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Egyptian Voters on the Promise, and Limits, of Historic Election: Sharif Kouddous Reports from Cairo
StoryMay 24, 2012
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Sharif Abdel Kouddous
independent journalist and Democracy Now! correspondent based in Cairo, Egypt.
Hany Massoud
Democracy Now! video producer.
See all of Democracy Now!'s reports on Egypt
Follow Sharif Abdel Kouddous on Twitter: @sharifkouddous
Voters are heading to the polls for the second day in Egypt's first competitive presidential election following the ouster of Hosni Mubarak 15 months ago. The first day of voting saw numerous reports of minor violations, but was largely hailed as free of fraud and violence. Ahmed Shafik, Mubarak's last prime minister and now a leading candidate, was swarmed by protesters outside his polling station who hurled shoes and debris at him. Democracy Now! correspondent Sharif Abdel Kouddous and videographer Hany Massoud spent the day traveling to polling stations around Cairo speaking to voters about their choices for president and their concerns in the election. [includes rush transcript]
StoryMay 23, 2012Egypt Votes: Sharif Abdel Kouddous Reports from Cairo on Historic Post-Mubarak Election
This is a rush transcript. Copy may not be in its final form.
JUAN GONZÁLEZ: We end today's show in Egypt, where voters are heading to the polls for the second day in the country's first competitive presidential election following the ouster of Hosni Mubarak 15 months ago. Authorities have declared today a public holiday to allow public sector workers to vote. Results are expected within the next few days. If no candidate wins above 50 percent of the vote, a runoff between the top two contenders is scheduled for June 16th.
AMY GOODMAN: Democracy Now! correspondent Sharif Abdel Kouddous spent yesterday traveling to polling stations around Cairo speaking to voters about their choices for president and their concerns in the election. He filed this report.
SHARIF ABDEL KOUDDOUS: Election day in Egypt. Millions of voters head to the polls in the country's first-ever competitive presidential race. Fifteen months after 30-year autocrat Hosni Mubarak was forced out of office in a popular uprising, the people of Egypt are deciding who will replace him.
TAREK HAMDI: Actually, this is the very first time in Egyptian history, since pharaohs and so on, that we do vote for our president.
SHARIF ABDEL KOUDDOUS: But with the enthusiasm comes anxiety, as well. The Supreme Council of Armed Forces has led the country through a turbulent transition, and many questions remain about the military's future role. The president is being elected without a constitution and without clearly delineated powers. Yet, for the first time in history, the outcome of Egypt's presidential election is not a foregone conclusion, and it's anyone's guess who will win.
Among the leading front-runners is the Muslim Brotherhood's candidate, Mohamed Morsi. This man says he supports Mohamed Morsi because he will enact Sharia law and has an Islamic political program. Last fall, the Muslim Brotherhood captured more than 40 percent of the seats in parliamentary elections. Yet some of those who supported the group the first time 'round are not backing them for the presidency.
EGYPTIAN VOTER 1: [translated] I voted for the Brotherhood in the first phase, in the parliamentary elections. I can't vote for the Brotherhood now. I am saying my personal opinion. Why? Because we need to have a collection of people in government—Brotherhood and liberals and so on. That is what the Egyptian people are: the different communities we have. I voted for Hamdeen Sabahi because he's a moderate. He has no pressures. He wants to serve the Egyptian people.
SHARIF ABDEL KOUDDOUS: Also among the front-runners are two former members of the Mubarak regime: Amr Moussa, who served as Mubarak's foreign minister for 10 years, and Ahmed Shafik, who was his last prime minister. Voters cited their experience in the former government as reasons to back them.
This man says he's voting for Amr Moussa because he's spent his whole career in foreign diplomacy and understands how to deal with the country. Ahmed Shafik understands politics and can bring the country forward, says this voter.
Outside another Imbaba polling station, this woman says the incoming president should focus on what poor people want: a life of dignity. Fellow voters nearby nod in agreement. But when she says she voted for Ahmed Shafik, they burst into a heated argument.
EGYPTIAN VOTER 2: [translated] All of the crises the woman said, I agreed with, except for Ahmed Shafik. He's a part of the former regime. Ahmed Shafik is Mubarak, and Mubarak is Ahmed Shafik. The man is sticking his tongue out at us. He says he is Mubarak's representative. I vote by exclusion. I exclude Ahmed Shafik and Amr Moussa. And then, any one of the 11 left [works], any one of them. Personally, I support Mohamed Morsi, but if Abdel Moneim Aboul Fotouh came, I would be happy. If Hamdeen came, I would be happy. If Khaled Ali came, I would also be happy.
SHARIF ABDEL KOUDDOUS: Here in Imbaba, as people line outside of school to go into the polling station, some of the main issues of concern are the economy and security. But some of those who took part in protests in Tahrir Square are calling for a deeper change.
EGYPTIAN VOTER 3: [translated] There are people who against change, people who are against Egypt moving forward, people who are against a certain idea prevailing. People say, "Damn the revolution," that the revolution ruined the country. The revolution made people go down believing in someone. It doesn't matter who. What matters is that people went down to choose someone. The revolution made people think politically. My dad is 60 years old. He never thought politically in his entire life. Now he studies what is good, what is bad. There is awareness. It's like there was an electric shock that made people think. There's something called the economy, the state, security. The revolution made people rise up and opened their eyes.
SHARIF ABDEL KOUDDOUS: Revolutionary graffiti covers the walls of downtown Cairo, one of them showing the former president, Hosni Mubarak, and Field Marshal Hussein Tantawi, and behind them, two members of the former regime who are running for president today: Amr Moussa and Ahmed Shafik. This on the corner of the square that is considered the epicenter of the revolution, Tahrir. But some of the revolutionaries who were at the core of this struggle that caused these elections to happen are boycotting today. One of them is Pierre Sioufi.
PIERRE SIOUFI: I'm not voting today, because I have a problem with voting in a—without a constitution, voting without knowing what the president is going to come and do, what are his—what can he do, I mean, what are his responsibilities. So, I'm not voting. I have no idea why should anybody vote. I mean, the whole story is just picking a puppet to put in, and whoever the puppet is is not going to make any difference, I think.
SHARIF ABDEL KOUDDOUS: A puppet to whom?
PIERRE SIOUFI: A puppet to the military institution. It's not even—it's not even anymore the SCAF itself, it's the military institution as a whole.
SHARIF ABDEL KOUDDOUS: The neighborhood of Shubra lies just north of Tahrir Square. It's home to more than three million residents, including the largest concentration of Copts, Egypt's Christian minority. Residents spoke about their concerns on election day.
EGYPTIAN VOTER 4: [translated] I chose a civil candidate, because I saw that he would be fit to be president. I think that the most important thing in the country is that there is some justice, because there is a lot of sectarian racism because of religion and class differences. That's why no one gets their rights in the country. So we need someone who is not partial to a specific religion.
SHARIF ABDEL KOUDDOUS: Another Shubra resident explained why he was backing Amr Moussa.
EGYPTIAN VOTER 5: [translated] He is the most preferable, because he is a civil man. He is not from the military or the Brotherhood. The Brotherhood say things and then go back on their word. They said they only wanted some of parliament, but look how much they took. When it is a civil country, I will feel that it is not military nor Brotherhood, so I will be at peace.
SHARIF ABDEL KOUDDOUS: Just across the river from Shubra is the upper-scale neighborhood of Zamalek. Voters here also voiced concerns about the rise of Islamist politics in post-Mubarak Egypt.
EGYPTIAN VOTER 6: I do know want our leader to be someone who is biased or have any preferences. I want someone who is a representative of the Egyptian culture and the simple Egyptian mentality, who can represent and give minorities as well as majorities their rights, and someone who does not invest his time in things that are, at the end of the day, very private. Just, for me, that is religion.
EGYPTIAN VOTER 7: [translated] It is necessary I vote for Amr Moussa, because he has long experience in international, regional and Arab diplomacy. He is a man of the world and understands the administration of a country's affairs. It's possible to say he can balance between political Islam that is ruling our legislative branch in the People's Assembly and the Shura Council.
SHARIF ABDEL KOUDDOUS: Abdel Moneim Aboul Fotouh is another Islamist candidate who has gained support from across the political spectrum, from secular liberals to conservative Muslims. Tarek Hamdi says he's voting for him to reject the Muslim Brotherhood's candidate.
TAREK HAMDI: Let's say it's a strategic vote, because I think, honestly, the 13 candidates are not the best. We are choosing the best of the worst, if we may say so. So, in this case, definitely, I believe from the studies I have made and the people I have talked with that he has the biggest chances.
SHARIF ABDEL KOUDDOUS: Other voters are backing Hamdeen Sabahi, a longtime opponent of the Mubarak regime and the founder of a Nasserist political party.
EGYPTIAN VOTER 8: He is the only revolutionary candidate. We're definitely not voting for the old regime, and we're definitely not voting for the people vocaling a religious state. So the only candidate available for that option is Hamdeen Sabahi. I have very low hopes of the whole process. I see it as non-democratic. I see that the transformation has been deformed by the SCAF. The whole transformation had been built wrong. We should have had a constitution before a presidential elections. We're going to face this month a first phase of elections, second phase of elections, the court order for Mubarak, the handover of authority that we're clearly seeing signals of the military establishment not willing to transform this authority to a civilian rule. We're looking at a very tense month that can be extremely explosive. Hopefully it passes the right way, it passes the revolution way, it passes the way that serves the people the best.
SHARIF ABDEL KOUDDOUS: Vote tallying in Egypt's presidential election begins today shortly after the polls close. Speculation is rife over who will come out on top. If no candidate wins more than 50 percent, a likely runoff will be held next month, with the final result announced on June 21st. For the winner, a turbulent future awaits. For Democracy Now!, I'm Sharif Abdel Kouddous with Hany Massoud in Cairo, Egypt.
AMY GOODMAN: And that does it for our broadcast.
The original content of this program is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License. Please attribute legal copies of this work to democracynow.org. Some of the work(s) that this program incorporates, however, may be separately licensed. For further information or additional permissions, contact us.
Egypt Votes: Sharif Abdel Kouddous Reports from Cairo on Historic Post-Mubarak Election


More than 200 Houston firefighters will be laid off after the City Council approved the measure Wednesday during a heated debate. The decision comes as the city tries to implement voter-mandated pay parity between the local police and fire departments. Mayor Sylvester Turner has said that the measure will cost the city $307 million from fiscal year 2019 to fiscal year 2023.
This fire company was a staple in the community for 110 years. Now, it's gone. Millbourne Fire Company in Delaware County voted to shut down Tuesday night following five years of financial instability. Chief Joseph Artmont Jr. has been at the fire company since 1982 and became chief three years ago.
Listener Roberta "Bobbie" Brown recently emailed the Curiosity Desk, after reading a story in the paper about Boston firefighters. In the piece, the author referred to them as "jakes." She wanted to know where that term comes from and how it came to refer to firefighters in the region. Like our curious listener, I too have seen firefighters called "jakes" in the papers.
VIDEOS: A Union Pacific freight train carrying ethanol careened off its tracks in a Fort Worth neighborhood Wednesday and burst into flames, prompting an evacuation of nearby homes and a hazardous materials response. It happened at about 12:40 a.m. near the 3200 block of Yuma Street, on the city's southeast side and near Echo Lake.
More than 120 current and former Conway police and firefighters will receive money payment before the end of the year after the city agreed to settle the class action lawsuit for $1.15 million. The Conway City Council voted on Monday in a special-called meeting to give Mayor Bart Castleberry the OK to request a preliminary approval in the suit that was filed against the city in 2012, a case Castleberry inherited when he was voted in as mayor in 2016.
VIDEO: In 2016, Ashland hired three additional firefighters for the first time since 2001 in response to service demands. Due to a $2 million budget deficit, those brand new positions could be eliminated in the 2019-2021 budget. Ultimately, that would affect service levels in the city and reduce the number of staffed crews from about 10 members to a maximum of nine members.
VIDEO: Concerns over title transfers and insurance lead the City of Howardwick to file a lawsuit against the volunteer fire department that protects it. In a recent lawsuit, the City of Howardwick contends a former city secretary improperly transferred the titles of three fire trucks and an ambulance to the Howardwick Volunteer Fire Department instead of to the city.
The Talcott Volunteer Fire Department sits proudly with its engines in line. It was in June of 2016 when the department was closed by state officials who cited lack of crew members and equipment testing. As the community had to rely on other departments as far away as Alderson to respond to calls, Chief Bill Costomiris and board members were forced to build from the ground up to get recertified.
The National Transportation Safety Board has been unable to determine the ignition source that sparked the explosion in a Silver Spring apartment complex meter room, where natural gas had been building up for at least three hours. Seven people were killed and 65 were sent to the hospital after the explosion and fire at the Flower Branch Apartments on Aug.
VIDEO: The wife of a fallen police officer thought she would be getting help - but found out her family won't be getting the benefit of help with tuition bills. A law in the books is supposed to take care of in-state college tuition for the families of fallen officers and firefighters, but there's a loophole that two freshman lawmakers want to tie up to make sure no family is left behind.


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Q: high percentage of network I/O in Dynatrace when using pgbouncer with Postgres 12 We are running some tests in Google Cloud and we are using CloudSQL Postgres 12 with cloudsqlproxy and pgbouncer.
[x ] bug report
Describe the issue
When we run tests in Google cloud using K8s and CloudSQL Postgres 12 with cloudsql authproxy and pgbouncer we are observing lot of Network I/O while executing queries.
Driver Version?
42.5.0
DT method hotspots
DT Network IO
Java Version?
11
OS Version?
Centos
PostgreSQL Version?
12
To Reproduce
Steps to reproduce the behaviour:
Happening almost all the time when using both pgbouncer and Postgres. Didn't observe earlier when directly using Postgres.
Logs
There are no error logs it's just that queries are taking lot of time to execute. Even queries as simple as which get by primary key are just hung up on SocketRead.
Any ideas will be really helpful. Let me know if more information need to be shared. What could be possible reasons of so much of network I/O? And it only seem to be an issue when working with pgbouncer.
We looked into pgbouncer logs and found nothing concrete there. There isn't any waiting showing up in the logs. We played around with configuration in pgbouncer.ini as well but nothing worked. If anyone has faced similar issue then share if anything worked.

redpajama_set_name: RedPajamaStackExchange


text:
How, When and Why You'll Want to Implement this Architecture
Software-Defined WAN Summit
9 a.m. PDT/12 p.m. EDT, SD-WAN 101:
Understanding the Basics of SD-WAN
Demand for SD-WAN solutions is growing by leaps and bounds with some estimates calling for the percentage of enterprises implementing SD-WAN to roughly triple over the next few years. In this session, we dig into what all the fuss is about. Attend this session to find out:
The trends in enterprise computing driving the demand for SD-WAN.
The networking architecture and expenses that SD-WAN replaces.
What SD-WAN allows in terms of network and application management.
How SD-WAN works. * An overview of the different ways the technology can be implemented.
Don't miss this session to understand what SD-WAN can do for your organization.
10 a.m. PDT/1 p.m. EDT, SD-WAN 102:
Top SD-WAN Benefits and Architecture Considerations
No technology is for everyone, but the appeal of SD-WAN is pretty broad. Due to the technology's efficiencies and advantages, it has a wider appeal than traditional WAN technology did and its software-defined flexibility starts to deliver functionality traditionally offered in adjacent technology areas. In this session, we'll help you determine:
How many branch offices you need for an SD-WAN to be cost effective.
How much organizations save by reducing their reliance on MPLS circuits via SD-WAN.
The types of application performance improvements that are possible with SD-WAN.
How to make cloud applications perform better using SD-WAN.
Best practices for deploying applications via SD-WANs.
How to improve security by using SD-WAN.
After attending this session, you'll have a much better sense of how to deploy SD-WAN for maximum benefit in your organization.
The Role of SD-WAN in a Remote Work Era
A big part of the SD-WAN story is optimizing the branch office connection to corporate headquarters and cloud applications. But what happens when the branch office is removed from the equation? In this timely session, get an expert look at the role that SD-WAN infrastructure plays when all those branch office employees, and headquarters employees for that matter, are working from home. We'll look at:
Scalability considerations of SD-WAN architectures and solutions when nearly 100% of the employees are hitting the network from home offices.
The role the branch office plays in SD-WAN architecture during the pandemic.
SD-WAN architectural and equipment options for home-based workers during the pandemic.
How to future proof your investments and architectures around SD-WAN for post-pandemic situations.
Software-defined wide-area networking (SD-WAN) is one of the fastest-growing SDx's out there. Attend this Virtualization & Cloud Review virtual summit to find out why that is and to find out what SD-WAN can do for your organization. Our independent industry experts will give you a high-level view of the basic benefits that SD-WAN delivers, and then will drill down into specific architectural considerations. They'll provide key insights that will help ensure your SD-WAN deployment delivers maximum value and performance and helps your organization cut unnecessary expenses.
Win a $500 Amazon Gift Certificate!
We're giving away a $500 Amazon Gift Certificate to one lucky attendee who registers and attends on the day of the event! ! SIGN UP TODAY!

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to configure lib path for apache utilities? I juste installed apxs bin with the httpd24-httpd-devel-2.4.6-16.el6.x86_64 package.
I need it to compile PHP with Apache module.
When I launch the command  apxs --help I get the following error:
cannot open /httpd/build/config_vars.mk: No such file or directory at /opt/rh/httpd24/root/usr/bin/apxs line 214

I locate this file into /opt/rh/httpd24/root/usr/lib64/httpd/build/, should I add this folder to my PATH or is there a way to tell Apache to look at this directory?

A: This is the problem with SCL package httpd24-httpd-devel. There was already bug for standard CentOS/RedHat rpm which was fixed.
Since apxs is a perl script if you take a look inside it, this is the responsible code:
my $libdir = `pkg-config --variable=libdir apr-1`;
chomp $libdir;

my $installbuilddir = $libdir . "/httpd/build";

What you also can do is change the $installbuilddir variable to:
my $installbuilddir = '/opt/rh/httpd24/root/usr/lib64' . "/httpd/build";

Just make sure you modify the httpd24 apxs, which is available at /opt/rh/httpd24/root/usr/bin/apxs, and not the orig one (/usr/sbin/apxs).
If you don't want to modify apxs, this quick fix would work also:
cd /
ln -s /opt/rh/httpd24/root/usr/lib64/httpd .

A: Are you doing sudo make install from inside a scl enable httpd24 bash? This resets $PATH to the default. You need to instead do:
sudo scl enable httpd24 perl516 'make install'


text:
Jamie Kay Ready Have Fun, Remain Competitive with Outlaw Midgets
By ladybug388 on April 3, 2022 • ( Leave a comment )
Growing up going to a mix of both dirt and asphalt events, Jamie Kay found himself looking to get behind the wheel of a racecar. The search began, and the Outlaw Midgets immediately came to the fore front of the discussion due to the price point.
"I was looking for a car that was affordable, fun to drive, and decently fast," Jamie Kay recalled. "I remember Mike Bradley said last year where you can buy a car this car, this size, and do lap times similar to a late model at the tracks that we go to – and it's a ton of fun for a tenth of a price for those vehicles. From that standpoint, it's a riot to drive one of these things."
Kay's search took him to marketplace where he scouted a couple different cars that were for sale. However, he became more impressed in seeing how many of those within the club were willing to share information and be helpful in purchasing the right racecar for him and his family.
Flash forward a year later, Kay is one of the faces of the Outlaw Midgets and is getting ready for his second year of competition in 2022. Admittedly while the goal was Rookie of the Year in 2021 which he accomplished, he admits not having any set expectations for the sophomore campaign.
"This year, I truly don't know what exactly the focus is this year, other than to have a fun with the group of people that we spend time with," he commented. "There's so many characters in the group of outlaws and we have so much fun with all the guys on a Saturday night / Sunday afternoon. I don't think we could ever say we could aim for a title or anything like that – that's naive. We're just in our second year of embarking on auto racing, but it's been a great experience and couldn't ask for a better experience with the boys."
The experience has included getting o travel around the province to the variety of speedways, and unique characteristics of each.
"There's times that I love the high speeds, banks like Sunset Speedway – then there's other times I like the flat racing at like Sauble," he shared. "The only track that is somewhat close to us – we're on the other side of Belleville, but the closest track is Peterborough and I always feel like that's kind of like my home track if you will. Every one of them is so unique and every track has been great to deal with. Sauble, my first ever track that I laid rubber on the track, and they were great to us because we were brand new and didn't know what was going on. JP at Peterborough – I can't say enough about him. Every time we want to rent the track and get used to the car, he's never said no; he always has the door open.
"I am just looking forward to every one of them – even Flamboro. I've never been there, a lot of people in the club have been there and I hear it's a different challenge than the other tracks. I am looking forward to the challenge of figuring out how to set-up for that one."
As he continues his endeavors, there is a focus on building upon his reputation of a clean racecar driver, saying that was important from the day he got involved.
"But as far as a the rookie of the year in the rearview, I think that's a lot of people that are looking for me to step up and go a little bit this year," he added. "But realistically, there's a lot of good drivers in the club – there's so many good ones that they can just show up and be miles above the rest in the club. At no point last year did I ever feel our car was miles above anything else; I always thought we were middle, front of the pack. So for me, it's about consistently getting everything out of the car that I've got and not having those ups and downs that you have through a season."
While looking forward to the racing on-track, there's also excitement in getting to experience more of the family atmosphere that he spoke of in the pits each week.
"It's vast fully different than the racing that I'm used to as I come from a standard bred race horse background," he commented. "Everybody is friendly – you say hello, you say goodbye – but that's the end of the conversation. Ultimately, you're hoping the other guy's horse falls down. in what I've found in this group of people, everybody is there to help and what I find is amazing is in your time of need, everybody jumps in to help change a tire, reset the chain, fix your belt tension, and make you better and more competitive.
"At the end of the day, that's what it's about – the product we put on the track. Every promoter needs the fans in the stands, and the better show we put on as a group, it'll just help everybody. If you have a club with one or two same characters at the front every night, it doesn't help anyone and it's not entertaining for the fans at the end of the day."
Entertaining the fans, improving the product involves making sure car counts stay strong, which seem to appear to be the case in 2022 with the Outlaw Midgets. There are several rookies who are looking to follow in Kay's footsteps, to which he says the biggest piece of advice to show up as often as you can.
"It's the only way you're going to learn," he commented. "It is a great group of people. Up and down the whole paddock, it's a fantastic group of people that are there for the love of racing. Let's be honest – if you're racing in the midgets, you're not racing to get rich. You're not doing it for the money but because you like the racing, and in some cases, these guys love the racing.
"If you want to learn to race and drive a car to start with and why setting up a car matters and getting that knowledge, this is the group that you want to be with. If you go above this, your learning curve grows exponentially. You're just going to waste a lot of time and money learning hard lessons. At this level, you can learn those lessons at an acceptable price point and have a lot of fun along the way."
Categories: Interviews, Outlaw Midgets, Press Release
Tagged as: featured, jamie kay, Outlaw Midgets, Press Release
2022 Mini Stock Drivers to Watch – Part 3
Tim Norris Named Race Director at Flamboro Speedway


SMi Group's Global MilSatCom 2017 Conference & Exhibition will take place on 7-9 November at the Park Plaza Riverbank Hotel in London.
Europe's leading event for defence and military satellite professionals will open doors next week, with an expected attendance of over 500 delegates from nations and organisations including Belgium, Brazil, Canada, Canada, China, Czech Republic, Denmark, Ethiopia, France, Germany, Ghana, Israel, Italy, Japan, Luxembourg, Netherlands, Nigeria, Norway, Poland, Singapore, South Korea, Spain, Sweden, Turkey, Ukraine, United Arab Emirates, United Kingdom, USA and more.
The full roster of speakers and presentations can be found on www.globalmilsatcom.com.


The Central Arts District, EDGE District, Grand Central District, Warehouse Arts District and downtown Waterfront District unite once a month for the St. Petersburg Second Saturday ArtWalk. Studios and galleries stay open late on Saturday, April 13 and welcome visitors from 5-9 pm.
Many of our ArtWalk studios/galleries celebrate the opening of brand new exhibitions during ArtWalk. Discover wonderful surprises, visit your favorite artist or find a new one. All are waiting for your exploration, conversation, purchases and inspiration. Make an evening of it and stop for dinner, drinks or dessert at a downtown restaurant.


text:
\section{Introduction}

Compressed sensing is a new data acquisition theory based on the
discovery that one can exploit sparsity or compressibility when
acquiring signals of general interest, and that one can design
nonadaptive sampling techniques that condense the information in a
compressible signal into a small amount of
data~\cite{CRT06:Robust-Uncertainty,CT04:Near-Optimal,Don06:Compressed-Sensing}.
In a nutshell, reliable, nonadaptive data acquisition, with far fewer
measurements than traditionally assumed, is possible. By now,
applications of compressed sensing are abundant and range from imaging
and error correction to radar and remote sensing, see
\cite{IEEEMag:25-2,IEEEMag:24-4} and references therein.

In a nutshell, compressed sensing proposes acquiring a signal $x \in
\mathbb{R}^n$ by collecting $m$ linear measurements of the form $y_k = \langle
a_k, x\rangle + z_k$, $1 \le k \le m$, or in matrix notation,
\begin{equation}
\label{eq:model}
y = Ax + z;
\end{equation}
$A$ is an $m \times n$ sensing matrix with $m$ typically smaller than
$n$ by one or several orders of magnitude (indicating some significant
undersampling) and $z$ is an error term modeling measurement
errors. Sensing is nonadaptive in that $A$ does not depend on
$x$. Then the theory asserts that if the unknown signal $x$ is
reasonably sparse, or approximately sparse, it is possible to recover
$x$, under suitable conditions on the matrix $A$, by convex
programming: we simply find the solution to
\begin{equation}\tag{$L_1$}
  \min_{\tilde x \in \mathbb{R}^n} \|\tilde x\|_1 \quad\text{subject to}\quad \|A\tilde x - y\|_2 \leq \varepsilon,
\end{equation}
where $\|\cdot\|_2$ denotes the standard Euclidean norm, $\|x\|_1 =
\sum |x_i|$ is the $\ell_1$-norm and $\varepsilon^2$ is a likely upper bound
on the noise power $\|z\|_2^2$. (There are other algorithmic
approaches to compressed sensing based on greedy algorithms such as
Orthogonal Matching
Pursuit~\cite{MZ93:Matching-Pursuits,TG07:Signal-Recovery}, Iterative
Thresholding~\cite{BD08:Iterative,FR07:Iterative}, Compressive
Sampling Matching Pursuit~\cite{NT08:Cosamp}, and many others.)

Quantitatively, a concrete example of a typical result in compressed
sensing compares the quality of the reconstruction from the data $y$
and the model \eqref{eq:model} with that available if one had an
oracle giving us perfect knowledge about the most significant entries
of the unknown signal $x$.  Define -- here and throughout -- by $x_s$
the vector consisting of the $s$ largest coefficients of $x \in \mathbb{R}^n$
in magnitude:
\begin{equation}
\label{eq:xs}
  x_s = \argmin_{\|\tilde x\|_0 \le s} \|x - \tilde x\|_2,
\end{equation}
where $\|x\|_0 = |\{i : x_i \neq 0\}|$. In words, $x_s$ is the best
$s$-sparse approximation to the vector $x$, where we shall say that a
vector is $s$-sparse if it has at most $s$ nonzero entries.  Put
differently, $x-x_s$ is the tail of the signal, consisting of the
smallest $n - s$ entries of $x$. In particular, if $x$ is $s$-sparse,
$x - x_s = 0$. Then with this in mind, one of the authors
\cite{Can08:Restricted-Isometry} improved on the work of Cand\`es,
Romberg and Tao~\cite{CRT06:Stable} and established that $(L_1)$
recovers a signal $\hat{x}$ obeying
\begin{equation}\label{eq:l1}
  \|\hat{x} - x\|_2 \leq C_0\frac{\|x-x_s\|_1}{\sqrt{s}} + C_1\varepsilon,
\end{equation}
provided that the {\em $2s$-restricted isometry constant} of $A$ obeys
$\delta_{2s} < \sqrt{2} - 1$.  The constants in this result have been
further improved, and it is now known to hold when $\delta_{2s} <
0.4652$~\cite{F10:Anote}, see also \cite{FL08:Sparsest}.  In short,
the recovery error from $(L_1)$ is proportional to the measurement
error and the tail of the signal.  This means that for compressible
signals, those whose coefficients obey a power law decay, the
approximation error is very small, and for exactly sparse signals it
completely vanishes.

The definition of restricted isometries first appeared in
\cite{CT05:Decoding} where it was shown to yield the error bound
\eqref{eq:l1} in the noiseless setting, i.~e.~when $\varepsilon = 0$ and $z =
0$.
\begin{definition}
  For an $m\times n$ measurement matrix $A$, the
  $s$-\textit{restricted isometry constant} $\delta_s$ of $A$ is the
  smallest quantity such that
$$
(1-\delta_s)\|x\|_2^2 \leq \|Ax\|_2^2 \leq (1+\delta_s)\|x\|_2^2
$$
holds for all $s$-sparse signals $x$.
\end{definition}

With this, the condition underlying \eqref{eq:l1} is fairly natural
since it is interpreted as preventing sparse signals from being in the
nullspace of the sensing matrix $A$. Further, a matrix having a small
restricted isometry constant essentially means that every subset of
$s$ or fewer columns is approximately an orthonormal system.  It is
now well known that many types of random measurement matrices have
small restricted isometry
constants~\cite{CT04:Near-Optimal,MPJ06:Uniform,RV08:sparse,BDDW07:Johnson-Lindenstrauss}.
For example, matrices with Gaussian or Bernoulli entries have small
restricted isometry constants with very high probability whenever the
number of measurements $m$ is on the order of $s\log(n/s)$.  The fast
multiply matrix consisting of randomly chosen rows of the discrete
Fourier matrix also has small restricted isometry constants with very
high probability with $m$ on the order of $s (\log n)^4$.

\subsection{Motivation}
The techniques above hold for signals which are sparse in the standard
coordinate basis or sparse with respect to some other {\em orthonormal
  basis}.  However, there are numerous practical examples in which a
signal of interest is not sparse in an orthonormal basis.  More often
than not, sparsity is expressed not in terms of an orthonormal basis
but in terms of an \textit{overcomplete} dictionary.  This means that
our signal $f \in \mathbb{R}^n$ is now expressed as $f = Dx$ where $D \in
\mathbb{R}^{n \times d}$ is some overcomplete dictionary in which there are
possibly many more columns than rows.  The use of overcomplete
dictionaries is now widespread in signal processing and data analysis,
and we give two reasons why this is so. The first is that there may
not be any sparsifying orthonormal basis, as when the signal is
expressed using curvelets~\cite{CD02:New-Tight,CDDY05:Fast} or
time-frequency atoms as in the Gabor
representation~\cite{FS98:Gabor-Analysis}. In these cases and others,
no good orthobases are known to exist and researchers work with tight
frames. The second reason is that the research community has come to
appreciate and rely on the flexibility and convenience offered by
overcomplete representations. In linear inverse problems such as
deconvolution and tomography for example -- and even in straight
signal-denoising problems where $A$ is the identity matrix -- people
have found overcomplete representations to be extremely helpful in
reducing artifacts and mean squared error
(MSE)~\cite{SED04:Redundant,SFM07:uwd}.  It is only natural to expect
overcomplete representations to be equally helpful in compressed
sensing problems which, after all, are special inverse problems.

Although there are countless applications for which the signal of
interest is represented by some overcomplete dictionary, the
compressed sensing literature is lacking on the subject.  Consider the
simple case in which the sensing matrix $A$ has Gaussian (standard
normal) entries.  Then the matrix $AD$ relating the observed data with
the assumed (nearly) sparse coefficient sequence $x$ has independent
rows but each row is sampled from $\mathcal{N}(0, \Sigma)$, where
$\Sigma = D^*D$.  If $D$ is an orthonormal basis, then these entries
are just independent standard normal variables, but if $D$ is not
unitary then the entries are correlated, and $AD$ may no longer
satisfy the requirements imposed by traditional compressed sensing
assumptions.  In~\cite{RandThes} recovery results are obtained when the sensing
matrix $A$ is of the form $\Phi D^*$ where $\Phi$ satisfies the restricted
isometry property.  In this case the sampling matrix must depend on the
dictionary $D$ in which the signal is sparse.  We look for a universal result
which allows the sensing matrix to be independent from the signal and its
representation.  To be sure, we are not aware of any such results in the
literature guaranteeing good recovery properties when the columns may
be highly -- and even perfectly -- correlated.

Before continuing, it might be best to fix ideas to give some examples
of applications in which redundant dictionaries are of crucial
importance.
\begin{description}
\item [Oversampled DFT] The Discrete Fourier Transform (DFT) matrix is
  an $n \times n$ orthogonal matrix whose $k$th column is given by
\[
d_{k}(t) = \frac{1}{\sqrt{n}} e^{-2\pi i kt/n},
\]
with the convention that $0 \le t, k \le n-1$. Signals which are
sparse with respect to the DFT are only those which are superpositions
of sinusoids with frequencies appearing in the lattice of those in the
DFT.  In practice, we of course rarely encounter such signals.  To
account for this, one can consider the oversampled DFT in which the
sampled frequencies are taken over even smaller equally spaced
intervals, or at small intervals of varying lengths.  This leads to an
overcomplete frame whose columns may be highly correlated

\item [Gabor frames] Recall that for a fixed function $g$ and positive
  time-frequency shift parameters $a$ and $b$, the $k$th column (where
  $k$ is the double index $k = (k_1, k_2)$) of the Gabor frame is
  given by
\begin{equation}\label{eq:gabor}
G_{k}(t) = g(t-k_2a) e^{2\pi ik_1bt}.
\end{equation}
Radar and sonar along with other imaging systems appear in many
engineering applications, and the goal is to recover pulse trains
given by
\[
f(t) = \sum_{j=1}^k \alpha_j
w\Big(\frac{t-t_j}{\sigma_j}\Big)\e^{i\omega_j t}.
\]
Due to the time-frequency structure of these applications, Gabor
frames are widely used~\cite{Mal99:Wavelet-Tour}.  If one wishes to
recover pulse trains from compressive samples by using a Gabor
dictionary, standard results do not apply

\item [Curvelet frames] Curvelets provide a multiscale decomposition of
  images, and have geometric features that set them apart from
  wavelets and the likes.  Conceptually, the curvelet transform is a
  multiscale pyramid with many directions and positions at each length
  scale, and needle-shaped elements at fine scales~\cite{CD02:New-Tight}.
  The transform gets its name from the fact that it approximates well
  the curved singularities in an image.  This transform has many
  properties of an orthonormal basis, but is overcomplete.  Written in
  matrix form $D$, it is a tight frame obeying the Parseval relations
$$
f = \sum_k \< f, d_k \> d_k \quad\text{and}\quad \|f\|_2^2 = \sum_k | \< f, d_k \> |^2,
$$
where we let $\{d_k\}$ denote the columns of $D$.  Although columns of
$D$ far apart from one another are very uncorrelated, columns close to
one another have high correlation.  Thus none of the results in
compressed sensing apply for signals represented in the curvelet
domain

\item [Wavelet Frames] The undecimated wavelet transform (UWT) is a
  wavelet transform achieving translation invariance, a property that
  is missing in the discrete wavelet transform (DWT)~\cite{D89:uwt}.
  The UWT lacks the downsamplers and upsamplers in the DWT but
  upsamples the filter coefficients by a factor of $2^m$ at the
  $(m-1)$st level -- hence it is overcomplete.  Also, the
   Unitary Extension Principle of Ron and Shen~\cite{RS97:Affine} facilitates
   tight wavelet frame constructions for $L^2(\mathbb{R}^d)$ which
   may have many more wavelets than in the orthonormal case.
   This redundancy has
  been found to be helpful in image processing (see
  e.g.~\cite{SED04:Redundant}), and so one wishes for a recovery
  result allowing for significant redundancy and/or correlation

\item [Concatenations] In many applications a signal may not be sparse
  in a single orthonormal basis, but instead is sparse over several
  orthonormal bases.  For example, a linear combination of spikes and
  sines will be sparse when using a concatenation of the coordinate
  and Fourier bases.
  One also benefits by exploiting geometry and pointwise singularities
  in images by using combinations of tight frame coefficients such as
  curvelets, wavelets, and brushlets.  However, due to the correlation
  between the columns of these concatenated bases, current compressed
  sensing technology does not apply.
\end{description}

These and other applications strongly motivate the need for results
applicable when the dictionary is redundant and has correlations.
This state of affair, however, exposes a large gap in the literature
since current compressed sensing theory only applies when the
dictionary is an orthonormal basis, or when the dictionary is
extremely uncorrelated (see
e.g.~\cite{GMS03:Approximation-Functions,RSV08:Redundant,BCJ10:Why}).

\subsection{Do we really need incoherence?}

Current assumptions in the field of compressed sensing and sparse
signal recovery impose that the measurement matrix have uncorrelated
columns.  To be formal, one defines the \textit{coherence} of a matrix
$M$ as
$$
\mu(M) = \max_{j<k} \frac{|\< M_j, M_k \> |}{\|M_j\|_2 \|M_k\|_2},
$$
where $M_j$ and $M_k$ denote columns of $M$.  We say that a dictionary
is \textit{incoherent} if $\mu$ is small.  Standard results then
require that the measurement matrix satisfy a strict incoherence
property \cite{Tro04:Greed-Good,CP07:Near}, as even the RIP imposes
this.  If the dictionary $D$ is highly coherent, then the matrix $AD$
will also be coherent in general.

Coherence is in some sense a natural property in the compressed
sensing framework, for if two columns are closely correlated, it will
be impossible in general to distinguish whether the energy in the
signal comes from one or the other.\footnote{ Recall that when the
  dictionary $D$ is sufficiently incoherent, standard compressed
  sensing guarantees that we recover $x$ and thus $f=Dx$, provided $x$
  is $s$-sparse with $s$ sufficiently small. }  For example, imagine
that we are not undersampling and that $A$ is the identity so that we
observe $y = Dx$. Suppose the first two columns are identical, $d_1 =
d_2$. Then the measurement $d_1$ can be explained by the input vectors
$(1, 0, \ldots, 0)$ or $(0, 1, 0, \ldots, 0)$ or any convex
combination.  Thus there is no hope of reconstructing a unique sparse
signal $x$ from measurements $y = ADx$.  However, we are \textit{not}
interested in recovering the coefficient vector $x$, but rather the
actual signal $Dx$.  The large correlation between columns in $D$ now
does not impose a problem because although it makes it impossible to
tell apart coefficient vectors, this is not the goal. This simple
example suggests that perhaps coherence is not necessary.  If $D$ is
coherent, then we clearly cannot recover $x$ as in our example, but we
may certainly be able to recover the signal $f = Dx$ from measurements
$y=Af$ as we shall see next.

\begin{figure}[ht]
\begin{center}
  \includegraphics[scale=0.6]{domains}
  \caption{The compressed sensing process and its domains. This
    distinguishes the domains in which the measurements, signals, and
    coefficients reside}\label{fig:domains}
  \end{center}
\end{figure}

\subsection{Gaussian sensing matrices}

To introduce our results, it might be best for pedagogical purposes to
discuss a concrete situation first, and we here assume that the
sensing matrix has iid Gaussian entries. In practice, signals are
never exactly sparse, and dictionaries are typically designed to make
$D^*f$ for some classes of $f$ as sparse as possible. Therefore, in
this paper, we propose a reconstruction from $y = A f + z$ by the
method of $\ell_1$-analysis:
\begin{equation}\tag{$P_1$}
  \hat{f} = \argmin_{\tilde f \in \mathbb{R}^n} \|D^* \tilde f\|_1 \quad\text{subject to}\quad \|A \tilde f - y\|_2 \leq \varepsilon,
\end{equation}
where again $\varepsilon$ is a likely upper bound on the noise level
$\|z\|_2$. Empirical studies have shown very promising results for the
$\ell_1$-analysis problem.  Its geometry has been
studied~\cite{EMR07:Analysis} as well as its applications to image
restoration~\cite{COS09:Split}.  However, there are no results in the
literature about its performance.

Our main result is that the solution to $(P_1)$ is very accurate
provided that $D^*f$ has rapidly decreasing coefficients. Our result
for the Gaussian case is below while the general theorem appears in
Section~\ref{sec:axiom}.
\begin{theorem}\label{thm:Gauss}
  Let $D$ be an arbitrary $n \times d$ tight frame and let $A$ be a
  $m\times n$ Gaussian matrix with $m$ on the order of $s\log(d/s)$.
  Then the solution $\hat{f}$ to $(P_1)$ obeys
$$
\|\hat{f} - f\|_2 \leq C_0\varepsilon + C_1\frac{\|D^*f - (D^*f)_s\|_1}{\sqrt{s}},
$$
for some numerical constants $C_0$ and $C_1$, and where $(D^*f)_s$ is the vector consisting of the largest $s$ entries of $D^*f$ in magnitude as in~\eqref{eq:xs}.
\end{theorem}

We have assumed that $D$ is a tight frame although this is simply to
make the analysis easier and is of course not necessary.  Having said
this, our results proves not only that compressed sensing is viable
with highly coherent dictionaries, but also that the $\ell_1$-analysis
problem provably works in this setting.  We are not aware of any other
result of this kind. To be sure, other methods for redundant
dictionaries such as~\cite{RSV08:Redundant,BCJ10:Why,Tro04:Greed-Good}
force incoherence on the dictionary $D$ so that the matrix $AD$
conforms to standard compressed sensing results.  The method in~\cite{RandThes}
requires that the sensing matrix depend on the dictionary.  These are drastically
different from the setting here, where we impose no such
properties on the dictionary.  We point out that our result holds even
when the coherence of the dictionary $D$ is maximal, meaning two
columns are completely correlated.  Finally, we also note that the
dependence on the noise level is optimal and that the tail bound in
the error is analogous to previous bounds in the non-redundant case
such as~\eqref{eq:l1}.

\subsection{Implications}

As we mentioned, the dependence on the noise in the error given by
Theorem~\ref{thm:Gauss} is optimal, and so we need only discuss how
the second term affects the estimation error.  This term will of
course be negligible when the norm of the tail, $D^*f - (D^*f)_s$, is
small.  Hence, the result says that for any dictionary, signals $f$
such that $D^*f$ decays rapidly can be approximately reconstructed
using $\ell_1$-analysis from just a few random measurements.  This is
exactly the case for many dictionaries used in practice and many
classes of signals as discussed earlier. As a side remark, one can
also guarantee rapid decay of $D^*f$ (we assume the signal expansion
$f = Dx$) when $D^*D$ is well behaved and the coefficient vector $x$
is nearly sparse. To see why this is true, suppose $D$ is a tight
frame so that $D^* f = D^* D x$. A norm commonly used to quantify
sparsity is the quasi $p$-norm with $p \le 1$ defined via $\|x\|_p^p
= \sum_i |x_i|^p$ (sparser signals with unit $2$-norm have
smaller $p$-norms). Now a simple calculation shows that
\[
\|D^* f\|_p \le \Bigl[ \max_j \sum_i |(D^*D)_{ij}|^p \Bigr]^{1/p} \,
\|x\|_p.
\]
In words, if the columns of the Gram matrix are reasonably sparse and
if $f$ happens to have a sparse expansion, then the frame coefficient
sequence $D^* f$ is also sparse. All the transforms discussed above,
namely, the Gabor, curvelet, wavelet frame, oversampled Fourier
transform all have nearly diagonal Gram matrices -- and thus, sparse
columns.

We now turn to the implications of our result to the applications we
have already mentioned, and instantiate the theorem in the noiseless
case due to the optimality of the noise level in the error.

\begin{description}

\item[Multitone signals] To recover multitone signals, we use an
  oversampled DFT, which is not orthonormal and may have very large
  coherence. However, since each ``off-grid'' tone has a rapidly
  decaying expansion, $D^*f$ will have rapidly decaying
  coefficients.\footnote{In practice, one smoothly localizes the data
    to a time interval by means of a nice window $w$ to eliminate
    effects having to do with a lack of periodicity. One can then
    think of the trigonometric exponentials as smoothly vanishing at
    both ends of the time interval under study.} Thus our result
  implies that the recovery error is negligible when the number of
  measurements is about the number of tones times a log factor

\item[Radar] For radar and sonar applications using Gabor
  dictionaries, our result similarly implies a negligible error.
  Indeed, with notation as in~\eqref{eq:gabor}, one sees that the
  sequence $\{\< w(t)\e^{i\omega t}, G_k(t)\rangle\}_k$ decays quickly
  (each pulse has a rapidly decaying expansion).  Therefore, our
  result implies a negligible error when the number of measurements is
  roughly the number of pulses in the pulse train, up
  to a log factor

\item[Images] Roughly speaking, the curvelet coefficient sequence of
  an arbitrary image, which is discontinuous along piecewise-$C^2$
  edges but is otherwise smooth, decays like $k^{-3/2}$ -- up to a log
  factor -- when arranged in a decreasing order of magnitude. Hence,
  our theorem asserts that one can get an $\ell_2$ error of about
  $s^{-1}$ from about $s \log n$ random samples of $f$. This is
  interesting since this is the approximation error one would get by
  bandlimiting the image to a spatial frequency about equal to $s$ or,
  equivalently, by observing the first $s^2$ Fourier coefficients of
  $f$. So even though we do not know where the edges are, this says
  that one can sense an image nonadaptively $m$ times, and get a
  recovery error which is as good as that one would traditionally get
  by taking a much larger number -- about $m^2$ -- of samples. This is
  a considerable gain. Of course, similar implications hold when the
  undecimated wavelet transform of those images under study decay
  rapidly

\item[Concatenations] When working with signals which are sparse over
  several orthonormal bases, it is natural to use a dictionary $D$
  consisting of a concatenation of these bases.  For example, consider
  the dictionary $D$ which is the concatenation of the identity and
  the Fourier basis (ignoring normalizations for now).  Then $D^*D$ is
  made up of four blocks, two of which are the identity and two of
  which are the DFT, and does not have sparse columns.  Then even when
  $f$ is sparse in $D$, the coefficients of $D^*f$ may be spread.  If
  this is the case, then the theorem does not provide a good error
  bound.  This should not be a surprise however, for if $D^*f$ is not
  close to a sparse signal, then we do not expect $f$ to be the
  minimizer of the $\ell_1$-norm in $(P_1)$.  In this case,
  $\ell_1$-analysis is simply not the right method to use.
\end{description}

To summarize, we see that in the majority of the applications, our
theorem yields good recovery results.  As seen in the last example,
the $\ell_1$-analysis method only makes sense when $D^*f$ has quickly
decaying coefficients, which may not be the case for concatenations of
orthonormal bases.  However, this is not always the case, as we see in
the following.

\textbf{An easy example.}  As above, let $D$ be the $n \times 2n$
dictionary consisting of a concatenation of the identity and the DFT,
normalized to ensure $D$ is a tight frame (below, $F$ is the DFT
normalized to be an isometry):
\[
D = \frac{1}{\sqrt{2}}[ I\quad F ].
\]
We wish to create a sparse signal that uses linearly dependent columns
for which there is no local isometry. Assume that $n$ is a perfect
square and consider the Dirac comb
\[
f(t) = \sum_{j=1}^{\sqrt{n}} \delta(t - j\sqrt{n}),
\]
which is a superposition of spikes spread $\sqrt{n}$ apart.
Thus our signal is a
sparse linear combination of spikes and sines, something that by the
last example alone we would not expect to be able to recover.  However,
$D^*f = [f \quad f]/\sqrt{2}$ is exactly sparse implying that $\|D^*f
- (D^*f)_s\|_1 = 0$ when $s > 2\sqrt{n}$. Thus our result shows that
$\ell_1$-analysis can exactly recover the Dirac comb consisting of
spikes and sines from just a few general linear functionals.

\subsection{Axiomization}\label{sec:axiom}
We now turn to the generalization of the above result, and give broader conditions about the sensing matrix under which the recovery algorithm performs well.   We will impose a natural property on the measurement matrix, analogous to the restricted isometry property.

\begin{definition}[D-RIP]
  Let $\Sigma_s$ be the union of all subspaces spanned by all subsets
  of $s$ columns of $D$.  We say that the measurement matrix $A$ obeys
  the \textit{restricted isometry property adapted to $D$}
  (abbreviated D-RIP) with constant $\delta_s$ if
$$
(1-\delta_s)\|v\|_2^2 \leq \|Av\|_2^2 \leq (1+\delta_s)\|v\|_2^2
$$
holds for all $v \in \Sigma_s$.
\end{definition}

We point out that $\Sigma_s$ is just the image under $D$ of all
$s$-sparse vectors.  Thus the D-RIP is a natural extension to the
standard RIP.  We will see easily that Gaussian matrices and other
random compressed sensing matrices satisfy the D-RIP.  In fact any $m
\times n$ matrix $A$ obeying for fixed $v\in\mathbb{R}^n$,
\begin{equation}\label{eq:cond}
\mathbb{P}\big( (1-\delta)\|v\|_2^2 \leq \|Av\|_2^2 \leq (1+\delta)\|v\|_2^2 \big) \leq C\e^{-\gamma m}
\end{equation}
($\gamma$ is an arbitrary positive numerical constant) will satisfy
the D-RIP with overwhelming probability, provided that $m \gtrsim
s\log (d/s)$.  This can be seen by a standard covering argument (see
e.g.~the proof of Lemma 2.1 in~\cite{RSV08:Redundant}).  Many types of
random matrices satisfy~\eqref{eq:cond}.  It is now well known that
matrices with Gaussian, subgaussian, or Bernoulli entries
satisfy~\eqref{eq:cond} with number of measurements $m$ on the order
of $s\log (d/s)$ (see e.g.~\cite{BDDW07:Johnson-Lindenstrauss}).  It
has also been shown~\cite{MPJ06:Uniform} that if the rows of $A$ are
independent (scaled) copies of an isotropic $\psi_2$ vector, then $A$
also satisfies~\eqref{eq:cond}.  Recall that an isotropic $\psi_2$
vector $a$ is one that satisfies for all $v$,
$$
\mathbb{E} |\< a, v \> |^2 = \|v\|^2 \quad\text{and}\quad \inf \{t : \mathbb{E}\exp(\< a, v\> ^2 / t^2) \leq 2\} \leq \alpha\|v\|_2,
$$
for some constant $\alpha$.  See~\cite{MPJ06:Uniform} for further
details.  Finally, it is clear that if $A$ is any of the above random
matrices then for any fixed unitary matrix $U$, the matrix $AU$ will
also satisfy the condition.

The D-RIP can also be analyzed via the Johnson-Lindenstrauss lemma (see e.g.~\cite{HV09:johnson,AC09:fast}).  There are many results that show certain types of matrices satisfy this lemma, and these would then satisfy the D-RIP via~\eqref{eq:cond}.  Subsequent to our submission of this manuscript, Ward and Krahmer showed that randomizing the column signs of any matrix that satisfies the standard RIP yields a matrix which satisfies the Johnson-Lindenstrauss lemma~\cite{KW10:NewAnd}.  Therefore, nearly all random matrix constructions which satisfy standard RIP compressed sensing requirements will also satisfy the D-RIP.  A particularly important consequence is that because the randomly subsampled Fourier matrix is known to satisfy the RIP, this matrix along with a random sign matrix will thus satisfy D-RIP.  This gives a fast transform which satisfies the D-RIP.  See Section~\ref{sec:Fast} for more discussion.

We are now prepared to state our main result.

\begin{theorem}\label{thm:main}
Let $D$ be an arbitrary tight frame and let $A$ be a measurement matrix satisfying D-RIP with $\delta_{2s} < 0.08$.  Then the solution $\hat{f}$ to $(P_1)$ satisfies
$$
\|\hat{f} - f\|_2 \leq C_0\varepsilon + C_1\frac{\|D^*f - (D^*f)_s\|_1}{\sqrt{s}},
$$
where the constants $C_0$ and $C_1$ may only depend on $\delta_{2s}$.
\end{theorem}

{\em Remarks.} We actually prove that the theorem holds under the
weaker condition $\delta_{7s} \leq 0.6$, however we have not tried to
optimize the dependence on the values of the restricted isometry
constants; refinements analagous to those in the compressed sensing
literature are likely to improve the condition.  Further, we note that
since Gaussian matrices with $m$ on the order of $s\log(d/s)$ obey the
D-RIP, Theorem~\ref{thm:Gauss} is a special case of
Theorem~\ref{thm:main}.

\subsection{Organization}
The rest of the paper is organized as follows.  In
Section~\ref{sec:proof} we prove our main result,
Theorem~\ref{thm:main}.  Section~\ref{sec:nums} contains numerical
studies highlighting the impact of our main result on some of the
applications previously mentioned.  In Section~\ref{sec:disc} we
discuss further the implications of our result along with its
advantages and challenges.  We compare it to other methods proposed in
the literature and suggest an additional method to overcome some
impediments.

\section{Proof of Main Result}\label{sec:proof}
We now begin the proof of Theorem~\ref{thm:main}, which is inspired by
that in~\cite{CRT06:Stable}.  The new challenge here is that although
we can still take advantage of sparsity, the vector possessing the sparse
property is not being multiplied by something that satisfies the RIP, as in
the standard compressed sensing case.  Rather than bounding
the tail of $f-\hat{f}$ by its largest coefficients as in~\cite{CRT06:Stable}, we bound a portion of $D^*h$ in
an analagous way.  We then utilize the D-RIP and the fact that $D$ is a tight
frame to bound the error, $\|f-\hat{f}\|_2$.

Let $f$ and $\hat{f}$ be as in the
theorem, and let $T_0$ denote the set of the largest $s$ coefficients
of $D^*f$ in magnitude.  We will denote by $D_T$ the matrix $D$ restricted to the
columns indexed by $T$, and write $D_T^*$ to mean $(D_T)^*$.   With $h = f - \hat{f}$, our goal is to bound
the norm of $h$.  We will do this in a sequence of short lemmas.  The first
is a simple consequence of the fact that $\hat{f}$ is the minimizer.

\begin{lemma}[Cone Constraint]\label{cone}
The vector $D^*h$ obeys the following cone constraint,
\begin{equation*
\|D_{T_0^c}^*h\|_1 \leq 2\|D_{T_0^c}^*f\|_1 + \|D_{T_0}^*h\|_1.
\end{equation*}
\end{lemma}

\textit{Proof. }
Since both $f$ and $\hat{f}$ are feasible but
$\hat{f}$ is the minimizer, we must have $\|D^*\hat{f}\|_1 \leq
\|D^*f\|_1$.   We then
have that
\begin{align*}
\|D_{T_0}^*f\|_1 + \|D_{T_0^c}^*f\|_1 = \|D^*f\|_1
&\geq \|D^*\hat{f}\|_1 \\
&= \|D^*f - D^*h\|_1 \\
&\geq \|D_{T_0}^*f\|_1 - \|D_{T_0}^*h\|_1 - \|D_{T_0^c}^*f\|_1 + \|D_{T_0^c}^*h\|_1.
\end{align*}
This implies the desired cone constraint.
\hfill {\vrule height6pt width6pt depth0pt}\medskip

We next divide the coordinates $T_0^c$ into sets of size $M$ (to be
chosen later) in order of decreasing magnitude of $D_{T_0^c}^*h$.
Call these sets $T_1, T_2, \ldots$, and for simplicity of notation set
$T_{01} = T_0 \cup T_1$.  We then bound the tail of $D^*h$.

\begin{lemma}[Bounding the tail]\label{thirteen}
Setting $\rho = s/M$ and $\eta = 2\|D_{T_0^c}^*f\|_1 / \sqrt{s}$, we have the following bound,
\begin{equation*
\sum_{j\geq 2} \|D^*_{T_j}h\|_2 \leq \sqrt{\rho}(\|D_{T_0}^*h\|_2 + \eta).
\end{equation*}
\end{lemma}
\textit{Proof. }
 By construction of the sets $T_j$, we have that each
coefficient of $D_{T_{j+1}}^*h$, written $|D_{T_{j+1}}^*h|_{(k)}$, is
at most the average of those on $T_j$:
$$
|D_{T_{j+1}}^*h|_{(k)} \leq \|D_{T_j}^*h\|_1 / M.
$$
Squaring these terms and summing yields
$$
\|D_{T_{j+1}}^*h\|_2^2 \leq \|D_{T_j}^*h\|_1^2 / M.
$$
This along with the cone constraint in Lemma~\ref{cone} gives
\[
\sum_{j\geq 2} \|D^*_{T_j}h\|_2 \leq \sum_{j\geq 1} \|D_{T_j}^*h\|_1 /
\sqrt{M} = \|D_{T_0^c}^*h\|_1 / \sqrt{M}.
\]
With $\rho = s/M$ and $\eta = 2\|D_{T_0^c}^*f\|_1 / \sqrt{s}$,
it follows from Lemma~\ref{cone} and the
Cauchy-Schwarz inequality that
\begin{equation*
\sum_{j\geq 2} \|D^*_{T_j}h\|_2 \leq \sqrt{\rho}(\|D_{T_0}^*h\|_2 + \eta),
\end{equation*}
as desired.
\hfill {\vrule height6pt width6pt depth0pt}\medskip

Next we observe that by the feasibility of $\hat{f}$, $Ah$ must be
small.
\begin{lemma}[Tube Constraint]\label{Ah}
The vector $Ah$ satisfies the following,
$$
\|Ah\|_2 \leq 2\varepsilon.
$$
\end{lemma}
\textit{Proof. }
Since $\hat{f}$ is feasible, we have
\begin{equation*}
\|Ah\|_2 = \|Af - A\hat{f}\|_2 \leq \|Af - y\|_2 + \|A\hat{f} - y\|_2 \leq \varepsilon + \varepsilon = 2\varepsilon.
\end{equation*}
\hfill {\vrule height6pt width6pt depth0pt}\medskip

We will now need the following result which utilizes the fact that $D$ satisfies the D-RIP.

\begin{lemma}[Consequence of D-RIP]\label{sixteen}
The following inequality holds,
\begin{equation*
\sqrt{1 - \delta_{s+M}}\|D_{T_{01}}D_{T_{01}}^*h\|_2 - \sqrt{\rho(1 + \delta_M)}(\|h\|_2 + \eta) \leq 2\varepsilon.
\end{equation*}
\end{lemma}
\textit{Proof. }
Since $D$ is a tight frame, $DD^*$ is the identity, and this along with
the D-RIP and~Lemma~\ref{thirteen} then imply the following:
\begin{align*}
2\varepsilon &\geq \|Ah\|_2 = \|ADD^*h\|_2  \\
&\geq \|AD_{T_{01}}D_{T_{01}}^*h\|_2 - \sum_{j\geq 2} \|AD_{T_{j}}D_{T_{j}}^*h\|_2 \\
&\geq \sqrt{1 - \delta_{s+M}}\|D_{T_{01}}D_{T_{01}}^*h\|_2 - \sqrt{1 + \delta_M}\sum_{j\geq 2} \|D_{T_{j}}D_{T_{j}}^*h\|_2 \\
&\geq \sqrt{1 - \delta_{s+M}}\|D_{T_{01}}D_{T_{01}}^*h\|_2 - \sqrt{\rho(1 + \delta_M)}(\|D_{T_0}^*h\|_2 + \eta).
\end{align*}

Since we also have $\|D_{T_0}^*h\|_2 \leq \|h\|_2$, this yields the desired result.
\hfill {\vrule height6pt width6pt depth0pt}\medskip

We now translate these bounds to the bound of the actual error, $\|h\|_2$.

\begin{lemma}[Bounding the error]\label{side}
The error vector $h$ has norm that satisfies,
$$
\|h\|_2^2 \leq \|h\|_2\|D_{T_{01}}D_{T_{01}}^*h \|_2 + \rho(\|D_{T_0}^*h\|_2 + \eta)^2,
$$
\end{lemma}

\textit{Proof. }
Since $D^*$ is an isometry, we have
\begin{align*
\|h\|_2^2 &= \|D^*h\|_2^2 = \|D_{T_{01}}^*h\|_2^2 + \|D_{T_{01}^c}^*h\|_2^2\notag \\
&= \langle h, D_{T_{01}}D_{T_{01}}^*h \rangle + \|D_{T_{01}^c}^*h\|_2^2\notag \\
&\leq \|h\|_2\|D_{T_{01}}D_{T_{01}}^*h \|_2 + \|D_{T_{01}^c}^*h\|_2^2\notag \\
&\leq \|h\|_2\|D_{T_{01}}D_{T_{01}}^*h \|_2 + \rho(\|D_{T_0}^*h\|_2 + \eta)^2,
\end{align*}
where the last inequality follows from Lemma~\ref{thirteen}.
\hfill {\vrule height6pt width6pt depth0pt}\medskip

We next
observe an elementary fact that will be useful. The proof is omitted.
\begin{lemma}\label{lem}
  For any values $u$, $v$ and $c>0$, we have
$$
uv \leq \frac{cu^2}{2} + \frac{v^2}{2c}.
$$
\end{lemma}

We may now conclude the proof of Theorem~\ref{thm:main}. First we employ Lemma~\ref{lem} twice to the inequality given
by Lemma~\ref{side}
(with constants $c_1$, $c_2$ to be chosen later) and the bound
$\|D_{T_0}^*h\|_2 \leq \|h\|_2$ to get

\begin{align*}
  \|h\|_2^2 &\leq \frac{c_1\|h\|_2^2}{2} + \frac{\|D_{T_{01}}D_{T_{01}}^*h \|_2^2}{2c_1} + \rho(\|h\|_2 + \eta)^2\\
  & = \frac{c_1\|h\|_2^2}{2} + \frac{\|D_{T_{01}}D_{T_{01}}^*h \|_2^2}{2c_1} + \rho\|h\|_2^2 + 2\rho\eta\|h\|_2 + \rho\eta^2\\
  & \leq \frac{c_1\|h\|_2^2}{2} + \frac{\|D_{T_{01}}D_{T_{01}}^*h
    \|_2^2}{2c_1} + \rho\|h\|_2^2 + 2\rho\Big(\frac{c_2\|h\|_2^2}{2} +
  \frac{\eta^2}{2c_2} \Big) + \rho\eta^2.
\end{align*}
Simplifying, this yields
$$
\Big(1 - \frac{c_1}{2} - \rho - \rho c_2 \Big)\|h\|_2^2 \leq \frac{1}{2c_1}\|D_{T_{01}}D_{T_{01}}^*h \|_2^2 + \Big(\frac{\rho}{c_2} + \rho\Big)\eta^2.
$$

Using the fact that $\sqrt{u^2 + v^2} \leq u + v$ for $u,v\geq0$, we can further simply to get our desired lower bound,
\begin{equation}\label{lb}
\|D_{T_{01}}D_{T_{01}}^*h \|_2 \geq \|h\|_2\sqrt{2c_1\Big(1 - (\frac{c_1}{2}+\rho+\rho c_2) \Big)} - \eta\sqrt{2c_1\big(\frac{\rho}{c_2}+\rho\big)}.
\end{equation}

Combining~\eqref{lb} with Lemma~\ref{sixteen} implies
$$
2\varepsilon \geq K_1\|h\|_2 - K_2\eta,
$$
where
\begin{align*}
K_1 &= \sqrt{2c_1(1-\delta_{s+M})\Big(1 - (\frac{c_1}{2}+\rho+\rho c_2) \Big)} - \sqrt{\rho(1+\delta_{M})}, \quad \text{and}\\
K_2 &= \sqrt{2c_1(1-\delta_{s+M})(\rho/c_2 + \rho )} - \sqrt{\rho(1+\delta_{M})}.
\end{align*}

It only remains to choose the parameters $c_1$, $c_2$, and $M$ so that $K_1$ is positive.  We choose $c_1$=1, $M=6s$, and take $c_2$ arbitrarily small so that $K_1$ is positive when $\delta_{7s} \leq 0.6$.  Tighter restrictions on $\delta_{7s}$ will of course force the constants in the error bound to be smaller.  For example, if we set $c_1 = 1/2$, $c_2 = 1/10$, and choose $M=6s$, we have that whenever $\delta_{7s} \leq 1/2$ that $(P_1)$ reconstructs $\hat{f}$ satisfying
$$
\|f-\hat{f}\|_2 \leq 62\varepsilon + 30\frac{\|D_{T_0^c}^*f\|_1}{\sqrt{s}}.
$$
Note that if $\delta_{7s}$ is even a little smaller, say $\delta_{7s} \leq 1/4$, the constants in the theorem are just $C_1 = 10.3$ and $C_2 = 7.33$.  Note further that by Corollary 3.4 of~\cite{NT08:Cosamp}, $\delta_{7s} \leq 0.6$ is satisfied whenever $\delta_{2s} \leq 0.08$.  This completes the proof.
\hfill {\vrule height6pt width6pt depth0pt}\medskip

\section{Numerical Results}\label{sec:nums}

We now present some numerical experiments illustrating the
effectiveness of recovery via $\ell_1$-analysis and also compare the
method to other alternatives.  Our results confirm that in practice,
$\ell_1$-analysis reconstructs signals represented in truly redundant
dictionaries, and that this recovery is robust with respect to noise.

In these experiments, we test the performance on a simulated
real-world signal from the field of radar detection. The test input is
a superposition of six radar pulses. Each pulse has a duration of
about 200 ns, and each pulse envelope is trapezoidal, with a 20 ns
rise and fall time, see Figure~\ref{fig:input}. For each pulse, the
carrier frequency is chosen uniformly at random from the range 50 MHz
to 2.5 GHz. The Nyquist interval for such signals is thus 0.2
ns. Lastly, the arrival times are distributed at random in a time
interval ranging from $t = 0 \text{ s}$ to $t \approx 1.64 \,
\mu\text{s}$; that is, the time interval under study contains $n =
8192$ Nyquist intervals. We acquire this signal by taking $400$
measurements only, so that the sensing matrix $A$ is a Gaussian matrix
with $400$ rows. The dictionary $D$ is a Gabor dictionary with
Gaussian windows, oversampled by a factor of about $60$ so that $d \approx
60 \times 8,192 = 491,520$. The main comment about this setup is that
the signal of interest is not exactly sparse in $D$ since each pulse
envelope is not Gaussian (the columns of $D$ are pulses with Gaussian
shapes) and since both the frequencies and arrival times are sampled
from a continuous grid (and thus do not match those in the
dictionary).
\begin{figure}[ht]
\begin{center}
  \includegraphics[scale=0.4]{Input}
\end{center}
\caption{Input signal in the time and frequency domains.  The signal
  of interest is a superposition of 6 radar pulses, each of which
  being about 200 ns long, and with frequency carriers distributed
  between 50 MHz and 2.5 GHz (top plot).  As can be seen, three of
  these pulses overlap in the time domain.}
\label{fig:input}
\end{figure}

Figure~\ref{fig:timefreq} shows the recovery (without noise) by
$\ell_1$-analysis in both the time and frequency domains.  In the time
domain we see (in red) that the difference between the actual signal
and the recovered signal is small, as we do in the frequencey domain
as well.  These pulses together with the carrier frequencies are well
recovered from a very small set of measurements.
\begin{figure}[ht]
\begin{center}
  \includegraphics[scale=0.4]{L1Analysis}
\end{center}
\caption{Recovery in both the time (below) and frequency (above)
  domains by $\ell_1$-analysis. Blue denotes the recovered signal,
  green the actual signal, and red the difference between the two.}
\label{fig:timefreq}
\end{figure}

In practice, reweighting the $\ell_1$ norm often offers superior
results. We use the \textit{reweighted} $\ell_1$-analysis method,
which solves several sequential weighted $\ell_1$-minimization
problems, each using weights computed from the solution of the
previous problem~\cite{CWB08:Reweighted}.  This procedure has been
observed to be very effective in reducing the number of measurements
needed for recovery, and outperforms standard $\ell_1$-minimization in
many situations (see
e.g.~\cite{CWB08:Reweighted},~\cite{KXAH:2step},~\cite{N09:rw}).
Figure~\ref{fig:timefreq2} shows reconstruction results after just one
reweighting iteration; the root-mean squared error (RMSE) is
significantly reduced, by a factor between 3 and 4.
\begin{figure}[ht]
  \begin{center}
    \includegraphics[scale=0.4]{L1RW_OneIter}
\end{center}
\caption{Recovery in both the time (below) and frequency (above)
  domains by $\ell_1$-analysis after one reweighted iteration. Blue
  denotes the recovered signal, green the actual signal, and red the
  difference between the two. The RMSE is less than a third of that in
  Figure~\ref{fig:timefreq2}}\label{fig:timefreq2}
\end{figure}

Because $D$ is massively overcomplete, the Gram matrix $D^*D$ is not
diagonal. Figure~\ref{fig:notONB} depicts part of the Gram matrix
$D^*D$ for this dictionary, and shows that this matrix is ``thick''
off of the diagonal.  We can observe visually that the dictionary $D$
is not an orthogonal system or even a matrix with low coherence, and
that columns of this dictionary are indeed highly correlated. Having
said this, the second plot in Figure~\ref{fig:notONB} shows the rapid
decay of the sequence $D^*f$ where $f$ is the signal in
Figure~\ref{fig:input}.
\begin{figure}[ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.25]{DtDmatrix_zoom_log}  &
\includegraphics[scale=0.25]{CoefficientDecay}
\end{tabular}
\end{center}
\caption{Portion of the matrix $D^*D$, in $\log$-scale (left). Sorted
  analysis coefficients (in absolute value) of the signal from
  Figure~\ref{fig:input} (right).}\label{fig:notONB}
\end{figure}

Our next simulation studies the robustness of $\ell_1$-analysis with
respect to noise in the measurements $y = Af + z$, where $z$ is a
white noise sequence with standard deviation
$\sigma$. Figure~\ref{fig:noise} shows the recovery error as a
function of the noise level.  As expected, the relationship is linear,
and this simulation shows that the constants in Theorem~\ref{thm:main}
seem to be quite small.  This plot also shows the recovery error with
respect to noise using a \textit{reweighted} $\ell_1$-analysis;
reweighting also improves performance of $\ell_1$-analysis, as is seen
in Figure~\ref{fig:noise}.
\begin{figure}[ht]
  \begin{center}
 \includegraphics[scale=0.3]{ErrorVsSigma}
\end{center}
\caption{Relative recovery error of $\ell_1$-analysis as a function of
  the (normalized) noise level, averaged over $5$ trials. The solid
  line denotes standard $\ell_1$-analysis, and the dashed line denotes
  $\ell_1$-analysis with $3$ reweighted iterations. The $x$-axis is
  the relative noise level $\sqrt{m} \sigma/\|Af\|_2$ while the
  $y$-axis is the relative error $\|\hat f -
  f\|_2/\|f\|_2$.}\label{fig:noise}
\end{figure}

An alternative to $\ell_1$-analysis is $\ell_1$-synthesis, which we
discuss in Section~\ref{sec:alts}; $\ell_1$-synthesis minimizes in the
coefficient domain, so its solution is a vector $\hat{x}$, and we set
$\hat{f} = D\hat{x}$.  Our next simulation confirms that although we
cannot recover the coefficient vector $x$, we can still recover the
signal of interest.  Figure~\ref{fig:SynAn} shows the largest $200$
coefficients of the coefficient vector $x$, and those of $D^*f$ as
well as $D^*\hat{f}$ for both $\ell_1$-analysis and
$\ell_1$-synthesis.  The plot also shows that the recovery of
$\ell_1$-analysis with reweighting outperforms both standard
$\ell_1$-analysis and $\ell_1$-synthesis.
\begin{figure}[ht]
  \begin{center}
 \includegraphics[scale=0.3]{SynAnDecay}
\end{center}
\caption{The largest $200$ coefficients of the coefficient vector
  $D^*f$ (blue), $D^*\hat{f}$ from $\ell_1$-analysis (dashed green),
  $D^*\hat{f}$ from $\ell_1$-analysis with $3$ reweighting iterations
  (dashed red), $\hat x$ from $\ell_1$-synthesis (cyan), and
  $D^*\hat{f}$ from $\ell_1$-synthesis (magenta).}\label{fig:SynAn}
\end{figure}

Our final simulation compares recovery error on a compressible signal
(in the time domain) for the $\ell_1$-analysis, reweighted
$\ell_1$-analysis, and $\ell_1$-synthesis methods.  We see in
Figure~\ref{fig:comp} that the $\ell_1$-analysis and
$\ell_1$-synthesis methods both provide very good results, and that
reweighted $\ell_1$-analysis provides even better recovery error.
\begin{figure}[ht]
\begin{center}
  \includegraphics[scale=0.4]{SynAn}
\end{center}
\caption{Recovery (without noise) of a compressible signal in the time
  domain.  Blue denotes the actual signal, while green, red, and cyan
  denote the recovery error from $\ell_1$-analysis, reweighted
  $\ell_1$-analysis ($2$ iterations), and $\ell_1$-synthesis,
  respectively. The legend shows the relative error $\|\hat f -
  f\|_2/\|f\|_2$ of the three methods.}\label{fig:comp}
\end{figure}

\section{Discussion}\label{sec:disc}

Theorem~\ref{thm:main} shows that $\ell_1$-analysis is accurate when
the coefficients of $D^*f$ are sparse or decay rapidly.  As discussed
above, this occurs in many important applications.  However, if it is
not the case, then the theorem does not guarantee good recovery.  As
previously mentioned, this may occur when the dictionary $D$ is a
concatenation of two (even orthonormal) bases.  For example, a signal
$f$ may be decomposed as $f = f_1 + f_2$ where $f_1$ is sparse in the
basis $D_1$ and $f_2$ is sparse in a different basis, $D_2$.  One can
consider the case where these bases are the coordinate and Fourier
bases, or the curvelet and wavelet bases, for example.  In these
cases, $D^*f$ is likely to decay slowly since the component that is
sparse in one basis is not at all sparse in the
other~\cite{DK10:Microlocal}.  This suggests that $\ell_1$-analysis
may then not be the right algorithm for reconstruction in such
situations.

\subsection{Alternatives}\label{sec:alts}
Even though $\ell_1$-analysis may not work well in this type of setup,
one should still be able to take advantage of the sparsity in the
problem.  We therefore suggest a modification of $\ell_1$-analysis
which we call \textit{Split-analysis}.  As the name suggests, this
problem splits up the signal into the components we expect to be
sparse:
\begin{equation*}
  (\hat{f}_1, \hat{f}_2) = \argmin_{\tilde{f}_1, \tilde{f}_2} \|D_1^*\tilde{f}_1\|_1 + \|D_2^*\tilde{f}_2\|_1 \quad\text{subject to}\quad \|A(\tilde{f}_1 + \tilde{f}_2) - y\|_2 \leq \varepsilon.
\end{equation*}
The reconstructed signal would then be $\hat{f} = \hat{f}_1 +
\hat{f}_2$.  Some applications of this problem in the area of image restoration have been
studied in~\cite{COS09:Split}.  Since this is an analagous problem to $\ell_1$-analysis,
one would hope to have a result for Split-analysis similar to
Theorem~\ref{thm:main}.

An alternative way to exploit the sparsity in $f = f_1 + f_2$ is to
observe that there may still exist a (nearly) sparse expansion $f = Dx
= D_1 x_1 + D_2 x_2$.  Thus one may ask that if the coefficient vector
$x$ is assumed sparse, why not just minimize in this domain?  This
reasoning leads to an additional approach, called $\ell_1$-Synthesis
or Basis Pursuit (see also the discussion in~\cite{EMR07:Analysis}):
\begin{equation}\tag{$\ell_1$-synthesis}
  \hat{x} = \argmin_{\tilde{x}} \|\tilde x\|_1 \quad\text{subject to}\quad \|AD\tilde x - y \|_2 \leq \varepsilon.
\end{equation}
The reconstructed signal is then $\hat{f} = D\hat{x}$.  Empirical studies also show that $\ell_1$-synthesis often provides good recovery, however, it is fundamentally distinct from $\ell_1$-analysis.  The geometry of the two problems is analyzed in~\cite{EMR07:Analysis}, and there it is shown that because these geometrical structures exhibit substantially different properties, there is a large gap between the two formulations.  This theoretical gap is also demonstrated by numerical simulations in~\cite{EMR07:Analysis}, which show that the two methods perform very differently on large families of signals.

\subsection{Fast Transforms}\label{sec:Fast}
For practical reasons, it is clearly advantageous to be able to use
measurement matrices $A$ which allow for easy storage and fast
multiplication.  The partial DFT for example, exploits the Fast
Fourier Transform (FFT) which allows the sampling matrix to be
applied to a $n$-dimensional vector in $\mathrm{O}(n \log n)$ time, and
requires only $\mathrm{O}(m\log n)$ storage.  Since the partial
DFT has been proven to satisfy the RIP~\cite{CT04:Near-Optimal} (see also~\cite{RV08:sparse}), it is a
fast measurement matrix that can be used in many standard compressed
sensing techniques.

One of course hopes that fast measurement
matrices can be used in the case of redundant and coherent
dictionaries as well.  As mentioned, the result of Krahmer and Ward implies
that any matrix which satisfies the RIP will satisfy the D-RIP when multiplied
by a random sign matrix~\cite{KW10:NewAnd}.  Therefore, the $m\times n$ subsampled Fourier matrix with
$m = \mathrm{O}(s\log^4n)$ along
with the sign matrix will satisfy the D-RIP and provides the fast multiply.

A result at the origin of this notion was proved by Ailon and Liberty, also
after our initial submission of this paper~\cite{AL10:AnAlmost}.  Recall that in light of~\eqref{eq:cond}, we desire
a fast transform that satisfies the Johnson-Lindenstrauss lemma in the
following sense.  For a set $Q$ of $N$ vectors in $n$-dimensional space,
we would like a fast transform $A$ that maps this set into a space of dimension $\mathrm{O}(\log N)$
(possibly with other factors logarithmic in $n$) such that
\begin{equation}\label{JLwant}
(1-\delta)\|v\|_2^2 \leq \|Av\|_2^2 \leq (1+\delta)\|v\|_2^2 \quad\text{ for all $v\in Q$.}
\end{equation}
Note that the dimension $\mathrm{O}(\log N)$ will of course also depend on the constant $\delta$.  Due to standard covering
arguments (see e.g.~\cite{RSV08:Redundant,BDDW07:Johnson-Lindenstrauss}), this would yield
a $m \times n$ fast transform with optimal
number of measurements, $m = \mathrm{O}(s\log n)$, obeying the D-RIP.

 Ailon and Liberty show that the subsampled Fourier matrix multiplied by a random sign matrix does exactly this~\cite{AL10:AnAlmost}.
Thus in other words,
for a fixed $m$, this $m \times n$ construction satisfies the D-RIP up to sparsity level $s = \mathrm{O}(m/\log^4 n)$.
The cost of a matrix--vector
multiply is of course dominated by that of the FFT, $\mathrm{O}(n\log n)$.
Its storage requirements are also $\mathrm{O}(m\log n)$.
Their results can also be generalized to other transforms with the same type of fast multiply.

These results yield a transform with a fast multiply which satisfies the D-RIP.  The number of measurements
and the multiply and storage costs of the matrix are of the same magnitude as those that satisfy the RIP.
The D-RIP is, therefore, satisfied by matrices with the same benefits as those in standard
compressed sensing.  This shows that compressed sensing with redundant and coherent
dictionaries is viable with completely the same advantages as in the standard setting.

\subsection*{Acknowledgements}

This work is partially supported by the ONR grants N00014-10-1-0599
and N00014-08-1-0749, the Waterman Award from NSF, and the NSF DMS
EMSW21-VIGRE grant. EJC would like to thank Stephen Becker for
valuable help with the simulations.

\small

redpajama_set_name: RedPajamaArXiv


text:
La catedral de San Pedro o simplemente catedral de Lancaster  (en inglés: Cathedral Church of St Peter) es un edificio religioso que se localiza en Lancaster, Lancashire, Inglaterra en el Reino Unido. Fue solo una parroquia católica hasta 1924, cuando fue elevada a la categoría de una catedral. Comenzó como una iglesia de la misión en 1798, y la actual iglesia fue construida en un sitio diferente entre 1857 y 1859. Fue diseñada por EG Paley en el estilo neogótico. En 1901 un baptisterio fue añadido por Austin y Paley, y el extremo este fue reordenado en [1995]] por Francis Roberts. La catedral aparte de la misa mantiene otros servicios, conciertos y otros eventos, y está abierta a los visitantes. El edificio está registrado en la Lista del Patrimonio Nacional de Inglaterra como un edificio de Grado II.

Véase también
 Iglesia católica
 Cristianismo

Referencias

Lancaster
Lancaster
Lancaster
Lancashire
Edificios listados de Grado II de Inglaterra
Arquitectura de 1859
Reino Unido en 1859
Lancaster

redpajama_set_name: RedPajamaWikipedia


Oeuvres de Moliere. Drawn by Boucher. Engraved by Boilvin, Courtry, Gaucherel, Greux, Lerat, Martinez, Milius, Massard, Mongin, and Rajon.
Issued Paris, 1876 by Alphonse Lemerre. Paper covered boards, publishers price list on back cover.
Excellent lot of thirty-five antique 19th century finely engraved prints, printed on hand-made paper. All loose as issued within covers.
All show light signs of age, but remain mostly very clean and fresh. Attractive examples.Portfolio issued with string-ties and cloth spine, prints loosely set in, the spine & ties now lacking.
Minor light toning and as typical for printed images of this vintage, perhaps small minor paper flaws, but to our eyes and for most people, all remain entirely acceptable and nice looking examples.
Sheets measure c. 8" x 5 1/2".


As is the case each year, we aim to have seasonal services on by the May Long Weekend, but we cannot guarantee that weather will cooperate. We'll refund the applicable services portion of your fees for services not available during your stay, due to a later than expected Spring thaw.
With temperatures below freezing overnight, it's the time of the season for turning off the water until spring.
The water services on campsites and within the loops will be shut off at 2:00PM on Monday September 24.
Potable water remains available 24 hours a day at the lift station and Campers' Centre, all year long.
Appropriate discounts will be applied, at the frontdesk, for guests that paid for specific sites with seasonal services, which are not available during their stay.
Below is our planned service availability dates for services and loop availability. These dates are "weather permitting" and appropriate refunds or additional charges will be applied at check-out if service availability differs from any seasonal price paid. These dates adjust annually to accommodate weekends better, so check back each season for any changes.


text:
Q: Get the core name in query response Solr I am using a multicore search in solr, the results are not blended, In order to classify my results I need to get the corename from the response, consider the below response
{
  "responseHeader": {
    "status": 0,
    "QTime": 0,
    "params": {
      "fl": "title,core_name",
      "indent": "true",
      "q": "*:*",
      "_": "1383405269434",
      "wt": "json",
      "rows": "2"
    }
  },
  "response": {
    "numFound": 926,
    "start": 0,
    "docs": [
      {
        "title": "Main Page"
      },
      {
        "title": "Albert Einstein"
      }
    ]
  }
}

If I give a parameter say, core_name it should return the core name in every result entry, that is,
{
  "responseHeader": {
    "status": 0,
    "QTime": 0,
    "params": {
      "fl": "title,core_name",
      "indent": "true",
      "q": "*:*",
      "_": "1383405269434",
      "wt": "json",
      "rows": "2"
    }
  },
  "response": {
    "numFound": 926,
    "start": 0,
    "docs": [
      {
        "title": "Main Page"
        "core_name": "collection1"
      },
      {
        "title": "Albert Einstein"
        "core_name": "collection1"
      }
    ]
  }
}

Is there any solr variable to get the core name?

A: As you are using shards, you can add the DocTransformer [shard] as query field within the fl parameter. The brackets around shard are mandatory and will also occur in the response. This DocTransformer only works, when the shards parameter is present.
To do so you would need to alter your query like this
localhost:8983/solr/collection1/select?shards=localhost:8983/solr/collection3,localhost:8983/solr/collection2,localhost:8983/solr/collection1&q=*:*&wt=json&fl=title,[shard]
This will get you a response like this
{
  "responseHeader": {
    "status": 0,
    "QTime": 0,
    "params": {
      "fl": "title,[shard]",
      "indent": "true",
      "q": "*:*",
      "_": "1383405269434",
      "wt": "json",
      "rows": "2"
    }
  },
  "response": {
    "numFound": 926,
    "start": 0,
    "docs": [
      {
        "title": "Main Page"
        "[shard]": "localhost:8983/solr/collection1"
      },
      {
        "title": "Albert Einstein"
        "[shard]": "localhost:8983/solr/collection1"
      }
    ]
  }
}

Further reading

*

*http://wiki.apache.org/solr/CommonQueryParameters#Transformers

*http://wiki.apache.org/solr/DocTransformers


CAPE TOWN | The South African government attempted to assure the public Wednesday that the country is safe, as United States government facilities remained shut for a second day amid an unspecified security threat.
The U.S. Embassy in Pretoria, consulates in Johannesburg, Cape Town and Durban and aid and development offices were shut down Tuesday after the State Department said it had received a "credible" threat against the embassy.
The U.S. facilities are expected to remain shut until at least Friday.
State Department spokesman Ian Kelly said the closure was a precaution.
ARUSHA, Tanzania | A former Rwandan minister is accused of diverting funds from international donors to murderous militias during the 1994 genocide that killed more than 500,000 people, a prosecutor at a U.N.-backed court said Wednesday.
Augustin Ngirabatware was charged with genocide and crimes against humanity in October and pleaded not guilty.
But prosecutor Wallace Kapaya said he has proof that Mr. Ngirabatware stole money donated by the World Bank and International Monetary Fund as well as cash from several countries, including the United States.
The money was intended to go to the Rwandan Ministry for Planning, which Mr. Ngirabatware used to head, but Mr. Kapaya said that instead it was used to buy weapons and transport for the extremist Hutu militia known as the Interahamwe.
Mr. Ngirabatware is the son-in-law of Felician Kabuga, Rwanda's most wanted genocide suspect. The U.S. government has offered a $5 million bounty for Mr. Kabuga's capture.
DAR ES SALAAM | Tanzania's high court on Wednesday sentenced three men to hang for the murder of a 13-year-old albino boy, killed for his body parts in the country's northwest, media and a rights group said.
At least 53 albinos have been killed since 2007 in the East African nation and their body parts sold for use in witchcraft, especially in the remote northwest regions of Shinyanga and Mwanza where superstition is rife.
Witch doctors say the body parts of albinos - who lack pigment in their skin, eyes and hair - bring luck in love, life and business.
One of the accused was found with two legs belonging to the 13-year-old, Matatizo Dunia, local radio said.
Authorities have arrested more than 90 people, including four police officers, for their involvement in the killings or trade of albino body parts.
The killings have sullied Tanzania's reputation for relative calm in the region, and drawn condemnation from the United Nations and European Union.
In neighboring Burundi, at least 11 albinos have been killed since last year. So far 13 people have been convicted, including one who received a life sentence. Authorities in Burundi say people in Tanzania ordered the killings.


Apple and Sonos today announced that Apple Music, Apple's streaming music service, will be available on Sonos, allowing those who own Sonos speakers to use them with the Apple Music service for the first time since it launched in June. A public beta of the new feature is also available leading up to the official launch.
Sonos users will be able to access Apple Music on their Sonos speakers, streaming songs from For You, New, Radio, and My Music. The beta will allow users to access the Apple Music features through their dedicated Sonos apps on iPhone, iPad, iPod touch, Mac, and PC, with full support set to roll out in 2016.
Apple has launched a new (PRODUCT)RED leather case for the iPhone 6S and iPhone 6S Plus. Originally, (PRODUCT)RED was only offered in silicone cases made for the newest iPhones. The case is available for purchase right now on Apple's website.
The case was originally spotted at an Apple retail store by French tech site iGeneration [Google Translate], who took pictures of the case and uploaded them to their site.
The iPhone 6S (PRODUCT)RED leather case costs as much as the other leather cases do, at $45, while it's the same story with the 6S Plus's case at $49.
A portion of sales from Apple's entire (PRODUCT)RED lineup goes to the Global Fund to help fund AIDS programs in Africa. Most recently, the company introduced (PRODUCT)RED accessories for the Apple Watch, debuting a new collection and band in the bright red color option during its September "Hey Siri" media event.
Apple has seeded OS X El Capitan 10.11.2 beta 5 to all developers and public beta testers via the Mac App Store. If you're already running the OS X 10.11.2 beta, launch the Mac App Store and open the Updates tab to download and install the new release, then restart your Mac.
While previous versions haven't included any notable user facing changes, Apple has asked that testers focus on graphics, Mail, Wi-Fi, Calendar, USB, Notes, Photos, Spotlight, and networking in previous versions. When used with iOS 9.2 beta, the El Capitan beta supports Wi-Fi Calling with AT&T's NumberSync feature as well.
We'll update this post with any changes found to the OS.
According to a pretty sketchy report by Economic Daily News, the MacBook Air will be getting a major upgrade in mid 2016. And yes, you read correctly.
The report says the new computer will be thinner and lighter than the current model, with substantial internet component changes. The report also states a rough timeframe of mid 2016 for the computer's arrival into the world.
The MacBook Air has not seen a significant hardware change since 2010, so a major update is certainly due. Most people have been scratching their heads about the fate of the MacBook Air, what with the new 12 inch MacBook being thinner and lighter than the Air, having less speed and performance, and costing a lot more. This thin-and-light notebook has put Apple's previous thinnest MacBook in quite an awkward position, but it may seem to people that we're about to find out what exactly might happen to it.
It's unclear whether we'll see the 11 inch MacBook Air take on the rumored changes, with Apple, according to the report, considering a '13 inch and 15 inch MacBook Air' lineup, bringing the Air into the 15 inch screen size form factor for the first time and possibly knocking out the 11 inch size once and for all.
Although the report claims the next-generation MacBook Air will include new batteries, cooling modules, chassis and more, Apple is expected to stick to the same supply chain manufacturers, namely Quanta.
Finally! A reasonable price for one of Apple's most expensive products!
Of course, it's not just the Sport models that have been discounted. The stainless steel Apple Watch has been discounted too by $50, making a 38mm Apple Watch with Sport band just $499 and the 42mm model just $50 more ($549).
You can visit Best Buy's website to check out all of their Apple Watch models on sale.
Apple has added a new finish to the Beats SoLo2 Wireless and urBeats Headphones lineup which, unsurprisingly, is Rose Gold. The headphones now come in Space Gray, Silver, Gold, and Rose Gold to match the color lineup of the iPhone and select Apple Watch models.
The Solo2 Wireless on-ear headphones, $299.95, were released in November 2014 as the first Beats product since Apple's acquisition of the company officially closed. Beats launched new Solo2 Wireless headphones in Gold, Silver and Space Gray, colors that match several Apple products, last April.
Beats urBeats In-Ear Headphones, $99.95, were first released long before Apple acquired the company in 2014.
Apple has seeded two new betas to developers today; the first being iOS 9.2 beta 4. This build of the next update to iOS 9 brings changes to the Safari View Controller, which is what allows a pop-up Safari window to be used in third-party apps. In iOS 9.2, Safari View Controller supports third-party Action Extensions, letting apps like 1Password to be accessed in the Safari view of other apps like Tweetbot.
The fourth beta of iOS 9.2, build 13C5075, is available as an over-the-air update and through the Apple Developer Center. Public beta testers can also download the update via over the air.
The second beta Apple seeded today was the third build of tvOS 9.1. We don't yet know exactly what is included in the new version of tvOS, as most changes thus far appear to be under-the-hood updates. Apple's release notes for the update include references to networking and scrolling tweaks, and it's likely tvOS 9.1 will include fixes for bugs and issues that have been discovered in the software since the Apple TV's official launch.
The third tvOS beta is available as an over-the-air update to those who have installed the first two betas. Installing an initial tvOS beta is a more involved process than installing a beta update for iOS or OS X. The update requires the Apple TV to be connected to a computer with a USB-C to USB-A cable, with the software downloaded and installed via iTunes.
The new charging dock will be available for in-store pickup on November 20th according to the accessories's webpage. The fastest Apple can get this dock to you via online order is the same date.


Ticking away inside the 44-millimetre stainless-steel case is the robust 79320-calibre chronograph movement with a 44-hour power reserve. It facilitates stop times and aggregate timing up to 12 hours and is a reliable companion even on unusually long dives. Thanks to the more discreet colours and shapes, the new Aquatimer Chronograph is as much at home on a dive as at a business meeting.


It was very disappointing to hear today that Marks and Spencer's is to close its Commercial Road store. The council had no advance warning of this decision and so as things stand details on what will happen next are scant, although press reports suggest a food-only store will open in Ocean Park.
Debenham's has announced that it will close ten stores across the country and with the chain having branches in both Commercial Road and Palmerston Road, there is a real chance that one or the other will be on the hit list.
The fragility of Commercial Road as a retail centre is a direct result of the failure to get the Northern Quarter scheme built before the shift to online and out-of-town shopping became as embedded as it now has. High Street retailing in the 2020s and beyond will have to be tied in with leisure activities and in future shop spaces will increasingly be shop window units, with customer orders being placed online. This requires a new approach.
The council's ability to influence events is limited: all the units in the city centre are privately owned and so decisions on rents do not rest with the council; the same applies to the two main car parks at the Cascades and the Tricorn. As ever, the council just doesn't have millions of pounds to spend. It can however try and secure funding for the new city centre road, which is the key to getting development moving.
It's taken three years to do it, but enough money has now been saved up (an extra £11m) to be able to make a serious bid to government for the remaining cash. If approved, a mixed use regeneration scheme becomes viable. That will still take five years to plan and build though; what happens in the meantime is an open question.