+\vw$ is the observed noisy output vector. Here, the convolution matrix $\vH$ (which is Toeplitz) constructed from $\vh$ which is selected as a random sequence of $\pm 1$'s. For a randomly generated filter $\vh$ of length $3$, we generate $1000$ random structured perturbations for $\vH_0$ and $\vy_0$, where $\norm{\va}\leq 2 \norm{\vH_0}$, and plotted the sorted estimation errors in ascending order in Fig. 3.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.75\textwidth]{sysid_ustr_rls.eps}\\
  \caption{Sorted residual errors for rgrt-reg-LS, rbst-reg-LS, and LS estimators over $1000$ trials when $\delta_H = \delta_Y = 0.65$, $\mu = 0.5$, $m=3$, and $n=2$.}\label{fig:sysid_ustr_rls}
\end{figure}

The average residual errors, on the other hand, are observed as $1.5549$ for the structured regret LS estimator ``str-rgrt-LS" of Remark 4, $1.7141$ for the structured robust LS algorithm ``str-rbst-LS'', $2.0061$ for the TLS estimator, $2.1205$ for the LS estimator, and $2.6319$ for the structured least squares bounded data uncertainties estimator, labeled as ``SLS-BDU'' and presented in \cite{pilanci10}. Therefore, we observe that the str-rgrt-LS algorithm yields the smallest average residual error among its competitors. In addition, we observe that the str-rgrt-LS estimator has a smaller residual error in most of the trials compared to its well-known alternatives, owing to its novel regret formulation.

Finally, in Fig. 4, we provide errors sorted in ascending order for the algorithm in Theorem 2 as ``rgrt-reg-LS'', for the robust regularized LS algorithm in \cite{sayed02} as ``rbst-reg-LS'' and finally for the regularized LS algorithm as ``reg-LS'' \cite{Kailath:book}, where the experiment setup is the same as in the first experiment except the perturbation bounds are set to $0.65$ and the regularization parameter is chosen as $\mu=0.5$. In Fig. 4, we observe that the robustness and the performance tradeoff (between the rbst-reg-LS and the reg-LS algorithms) of the introduced rgrt-reg-LS algorithm.

When there is small perturbations on the data matrix and the output vector, i.e., in the best-case scenario, the residual error of the reg-LS estimator is $0.1045$, whereas it is $0.2416$ for the rgrt-reg-LS estimator and $0.4282$ for the rbst-reg-LS estimator. As can be observed from Fig. 4, for higher perturbations, the performance of the reg-LS estimator significantly deteriorates, whereas the rgrt-reg-LS and rbst-reg-LS algorithms provide a robust performance. On the other hand, the rgrt-reg-LS estimator significantly outperforms the rbst-reg-LS estimator in terms of the average error performance and achieves even a more desirable error performance compared to the reg-LS estimator. The average residual errors are calculated as $0.9059$ for the rgrt-reg-LS estimator, $0.9177$ for the reg-LS estimator, and $1.0316$ for the rbst-reg-LS estimator.

\section{Conclusion}\label{sec:conc}
In this paper, we introduce a robust approach to LS estimation problems under bounded data uncertainties based on a novel regret formulation. We study the robust LS estimation problems in the presence of unstructured and structured perturbations under residual and regularized residual error criteria. In all cases, the data vectors that minimize the worst-case regrets are found by solving certain SDP problems. In our simulations, we observed that the proposed estimation methods provide an efficient tradeoff between the performance and robustness, better than the best available alternatives in different signal processing applications.

redpajama_set_name: RedPajamaArXiv


At Dinafem Seeds, one of our main goals is to ensure that all our marijuana seeds are unique and fully meet the expectations of the most demanding cannabis growers. The amazing quality offered by all our genetics has nothing to do with luck, but with the many years of hard work and research spent by a team with ample experience in the field.
Inexperienced growers sometimes feel overwhelmed at the thought of having to decide which cannabis seed to pick. When that critical moment arrives, it's paramount to value some aspects concerning the cultivation, the yield or the taste. For doing so, we should pay close attention to each strain's demands (degree of difficulty, size, climate, space available, yield…) and to what we're really expecting to get from her in terms of effects, flavours, and aromas.
To make the decision easier and to provide a solution to the cannabis grower, our website features photos, videos, a detailed description and a technical data chart of every seed. Besides, our customer service team is always at your disposal at pedidosdinafem.org to help you on a one-on-one basis ready to address any issues you may have before, during or after your purchase. If you're still itching for more, visit our blog and find many cannabis-related posts.
Our seeds are a guarantee of quality, we know this for certain. Why? Because we have worked with stabilised genetics, because they have been meticulously selected by expert hands only and because we make sure they're stored and transported in durable containers specially designed to keep the optimal temperature and moisture conditions.


A French man was found dead in Donepamai village, Sisattanak district after he hanged himself at his house last Thursday. Investigative police say they are unsure as to the motive for the suicide but said it must have been for personal rather than financial reasons.
The investigative police have identified the man as a thirty-one-year -old French citizen who was working in Vientiane.
Police did not find anything untoward at his house in Donepamai village, Sisattanak district, apart from the dead man's body.
Deputy Chief of the Interrogative and Investigative Department Mr A M-ay Luangpakdy said that an investigation is ongoing and police are aiming to ascertain the motive behind this mysterious death.
Mr A M-ay said the security guard and his Lao girlfriend found him hanging from a piece of electrical cord at around 5:30pm. He explained that the man's girlfriend rang him seven times at around 2:30pm but there was no reply. Later, she decided to come to see the man at his house.
She knocked on the door but there was no response. The lady then looked in the window and saw the man hanging inside. She came down to call the security guard for help. They broke the window to rescue the man but he was already dead.
"The security guard told us that he did not suspect anything when the man returned home at around 1:30 pm," Mr A M-ay said, adding that everything was normal and he did not suspect anything to happen that afternoon.
Officials at the bank told the police the man had no problems with finance and he lived at the Lao style house in Donpamai that the bank rented for him.
"There was no sign of fighting. Everything was there and normal," Mr A M-ay said. He speculated that personal issues may have been to blame. However the police will look at all possibilities before reaching that conclusion, he assured.
There have been two similar cases in recent times. One was a Japanese citizen and the other was Australian.


text:
Tel: (205) 795-3677 | Tel: (877) 728-3537 | Fax: (205) 795-3676
The Larkin Law Firm, P.C.
About The Larkin Law Firm, P.C.
Our Birmingham, Tuscaloosa, and West Alabama Legal Team
Our team of Birmingham employees have years of experience working both in and out of Birmingham, Tuscaloosa, and West Alabama areas courtrooms, with clients of all backgrounds and ages. We concentrate in Civil Law which includes: Personal Injury, Products Liability, and Probate.
You can always count on us to take the time necessary to clearly explain your case's procedures and status so that you are up to date on every development at all times. As we understand that charges and matters of the law are often unplanned and happen without notice, once we take your case, we always ensure that someone is available to dedicate time to your case. The sooner you have Attorney Larkin on your side, the easier it is to build a strong case and work toward a successful resolution. .
Our Birmingham, Tuscaloosa, and West Alabama Law Office
The law office of The Larkin Law Firm, P.C. has been proudly serving the Birmingham, Tuscaloosa, and West Alabama areas communities for many years. With our main office in Birmingham, The Larkin Law Firm, P.C. easily serves those in need of legal services throughout Alabama, including the cities of Anniston, Gadsden, Huntsville, Mobile, Montgomery, and Tuscaloosa. We are extremely proud to be able today to offer legal advice, legal counsel and representation in most civil issues.
The Larkin Law Firm, P.C. welcomes your call or visit, by appointment, regardless of your legal situation or budget. Depending on the case and specific situation, we are in some cases able to represent you on a contingency fee basis. In most cases involving monetary compensation, you do not have to give us a penny unless and until we help you win your case. You have nothing to lose and everything to gain by calling us for advice or more information.
At The Larkin Law Firm, P.C., our utmost priority is to support and represent the community. No case is too big or too small for our dedicated team. We will work on your behalf until we are satisfied that we have done absolutely everything in our power to bring resolution to your case.
Tom L Larkin
Attorney Tom L. Larkin is the Founder and President of The Larkin Law Firm, P.C., a Birmingham-based law firm. In 1976, Attorney Larkin graduated from Cumberland School of Law at Samford University in Birmingham. He was admitted to practice in Alabama in September of 1976. Early in his career, Attorney Larkin was a general practitioner. He partnered with Arthur D. Shores, a renowned civil rights attorney to establish the law firm of Shores & Larkin. Attorney Larkin went on to become the first full-time African American professor at Cumberland School of Law in 1979. In 1991, Attorney Larkin teamed up with his law school friend, Steven Tipler, to form one of the first multi-racial plaintiffs' personal injury firms in Alabama. Attorney Larkin practiced with Attorney Steven Tipler in the law firm of Tipler Law Offices, later renamed Tipler Larkin Trial Lawyers, until 2009 when he opened the Larkin Law Firm, P.C.
Attorney Larkin is a life member of Omega Psi Phi Fraternity (1968) where he has held the offices of 7th District Counselor, State Counselor and District Social Action Chair. He currently serves as the District Rules Chair for the 7th District. Attorney Larkin is presently a board member of Children's Dance Foundation and F.O.C.U.S. (Females On Course Using Steps). Attorney Larkin is the immediate past Sire Archon of Sigma Pi (Boulé) Southeast Region.
2214 Second Avenue North, Suite 100,
Email: Tom@tlarkinlaw.com
Search Terms: Attorney, Lawyer, Legal Counsel, Personal Injury, Malpractice, Product Liability, Probate
The following language is required pursuant to Alabama Rules of Professional Conduct Rule 7.2. No representation is made that the quality of legal services to be performed is greater than the quality of legal services performed by other lawyers. This web site is designed for general information only. The information presented at this site should not be construed to be formal legal advice nor the formation of an attorney-client relationship.
All information provided is provided for information purposes only and does not constitute a legal contract between The Larkin Law Firm, P.C. and any person or entity unless otherwise specified. Information is subject to change without prior notice. Although every reasonable effort is made to present current and accurate information, LinkNow™ Media makes no guarantees of any kind.


text:
Q: Pymongo aggregate does not return a cursor but an object I am writing a code using pymongo which uses the aggregation framework to save some data in other collection.
The code is this:
from pymongo import MongoClient

def makeAggregate():
  print 'Making aggregation of commits..'

  commitsCollection = MongoClient("mongo-srv", 27017).gt.commits
  rankingCollection = MongoClient("mongo-srv", 27017).gt.commitsRanking

  pipe = [{'$unwind': '$commits'},{'$group':{"_id":"$_id", "picture": {"$first": "$picture"},'a':{'$sum':'$commits.a'},'d':{'$sum':'$commits.d'},'c':{'$sum':'$commits.c'}}}]
  cursor = commitsCollection.aggregate(pipeline=pipe)

  obj = next(cursor, None)
  while obj:
    rankingCollection.save(obj)
    obj = next(cursor, None)

makeAggregate()

The code works fine on my computer, but when I moved the script to a server, then the script failed, saying:
Traceback (most recent call last):
  File "aggregate.py", line 17, in <module>
    makeAggregate()
  File "aggregate.py", line 12, in makeAggregate
    obj = next(cursor, None)
TypeError: dict object is not an iterator

The command python --version returns
On my computer:
Python 2.7.3
On the server
Python 2.7.6
The command pip show pymongo returns
On my computer:
Usage: pip COMMAND [OPTIONS]
pip: error: No command by the name pip show
  (maybe you meant "pip install show")

(Executed pip install show but keeps saying this when running show..)
On the server:
Name: pymongo
Version: 2.7
Location: /usr/local/lib/python2.7/dist-packages/pymongo-2.7-py2.7-linux-x86_64.egg
Requires:

Running pymongo.version inside python gives me:
In my computer:
3.0
In the server
2.7
Maybe I have to update this? How can I do that?

A: Yes thats the issue,  Different version of Pymongo for Development and Production environment
In PyMongo 2.7 it returns : Dictionary
{u'ok': 1.0, u'result': [{u'count': 3, u'_id': u'cat'}, {u'count': 2, u'_id': u'dog'}, {u'count': 1, u'_id': u'mouse'}]}

Whereas in PyMongo 3.0 it returns : Cursor Object
{u'count': 3, u'_id': u'cat'}, {u'count': 2, u'_id': u'dog'}, {u'count': 1, u'_id': u'mouse'}

Refer Pymongo 2.7 Documentation
Refer Pymongo 3.0 Documentation
Changes made from PyMongo 2.7 to PyMongo 3.0
Pro-Tip :
Use Virtual Environment for Python and create a Requirements Text File.So as you can install the same version of Python Library and it Dependencies in Local Development and in Production.
Refer Virtual Environment Python Package


TNE 3.2 contains several new features and improvements. For instance, there are new capabilities for integration with other systems. Via the new feature "TNE Agent", external applications can be easily integrated with TNE in an effective and user-friendly way. The new module "Generate Feature Service" creates new and exciting options for data capture and bulk loading of objects from different types of data sources.
Support has been added in TNE's segmentation module for the OpenTNF format as well as new functionality to better meet the requirements identified when analyzing traffic accidents. There are also improved tools in the TNE Studio User Interface (UI) for users and administrators to manage, among other things, permissions and subscriptions.


text:
You've Got is a daily web series from AOL. The series premiered on AOL.com and AOL Video (video.aol.com) on November 1, 2010. It is now found on the AOL On Network (on.aol.com). The show has been described as an "open mic" for the web, presenting interesting personalities to the world, similar to Saturday Night Live's iconic hosted opening segments. The first person featured on the show was Kelly Ripa., followed by President Barack Obama. Other notable guests have included Ellen DeGeneres,Matt Damon,Elmo,Larry King,Will Ferrell,Joan Rivers and Kevin Bacon. The show's name is derived from the iconic welcome message from AOL Mail, "You've Got Mail." New episodes air every morning on AOL.com and AOL On.
AOL has revealed that You've Got generated 8 million views in its first month, putting it on pace with a top 10 web series.
Each episode of You've Got features a different notable personality, some famous, some not. Videos are typically 1–2 minutes in length and generally consist of a pointed message or a brief comedic sketch.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/You've_Got


Feel free to stop by the chapter house Friday night. We're always proud to have guest in the new house which was made possible by our generous alumni, family and friends. More info coming soon! House Tours: Drop by the house to have a tour— 5:00pm – 6:00 pm We recommend the following hotels which are close to the chapter house and venues for the weekend. Both would offer a MSU Friends and Family rate; This is based on availability and it is highly recommended that individuals book things early. The guest needs to reference "MSU FRIENDS AND FAMILY" rate when they call to make reservations.
Don't miss your chance to pick up this awesome white Adidas polo with MSU color stitching!
Purchase below and add to your cart.
Hole Sponsorships are $250.00. Contact Nick DeLong at DeLong117@live.missouristate.edu or 618-401-1317 to advertise your business to over 100 golfers and Phi Delt fans.
While your husband and/or son is enjoying the Jeffrey R. Nieman Golf Classic, moms and wives please stay tuned for details on a relaxed luncheon get-together.
As always this is a great event for family and friends to get together and applaud this highly motivated chapter. We'll also be honoring Silver Legionnaires for those initiated in 1992 (or earlier). A live and silent auction will take place. Tickets are $55. Includes dinner, beer and wine. If you have dietary restrictions, please email Sean Brinker at sdb@zvplaw.com. NOTE: Active brothers and phikeias need not purchase a ticket, or have one purchased on their behalf, as they will pay for their tickets through the normal mechanisms of chapter fees via OmegaFi.


The alleged "War on Christmas" is back with a vengeance and it has arrived thanks to Starbuck's new holiday red cup. The coffee store chain has reinvented its holiday cup without tree ornaments or snowflakes and its evangelical Christian customers have taken note.
The Seattle-based company has been at the centre of an online attack, with customers announcing on social media that they will no longer buy their drinks from the chain. According to The Washington Post, one pastor has urged those who do not want to boycott Starbucks to tell baristas their names are "Merry Christmas" when they order their drinks.
The backlash against Starbuck began with a Facebook video posted by Christian evangelist Joshua Feuerstein on 5 November who criticised the company for the lack of explicit holiday messages on the cups. Feuerstein called on Starbucks customers to "start a movement" with the hashtag #MerryChristmasStarbucks. At the time of publishing, the video had 13,385,630 views, 165,064 likes and 474,859 shares.
According to Amobee Brand Intelligence data, digital engagement around the company has seen a significant rise in the last five days. The Wall Street Journal reported that the marketing technology company found that there have been 40,455 tweets between 5 November and 12pm ET on 9 November around "Red Cup," the hashtag #MerryChristmasStarbucks or "Starbucks Red Cup".
However, Starbucks has said that claims it wants to wage a war on the holiday are unfounded. "In the past, we have told stories with our holiday cups designs," Starbucks vice president of design and content, Jeffrey Fields, said in a statement on 8 November. "This year we wanted to usher in the holidays with a purity of design that welcomes all of our stories."
The company added: "Creating a culture of belonging, inclusion and diversity is one of the core values of Starbucks, and each year during the holidays the company aims to bring customers an experience that inspires the spirit of the season." The Wall Street Journal noted that this is not the first time the company has courted controversy. Earlier this year, it faced backlash for attempting to stimulate a conversation on race by having baristas write "Race Together" on cups.


text:
Републикански път IIІ-5802 е третокласен път, част от Републиканската пътна мрежа на България, преминаващ изцяло по територията на Пловдивска област. Дължината му е 20 km.

Пътят се отклонява надясно при 43,9 km на Републикански път II-58 в североизточната част на село Тополово и се насочва на североизток, надолу по долината на река Мечка (десен приток на Марица) през най-западната част на Хасковската хълмиста област. Преминава през село Леново и в северозападната част на село Поройна се свързва с Републикански път III-667 при неговия 19,5 km.

Вижте също
Републиканска пътна мрежа на България

Източници

 Карта на републиканската пътна мрежа.
 Пътен атлас на България М 1:200000.
 Списък на републиканската пътна мрежа на България .

Републикански пътища в България
Област Пловдив

redpajama_set_name: RedPajamaWikipedia


Don't let the office cold or flue stop you from working out. Follow our top tips to survive flu season.
Each year, the typical adult can expect to contract two or three colds, according to the Centers for Disease Control and Prevention. Skip the annual flu vaccine and you set yourself up for a bout of that as well. But it doesn't have to be this way! Aside from good hand washing (with soap, for at least 20 seconds), there's a lot you can do to drastically cut your risk of getting sick. And even if you do catch a bug, you may be able to cut short the duration of your illness. Arm yourself with these tips from the experts, and make this cold and flu season your healthiest yet.
The same live cultures that help ease digestive distress can help stave off a cold. A 2011 study backs this up: Scientists found that people who consumed probiotics via supplements or fermented foods (think yogurt, kefir and kimchi) had 12 percent fewer upper respiratory infections.
You might as well lick a restroom door (ick). Not touching your face greatly cuts your odds of getting sick. But that's easier said than done: The average person puts a hand on her mouth or nose more than three times an hour. To break the habit, try sitting on your hands when they're idle.
Throughout cold season, add this to your nighttime routine: Rinse your nose using a neti pot with boiled (and cooled) salted water, or an over-the-counter nasal irrigator or saline solution. It will help clear out viral particles you've breathed in during the day before they take root in your system.
Fluids help thin out the mucus that your body makes when you're sick. And when that germ-filled mucus is thinner, it's easier to clear out of your system. We suggest downing at least 2 liters of water or other fluids a day.


Slipcovers For Chairs And Sofas. Davenport Sofa Slipcover Special Order Fabrics Ballard Designs With Regard To Slipcovers For Chairs And Sofas Decor 6. Charleston Furniture Slipcovers Pottery Barn For Chairs And Sofas Designs 4. Metro Sofas Sleepers Chair Slipcovers Modern Slipcovered With Regard To For Chairs And Ideas 0. Classic Slipcovers Brushed Twill Loveseat Slipcover Free Shipping Inside For Chairs And Sofas Inspirations 14. Sofa Slipcovers Sure Fit Home Decor With Regard To For Chairs And Sofas Design 7. Rowe Nantucket Sofa With Slipcover Jordan S Furniture Intended For Slipcovers Chairs And Sofas Designs 9. Slipcover Only For Willow Modern Slipcovered Twin Sleeper Sofa In With Slipcovers Chairs And Sofas Inspirations 2.
9 Best Sherry S Sofa Chair Slipcovers Images On Pinterest Ethan In For Chairs And Sofas Prepare 5. Sofa Slipcovers Sure Fit Home Decor Inside For Chairs And Sofas Decorations 3. Slipcovers For Sofas With Attached Cushions Can It Be Done Pertaining To Chairs And Designs 10. Loveseat Slipcovers Sure Fit Home Decor Pertaining To For Chairs And Sofas Design 15. Whitman Sofa Slip Cover Only Light Gray Farmhouse Slipcovers Inside For Chairs And Sofas Remodel 17.


Will be there! Can't wait, bringing one guest. Hope this is ok?
Thanks, Sabrina. We've forward your RSVP to Darryl & Co. See you there!
Thanks, Darla. Plus one is fine : ) We've forwarded your RSVP to Darryl & CO. See you there!