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In this Paper,we will discuss the concept of Azure AD pass-through authentication which will enable the organization to keep the users' password in on-premises and redirect all cloud authentications to be against local active directory.
Read about your paper.. Seems interesting I will follow it and read it. thanks for sharing it here.


Alice Fricke serves as Senior Client Portfolio Manager on the Global Equity team, explaining the team's investment philosophy, process and performance. Most of Alice's experience is in the Asian and European equity markets. She began her career as a banker resident in Asia with Deutsche Bank and then, after returning to the U.S., covered the equity markets as a stockbroker and analyst at various firms before joining the Firm.


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HomeLatest NewsTrump defends decision to withdraw troops from Syria…and Resignations continue..
Trump defends decision to withdraw troops from Syria…and Resignations continue..
December 22, 2018 Ahmed Rashed Latest News, Syria Latest News
Donald Trump has been defending his decision to withdraw all US troops from Syria, and end an air campaign against ISIS.
After historic victories against ISIS, it's time to bring our great young people home! pic.twitter.com/xoNjFzQFTp
US Department of Defense resignations continue..
US envoy for anti-ISIS coalition McGurk resigns: reports
Brett McGurk, the top US envoy in ISIL fight, quits over Trump's decision to withdraw troops from Syria, US media report
Brett McGurk, the US envoy to the global coalition fighting ISIL, resigned this week over his disagreement with President Donald Trump's surprise decision to pull US troops out of Syria, US media reported on Saturday, citing unnamed sources.
Both the Associated Press and CBS news reported the departure, saying McGurk submitted his resignation letter to Secretary of State Mike Pompeo on Friday, a day after Defense Secretary Jim Mattis said he was leaving his post.
In the letter, which was described to the Associated Press by an official familiar with its contents, McGurk said fighters with the Islamic State of Iraq and the Levant (ISIL, also known as ISIS) were on the run, but not yet defeated as Trump had said.
He added that the premature pull-out of American forces from Syria would create the conditions that gave rise to ISIL.
McGurk, whose resignation is effective on December 31, was planning to leave the job in mid-February after a US-hosted meeting of foreign ministers from the coalition countries, but he felt he could continue no longer after Trump's decision to withdraw from Syria and Mattis's resignation, according to the official speaking to the Associated Press.
One day before, Jim Mattis, Defense Secretary, Resigns in Rebuke of Trump's Worldview
Defense Secretary Jim Mattis, whose experience and stability were widely seen as a balance to an unpredictable president, resigned Thursday in protest of President Trump's decision to withdraw American forces from Syria and his rejection of international alliances.
Mattis had repeatedly told friends and aides over recent months that he viewed his responsibility to protect the United States' 1.3 million active-duty troops as worth the concessions necessary as defense secretary to a mercurial president. But on Thursday, in an extraordinary rebuke of the president, he decided that Trump's decision to withdraw roughly 2,000 American troops from Syria was a step too far.
Donal Trump
US defense secretary
Sudan Protests Over Rising Prices Continue for Fourth Day
Al-Sisi's Regime is Planning Mass Arrests on January Revolution Anniversary
Israel Latest News
US Cuts Aid to 'Palestine-Israel' Conflict Resolution Programs
September 18, 2018 Ahmed Rashed Israel Latest News, Palestine Latest News
Money earmarked for Palestinians in West Bank and Gaza will instead be redirected to programs within Israel's borders. The US has cut additional aid to the Palestinians for programmes supporting conflict resolution with Israelis, adding [ More … ]
Saudi Crown Prince: The Kingdom has received around 2.5 million Syrian citizens
September 20, 2016 MEO Staff Saudi Arabia News
Speaking at the 71st annual session of the UN General Assembly in New York on Monday, he Saudi crown prince, Muhammad Bin Naif, said the kingdom ranked as the third country worldwide in terms of [ More … ]
Iraqi lawmakers criticize Trump visit as blow to Iraqi sovereignty
December 27, 2018 Ahmed Rashed Iraq Latest News
Iraqi political and militia leaders condemned U.S. President Donald Trump's surprise visit to U.S. troops in Iraq on Wednesday as a violation of Iraq's sovereignty, and lawmakers said a meeting between Trump and Iraqi Prime [ More … ]

redpajama_set_name: RedPajamaCommonCrawl


In the, accept Payments section, open the, select an Additional Payment Method dropdown from the. But we must test. I prefer Coinbase and have used for client stores. Once..
Auch hier gibt es weitere Börsen, welche einen Einstieg in Kryptowährungen bieten. Zurück zur Frage, ob sich die Investition lohnt. Kraken hat es extra für Sie gedreht, da die..
So if you wish to upgrade from.13.0.13.1, you would just need to use the same command, but updating to something like.14.x would require switching branches. Reboot to have it..
already mit was kann man leicht geld verdienen started social trading? Commitments of Traders Strategy learn more about Forex trading strategy that is based on following the cftc CoT reports. So whats the closest thing we can get our hands on to see the state of the market and how the big players are moving their money? Commitments of Traders An Ultimate Guide for a Forex How to use it in trading. Read more, trading Strategy: COT Report Analysis - Forexsites. Complete Trading for a Living, hadady,. Read more, how to use cftc reports in trading. Just like players in a team sport, each group has its unique characteristics and roles. Get more information from our, currency Strength Meter! The, commitments of Traders (CoT) widget offers a clear graphical representation of the weekly CoT reports published by the Commodity Futures Trading Commission (cftc).
Youre probably asking yourself, Why the heck do I need to use data from the FX futures market? The Visitor undertakes to pay the service fee to the operator of the Website through PayPal. The operator of the Website may amend the content of these terms and conditions at its own discretion. The Visitor undertakes to hold the operator of the Website harmless from any damage, cost, expenses, financial obligations resulting from claims of third parties directly or indirectly against the operator of the Website arising from the use and/or application of the guidance and information available. I personally use the Commitments of Traders Report (COT) report in all my swing trading. Read more, using the Commitment of Traders Report From the cftc to Home / Forex / Follow The Smart Money In Forex Trading The 4 Best Economic Indicators.


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Your West Chesire
Romy's Rhino
The Lache Community Centre Rhino has had a face lift thanks to budding young designer Romy Redston.
'African Rhino' was originally part of the Chester Rhino Mania public art trail in 2010. One of 62 decorated sculptures that was sold at auction after the art trail ended and spent seven years at Liverpool John Lennon Airport.
Last year Liverpool Airport kindly gifted the rhino to the Lache Community. Originally painted by Cheshire artist Mike Kilkelly as a tiger, the rhino has now had a total face lift following a competition with local school children.
Said Councillor Louise Gittins, Cabinet Member Communities and Wellbeing: "We thought this would be a great opportunity to give the rhino a new coat. We worked with the two local primary schools, Lache Primary and St Clare's Catholic Primary to hold a design competition with the children, the winning design being brought to life on the rhino by local artist, Lucy Elizabeth Jones , who was part of the original art team for Rhino Mania.
Mrs Marian Ryder, St Clare's Catholic Primary School said: "Pupils in St Clare's were delighted when they were invited to design a new skin for the 'community rhino'. There were lots of excellent entries and after viewing all the wonderful, artistic pieces, the judges decided that Romy's design was the one.
"Romy is in Year Three at St Clare's and loves being creative and artistic. She was very honoured and excited about being asked to cut the ribbon on the newly decorated rhino and is thrilled with the final outcome. There was a buzz around school on Friday when friends from Romy's class accompanied her to the ceremony. All the children said they love the 'new look' rhino."
Original rhino design
Lache Primary and St Clare's Catholic Primary
Romy and class
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"This is a massive setback for all of South Yorkshire. We had a real chance to claw away powers from Westminster and make decisions closer to home. We had the chance to bring jobs and investment to South Yorkshire but internal party politics has got in the way.
"This deal has my City's name on it but it was for the whole of South Yorkshire. A key question is why couldn't Sheffield Leadership keep all parties at the table.
"We are witnessing the results of Julie Dore's Donald Trump style diplomacy.
This failure of this deal has made us into a laughing stock. Cities like Manchester and Birmingham are now looking at taking even more power from Westminster but our leaders have pushed us to the back of the queue.


***Please click here to download the application form for children ages 18 months – age 3.
An interview with the school director is required, as is a drop-in day so that teachers may evaluate applicant. A placement exam may be administered to determine the student's level of ability in English. Entrance to grade level will be determined by the school director. Admission is rolling, however, preference may be given to siblings of students already enrolled with San Juan del Sur Day School and to families with a long-term commitment to San Juan del Sur.


Now that the previous exercise have given a basic overview and understanding of the Tracker, you will process the Events that were fired on your new Business Object Card in the latest iteration.
In order to process any Event as a Team User, you need to acquire an understanding of the cause of the Event. Of course, in any migration the possible causes for any given Event can be endless and it can be quite a job figuring out what is going on in order to identify the root problem. No tool - migFx included - can eliminate this complexity. But as you will see in this exercise, migFx does provide you with efficient means of analyzing and tracing Events and provides great assistance to help you out.
Select the Events tab, BankId All and Business Object Card to see the list of Events that were fired on the Card Business Object.
Now let's process the Events one by one.
First of all, the Event prefix tells you which migration step fired the Event.
The list contains only one reference. Following the hierarchy chain from left to right, it is apparent that the Event is fired form the Card Business Object. More specifically if - in the Source Map of the Business Object - the Source Object CardHolder (a child of Source Object Src.DebitCard) is not found.
Armed with this information, you can double-click the reference line to open the Card Business Object, and locate the Not Found Flag on the CardHolder Source Object.
Right there, you have the root cause of the Event: Apparently for some rows in the Src.DebitCard table in the Staging Database, there is no row in the Vw.Customer table corresponding to the CardHolder of the Src.DebitCard.
One of the Card Business Object instances that fired the Event E0011 is now displayed in detail. Note that the Card belongs to BankId 30. For the Event E0011 the Customer number of the missing Customer is displayed in the Event message.
Let's see if that Customer has been processed in the migration.
The instance of the Customer Business Object is now found and has been migrated successfully. But looking carefully it does actually belong to BankId 40. This seems to be the problem: The Card belongs to BankId 30 while the Customer belongs to BankId 40.
You can do the same analysis on other occurrences of Event E0011. They all seem to caused by the same problem.
At this point there not much more to be done. It seems to be a data problem in the Source System. What you can do is to delegate this Event to the Partition Users and ask them to look into the problem in the Source System.
Now return to the Events list. You will notice that in the list the Level has changed to Delegate and the State to Waiting, because the Event has been delegated to the Partition Users and you and your team are now waiting for them to resolve the Event. In addition, if you hover the mouse over the Comments column, you can see the your Comment. Of course, the Partition Users will also see your Comment in this way.
This Event has Severity Warning. So nothing is being rejected, but still somebody should verify that the resulting names are acceptable, and - if not - edit the names in the Source System for the next iteration.
So again here, the correct action in Tracker is to delegate the Event to the Partition Users and provide guidance in a Comment. Click the Event message to go to the Event Editor.
Enter a suitable comment to explain the problem to the Partition users. For instance: Please verify that the edited names are acceptable. If not, the correct action is to edit the names in the Source System to avoid this warning. Mark the Event Accepted when done.
The severity is Information, so apparently the Event has no consequences and serves only to tell that something was done. Anyway, let's trace the Event and figure out why it is fired.
From the reference chain it is clear that the Event is fired when the value of the Interface Field CardInactiveDate of Business Object Card is calculated.
In the Flag Handling of the value assignment dialog, it is evident that the Event E0017 is fired when the Card is blocked. But why is the default date returned?
While the two first CardStatus children has a CardStatusEndDate, the third one does not. In addition it has a CardStatus Blocked.
We are getting close to an explanation. If you look back the exercise 2.5 - Map Business Object Card in the section Map Target Interface Field CardInactiveDate, you will see that this is a known difference between the Source and Target Systems.
So - again - let's just delegate this Event to the Partition Users to ensure they are aware of the Event and accept it for the final sign-off. In the Event List, click the Event Message to go to the Event Editor.
Add a suitable Comment, for instance: Please accept this event. If a card is blocked, the Target System requires a date. The Source System contains no such date. It has been agreed with the Business, that in case of a blocked card, we use first bank day after migration as default.
The I prefix of the Event signals that the it was fired by the Target Engine during the Import step. The Severity is Reject Root, so the Event is serious as it causes the entire Card Business Object instance to be rejected.
The Event is actually fired from 4 different locations, but only the last one concerns the Card Business Object, so let's concentrate on that one. Again looking at the reference chain, it seems the Event is fired on the Not found flag on the CardHolder relationship on the Card Business Object.
The Depending on items box shows that the Card Business Object Instance depends on a Customer and an Account Business Object instance, both of which has been rejected.
So the Customer is rejected by the Event E0006 which is fired if an AddressLine on the Customer Business Object instance is too long. In a later exercise, we will have a look at this Event. For now, we are satisfied that the cause of the Event I0011 on the Card Business Object Instance is that it is depending on a Customer Business Object instance that in turn is rejected by Event E0006.
From the intermediate Event List you can drill down to the other Card Business instance rejected by I0011 and see that this too is caused by the E0006 Event on a Customer Business Object Instance.
The Item Detail of both rejected Card Business Object instances reveals that it is not only the dependency on a rejected Customer, that is a problem. Both Cards also each depend on an Account Business Object instance, that has been rejected.
You can use the Tracker to trace these rejected Accounts to figure out that these Accounts are actually also rejected because they depend on rejected Customers. Furthermore that again it is the Event E0006 that is the root cause.
So by now it is clear that we must expect that resolving the Event E0006 will let the Customer Business Object instances succeed and in turn both the affected Account and Card Business Object instances should succeed and the Event I0011 will be resolved.
The Event I0023 closely resembles the previous Event I0011. In the Studio Target Map you can verify the the Event is fired in a relationship from Card to Account is not found.
The Account is rejected by Event E0010. You can analyze other instances of the Card Business Object rejected by Event I0023 and see that they all are caused by the rejection of an Account by E0010.
The Severity of this Event is Reject Root, so all the the Cards firing the Event are rejected by the Target Engine. Considering that this is over 5.000 Cards - around 1 third of the total count - it is rather serious. Let's figure out what the problem is.
Judging by the texts on the two Flags, they both signal that the CardStatus is unknown, the only difference between the two is whether the card itself is active or expired.
The Specification for the MappingRule GetCardStatus clearly states that the CardStatus is expected to be found in the ValueSet CardStatus - otherwise one of the 2 Flags is raised depending on the CardInActiveDate received as a parameter to the rule.
This needs some further analysis. Maybe the CardStatus Blocked is just missing in the ValueSet by mistake - or it is missing by design because this CardStatus does not exist in the Target System.
Another consideration is that the Severity Reject Root seems very harsh. After all, the Event is fired when importing the child Business Object CardStatus. Maybe is is not necessary to reject the entire Card just because one of the CardStatus children has an unknown CardStatus?
Add a comment, for instance: CardStatus Blocked is unknown. Further analysis is needed.
Working on these Events should have given you a good sense of how the Tracker and the Studio in combination gives a robust foundation for the analysis of a vide range of migration problems.
In any data migration there will initially be a myriad of problems of daunting variety. Indeed, that migFx is a generic solution that is deployed by different installations in any industry vertical only increases this fact exponentially.
In the next, final section of exercises you will use these capabilities combined with the extraordinary speed of iteration in migFx to fix some the events analysed in this exercise.


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New Restaurants in Bend, Oregon (2022)
The dining scene in Bend, Oregon is constantly evolving, as new restaurants, cafes and pubs open up to feed the community and serve as lively gathering spaces for breakfast, lunch or dinner out on the town. New restaurants are often under development and construction to keep up with Bend's growing population and visitor base, helping the region to maintain a vibrant foodie scene with dishes of various cuisines and establishments with a variety of concepts. In the mood for something new? Try one of these new Bend, Oregon restaurants. Check restaurant websites for the most up-to-date hours of operation.
Diners can enjoy a variety of options at The Grove, a market hall food court concept that opened in Bend's NorthWest Crossing in 2021. At The Grove, visitors will find Sebastian's Seafood Cafe, offering up fresh fish and chips and fish tacos adjacent to Sebastian's Seafood & Speciality Market. Also in The Grove is Sunny's Italian Joint, offering pizza, salads and other Italian-inspired eats. For authentic Thai street food, try ThaiPas, yet another option in The Grove. Outside of main entrees, The Grove is also home to Elly's Ice Cream, Thump Coffee and Waypoint—a cocktail bar by Bend Brewing Co.
BOSA Food & Drink
Soak up Bend's west side neighborhood vibes from a window seat at BOSA Food & Drink, located in the historic Bakery Building on lively Galveston Avenue. Enjoy regional Italian and French specialities, including housemade pasta and ingredients from both the Pacific Northwest and sources from Europe in a casual fine dining atmosphere. The space underwent a full remodel before opening in 2021, and now boasts a full bar and open kitchen. BOSA Food & Drink is open nightly at 1005 NW Galveston Ave. Reservations are recommended.
Walt Reilly's Eatery & Entertainment
Putt putt, pints and tasty dishes converge at Walt Reilly's Eatery & Entertainment, a family-friendly destination on Bend's west side. Enjoy smash burgers, savory appetizers and salads before or after a game of mini golf or series of virtual swings on over 190 top golf courses from around the world. Kids can enjoy their own menu, with options including a mac and cheese or a mini smash burger. Walt Reilly's Eatery & Entertainment is open daily at 225 SW Century Drive.
Nome Italiano
Joining the west side Bend dining scene in early 2022, Nome Italiano is the latest concept from Sixtop, a group that also brought the city Bos Taurus, Hablo Tacos and Miyagi Ramen. Enjoy oysters, pizza and entrees including steaks, chicken and seafood in an upscale, fine dining environment. Wines from throughout Italy are also on the menu. Nome Italiano is open Wednesday to Sunday at 1465 SW Knoll Ave.
The Cellar Underground Pub
Redmond's Porter Brewing Co. expanded to Bend in 2021, though passersby may not have noticed the new establishment, which is entirely underground in downtown Bend. The Cellar Underground Pub serves up English-style cask-conditioned ales that are hand pumped and served at cellar temperature. The family behind the brewery also runs Vi's Pies, a from-scratch bakery serving up Scottish meat pies, curry pies, mac and cheese pies and chicken pot pies, along with sausages and sausage bites and pretzels. The Cellar Underground Pub is open daily at 206 NW Oregon Ave., Suite 2.
In the Oxford Hotel downtown, Roam opened in 2021 and serves up breakfast, lunch, brunch and dinner. Enjoy charcuterie, small plates, pasta, seafood and meats, along with an eclectic list of wines, cocktails and beers. Get comfy and enjoy a seasonally changing menu. Roam is open daily at 10 NW Minnesota Ave., Suite 100.
Thai Hot Pot and Noodle House
The owners of the popular Wild Rose thai spot in downtown Bend have done it again with another tasty restaurant—this time a more upscale eatery featuring traditional Thai hot pot, along with noodles and street food dishes. Sen Thai Hot Pot and Noodle House overlooks the Deschutes River in a brand new building. Sen Thai Hot Pot and Noodle House is open Tuesdays through Sundays at 69 NW Newport Ave.
Anita's Kitchen and Gourmet Indian Food
Enjoy delicious Indian takeaway food from Anita's Kitchen, which opened off Third Street in southeast Bend in 2020. This food truck uses authentic Indian spices and herbs, whipped up into dishes including samosas, tandori chicken and chana bhatura. Sip on masala chai, fresh lemonade or lassi, a salty or sweet Indian yogurt drink. Anita's Kitchen is open Wednesdays to Sundays at 420 SE Third St.
Justy's
This bar, grill and sandbox (beach volleyball, anyone?) is situated on a bluff in southwest Bend, offering great mountain views. Justy's offers elevated pub food, including appetizers, salads, burgers and sandwiches, along with cocktails and a selection of local beers and ciders in a family-friendly environment. Reserve a court, work up an appetite playing volleyball and then relax with a filling meal and beverage. Justy's is open daily at 1787 SW Chandler Ave.
Meadowlark and Boxwood Kitchen
One of the newest restaurants in Bend, Meadowlark, opened in 2022, building off the tasty success of its sister restaurant, Boxwood Kitchen in the Old Mill District, which opened in early 2019. Meadowlark serves dinner, as well as weekend brunch, alongside handcrafted cocktails or a featured mocktail of the day. Meadowlark is open daily for dinner and also on weekends for brunch at 19570 Amber Meadow Drive. Meadowlark sits on the south end of Bend and is a great example of breaking away from the core of town to find a delicious restaurant option. For more great eats, try casting an even wider net to enjoy food from restaurants throughout Central Oregon.
Hot Springs in Central Oregon Near Bend and Beyond
Oregon is known for being an outdoor paradise; where else can travelers discover misty forests, snowy mountain peaks and vast desert expanses in one day? From the Cascade Mountains to the Painted Hills, volcanic activity has helped shape the many different landscapes of Oregon. Along with mountains and valleys, Oregon's volcanic past helped create a host of hot springs scattered across the state. These hot springs are natural, hot pools of water—natural hot tubs. Some are more developed and have lodging nearby, while others feel completely remote and undiscovered. So whether it's a soak under the stars in the high desert or a steamy pool in the forest, there's a hot spring calling.
Make the Most of Your Bend and Central Oregon Ski Vacation
Dreaming of skiing in Bend, Oregon? The Central Oregon region is an incredible destination for a ski getaway. While the average snowfall in Bend is a little more than thirty inches, Central Oregon ski resorts boast annual numbers of more than 450 inches. From Bend, visitors can drive to Mt. Bachelor resort in just 30 minutes or the Hoodoo Ski Area in less than an hour. Outside of the resorts, there are boundless backcountry trails for cross-country skiing to glide into pristine and untouched wilderness. With tons of snow, a plethora of winter activities in Central Oregon, and seemingly endless miles of terrain to explore, Bend and Central Oregon make the perfect ski getaway. Read on to discover how to make the most of your ski-cation in this winter wonderland.
The Ultimate Guide to Pumpkin Patches in Terrebonne, Oregon
Nothing signals the coming of fall quite like spending the day at a pumpkin patch. It's tough to not feel the cozy spirit of the season when you're solving a festive corn maze while bundled up and sipping a hot apple cider. Pumpkin patches in Central Oregon benefit from our desert climate and still get plenty of sunshine, even in late October. While smaller family farms all around the region will sometimes host fall events, the two most popular pumpkin patches both reside in Terrebonne: the Smith Rock Pumpkin Patch and the DD Ranch Pumpkin Patch.
Top 5 Tours to Take in Bend, Oregon
Bend is a nature-lover's paradise blended with a hop-head's dream; nowhere else will you find this much craft beer and this many outdoor activities. The city's been growing quickly, reaching 100,000 residents in 2020, and with the growth comes plentiful opportunities for adventure. From walking the very halls that brew your favorite beer to any number of adrenaline-inducing sports, we have you covered for the five best types of guided tours in Bend, Oregon.
Great Getaway: Sisters, Oregon
When pulling into Sisters from the West you're greeted by a main-street cowboy town filled with local shops, western storefronts and lots of charm. But as you reach the far end of that main street (it's actually U.S. Hwy. 20,) something different emerges. There's some upscale lodging, a brewpub, movie theater, and spa that gives Sisters a whole new attitude.
Casual Family Weekend in Sunriver
Imagine this: you're riding your bike through a rolling meadow, the Cascade mountain range in the background, and the only sound is from the whir of a prop plane taking off against a crisp blue sky. Up ahead, your children pull carrots from their pockets and present them to a gathering of friendly horses in an adjacent pasture.


With Lake San Marcos offering a variety of attached and detached homes and neighborhoods, it can be a little overwhelming! We'd love to schedule a lake tour with you to drive the community and share everything it has to offer. We can share the benefits of each neighborhood and the amenities offered to its owners. If available we can show homes on the market to get a better feeling of the layout and exterior space. Whether you live local or out of state if you're interested in finding out more about this hidden gem, contact us today.


It's a busy time of year for fans of technology, as the IFA trade show is officially underway in Berlin, Germany. Earlier today a new line of Acer Windows 10 PC's was announced at IFA, and now Lenovo has also lifted veils on its new line of Windows 10 devices. Overall, three new Lenovo Windows 10 devices were announced at IFA: the Yoga Book, Yoga 910 2-in-1 and the Miix 510 detachable. We've given the devices a look and have included the details of each new device for you below.
At 1.52 pounds and at 9.6mm thin when closed (tapering to 4.05mm at its slimmest edge,) Lenovo is calling the Yoga Book the world's thinnest and lightest 2-in-1 tablet. The tablet is available in an Android version for the starting price of $499 starting in October 2016. The Yoga Book with Windows, meanwhile is $549 and available starting in October 2016.
Powered by a powerful Intel Atom X5 processor and 4GB of RAM.
8500 mAh battery with 15 hours of battery life.
Instant Halo keyboard: A full touch screen backlit keyboard (no physical keys) that weaves software and hardware into one fluid interface. The touch screen is made with glass that was meticulously chosen to give a rough, matte feel and finish, along with anti-glare coating to ensure the best possible touch-typing experience. Haptic technology is also included and gives the user touch feedback to guide typing through built-in prediction and artificial learning software.
Real Pen Accessory: Roughly the size of a conventional ink pen, the real-pen accessory is powered by Wacom feel IT technologies to work with the state-of-the-art electro-magnetic resonance (EMR) film housed within the multi-use keyboard, which enables this real-time digitization. The pen holds real ink tips onto a piece of paper or notepad covering the multi-use keyboard panel, or as a stylus when applied straight onto the panel. It has 2,048 pressure levels and 100-degree angle detection and never has to be charged, as the tips can be replaced.
Smaller custom made three-axis Hinge constucted from over 130 mechanical pieces that offers up to four modes – Browse, Watch, Create and Type.
The Lenovo Yoga 910 is an ultra-thin convertible laptop which is just 14.3mm thin. Lenovo claims that this device will meet the demands of quickly evolving consumer lifestyles. Prices on The Yoga 910 start at $1299 and it will be available starting in October 2016.
A choice of 4K or 13.9 inch Full HD screen.
Near edgeless display (5mm bezel) that offers 10% more screen area, a 14% increase in pixel density compared to the previous model for sharper, clearer images and 100% sRGB color gamut coverage.
The Miix 510 detachable is designed for those use a device for both work and play. The detachable gives you a keyboard when you want it and the freedom to leave it behind when you're on the go. Available in October 2016 in a Platinum silver color, prices on The Miix 510 starts at $599.99. Key Specs and features on the device can be seen below.
Powered by the 6th gen Intel core I7 processor.
Front facing 2 MP camera.
So, there you have it. Those are the three new Lenovo Windows 10 devices were announced at IFA. What do you think of these devices? Will you pick any of them up once they are released later this Fall? Let us know what you think by dropping us a comment below!


Hoke used to be a linebacker at Ball State, and he was the head coach of their football team from 2003 to 2008. Hoke is currently looking for a job, as he has not found a home after being fired from Michigan at the end of 2014. Hoke was rumored to be interested in the East Carolina job, so this job would be about the same tier and would be a familiar place for him to coach.
Ball State is 33-29 in the past five seasons under Lembo. The Cardinals were 3-9 this season and 5-7 in 2014 after a 10-win year in 2013.
Bo was born in Atlanta, GA, and has always been a savvy sports follower. He attended Georgia Tech and has previously worked with ESPN TrueHoop, HawksHoop, and the Georgia Tech Newsroom. He covers national college football for FanBuzz.


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Bill G on Microsoft's biggest blunder... Was it Bing, Internet Explorer, Vista, the antitrust row?
Nope: It was not giving Android a run for its money...
By John Oates 24 Jun 2019 at 14:00
108 SHARE ▼
Bill Gates has said his biggest management miscalculation was failing to position Microsoft's Windows Phone as the primary rival mobile operating system to Apple's iOS.
Speaking at venture capital beanfeast Village Global last week, Gates said that although the company he founded had not missed mobile phones it had gone about things the wrong way. Although Gates stepped down as CEO in 2008, it is fair to say he retained an interest in the company he founded and remained chairman until 2014.
Gates, sporting a natty pink jumper, said: "The greatest mistake ever is whatever mismanagement I engaged in that caused Microsoft not to be what Android is – that is the standard phone platform – non-Apple phone platform. That was a natural thing for Microsoft to win."
Tens to be disappointed as Windows 10 Mobile death date set: Doomed phone OS won't see 2020
The grey-haired 63-year-old said the software world, especially for platforms, is all about winner-takes-all markets. He said a company with half as many apps or even 90 per cent as many apps was on its way to complete doom: "There's room for exactly one non-Apple operating system, and what's that worth? $400bn that would be transferred from company G to company M."
Gates said if Microsoft had got that right, it would be the leading company in the world, not just one of the leaders.
Microsoft's mobile strategy has been rebooted several times. It killed its Windows Mobile and brought in Windows Phone, along the way buying Nokia before finally leaping off the platform and throwing in the towel.
Steve Ballmer was not alone in doubting the chances of the iPhone making serious sales, especially in business, because it lacked a keyboard. But Microsoft moved to gesture and touch interfaces for mobile too slowly to succeed.
Microsoft is killing off support for Windows 10 Mobile on 10 December this year.
Gates has previously admitted not investing enough in Internet Explorer as his major error.
Google bought Android, based on the Linux kernel, to run on its phones in 2005. The core operating system is available via an open source licence, although Google also sells a set of proprietary software apps and services. This approach has made Android the world's largest mobile operating system. Although Microsoft has softened on its approach to open source software, such a strategy would have been a big shift for the company back in 2005.
Whether Gates is right that there is room for only one non-Apple mobile OS is also open to question – American pres Donald Trump's ban on Google and other American firms working with Huawei has seemingly forced the Chinese company, the world's second largest handset maker, to create its own platform. ®
Daddy, are we there yet? How Mrs Gates got Bill to drive the kids to school
Oh yeah? You're busier than Bill fricking Gates??
Bill Gates joined on stage by jar of poop as he confesses deep love for talking about toilets
Microsoft man opens three-day newfangled shithouse expo
Bill Gates declined offer to serve as Donald Trump's science advisor
Spontaneous offer made during chat about vaccines (which - phew! - Trump seemed to like)
Twenty years ago today: Windows 98 crashed live on stage with Bill Gates. Let's watch it again...
Video Relive that sphincter-loosening Blue Screen of Death
Cryptocurrencies kill people and may kill again, says Bill Gates
Q&A touches on tough topics like 'Tabs or Spaces?'
My name is Bill Gates and I am an Android user
Sir. Have you no, er, shame?
Bill Gates says he'd do CTRL-ALT-DEL with one key if given the chance to go back through time
Gives two-fingered salute to IBM designers for forcing us to use three-fingered salute
Bill Gates joins $170bn climate change investment club
Clean tech on the list


Victor Rice Mill is established in 2000 and has since been devoted to supplying top quality rice mill machine,rice huller machine,rice destoner,rice mill plant etc up to Certified European (CE) specifications. Owning a professional team with specialized theoretical and practical background, we can offer full scale service to our customers by developing projects from blueprint , customized solution, machinery production to installation and commission of complete paddy rice mill plant .
Why Choose Victor Rice Mill?
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text:
Karan Soni, on "How To Win An Oscar"
Loving all things Desi and American, seeing Karan on the promo spot piqued my curiosity so I found the video and Googled who was the actor.
The 89th Annual Academy Awards a.k.a. the Oscars are on this Sunday and I plan on putting the kid to bed early so I can not only watch the ceremony but listen to all the speeches and revel at the performances. It's always exciting to hear how those speeches pull from current events and might influence pop culture. I've long been a movie buff and with the Oscars being the ultimate "who's who" of actors and movies for the year, I make it a point to be at home, alone, to watch it every year. I've already told my husband that we're getting take out and not leaving the house, lest I miss even a moment!
In anticipation of the show, ABC has been running promo spots with host Jimmy Kimmel. One in particular, inspired this post. Jimmy Kimmel walks us thru his guide on how to win an Oscar while actor Karan Soni is acting the steps out in the background.
Loving all things Desi and American, seeing Karan on the promo spot piqued my curiosity so I found the video and Googled who was the actor. Turns out Karan has been quite busy in Hollywood and is going to be in many upcoming, high profile projects. He made his mark as the nerdy tech in a series of AT&T commercials then played small roles in Deadpool and Ghostbusters. Currently he's a series regular in Starz' Blunt Talk. He's also going to be in the next season of FX's breakout hit Fargo, due to be out later this year.
What a banner year with (my fav) Riz Ahmed and Dev Patel winning BAFTAs and hopefully Oscars too! I see good things for young Karan Soni as well.
Author radha824Posted on February 21, 2017 Categories Actor, Movies, TelevisionTags ABC, Academy Awards, actors, american, AT&T, BAFTA, Blunt Talk, ceremony, Commercial, Deadpool, desi, Dev Patel, Fargo, FX, Ghostbusters, Google, Hollywood, Jimm Kimmel, Karan Soni, movie buff, Movies, Oscar, pop culture, promo spots, Riz Ahmed, Starz, take outLeave a comment on Karan Soni, on "How To Win An Oscar"
Desi Theater in Chicago
Living in Chicago, I'm lucky enough that we have the talent, space, patrons, and interest to cultivate South Asian arts. One of the most special artistic outlets we have is Rasaka Theater Company, a Desi theater company that performs "South Asian stories" by "South Asian artists".
Rasaka Theater Company Logo
I first learned about Rasaka back in 2011 when I went to watch their production of Gruesome Playground Injuries, a play by Rajiv Joseph, starring Mouzam Makkar (who later went on to act on such TV shows as The Exorcist, Vampire Diaries, and upcoming Chicago Justice). I was serving on the associate board of a local non-profit and helped sell tickets to the play as a fundraiser. Ever the one to celebrate South Asians making the American experience their own, I was so excited to see a medium that isn't the most mainstream art, being showcased and accessible. Hence why I am choosing to write about them now.
Rasaka was founded in 2003 by a group of South Asian artists who were looking to put on a production and worked together to bring their gifts to the community. Check out their mission statement:
Rasaka Theatre Company is based in Chicago and is the Midwest's first South Asian American ensemble. Our goal is to increase diversity among artists and audience by engaging and illuminating the South Asian American experience. We are particularly dedicated to providing a platform for the artistic expression of South Asian artists.
Our vision encompasses the production of theatrical projects – newly created, adapted and already established. We aim to broaden, strengthen and enrich the experience of the theatre going audience thereby enhancing the local, regional, national and international reputation of Chicago theatre.
"Rasa" means the essence or inspiration of a work of art, and adding "ka" behind a word makes it "of" that, as in "of the essence of art." Rasaka's goal is to increase diversity among artists and audience by engaging and illuminating the South Asian American experience, with particular dedication to providing a platform for the artistic expression of South Asian artists.
If you're interested in supporting their mission, I encourage you to check out some of their productions. They are currently running a play based on Anton Chekhov's Uncle Vanya adapted by Lavina Jadhwani. It's playing at the Edge Theater in Uptown thru February 3rd and you can buy tickets here.
Vanya (or, 'That's Life!') Pictured from L to R: Puja Mohindra, Richard Costes, Rom Barkhordar, & Tiffany Renee Johnson
"Anton Chekhov's Uncle Vanya continues to be one of the greatest and continually relevant of plays – concerned with family relationships, broken hearts, and the rampant disappointment of midlife crisis. In this hilarious and sharp reinvention, there is great providence in the shattering of a teacup, time moves backwards, and our bored, sad Chekhovian friends have a tendency to lapse into singing American Indie folk tunes. What could be more Russian? Or American?"
Author radha824Posted on January 12, 2017 Categories Chicago, South Asian Arts, TheaterTags american, Anton Checkov, arts, associate board, Chicago, Chicago Justice, community, desi, Edge Theater, Exorcist, Gruesome Playground Injuries, Lavina Jadhwani, local, Mouzam Makkar, non-profit, Production, Puja Mohindra, Rajiv Joseph, Rasaka, Richard Costes, Rom Barkhordar, South Asian, Theater, theatre, Tiffany Renee Johnson, Uncle Vanya, Uptown, Vampire DiariesLeave a comment on Desi Theater in Chicago


text:
5-Piridoksat dioksigenaza (, 5-piridoksatna oksidaza) je enzim sa sistematskim imenom 5-piridoksat,-{NADPH}-:kiseonik oksidoreduktaza (deciklizacija). Ovaj enzim katalizuje sledeću hemijsku reakciju

 3-hidroksi-4-hidroksimetil-2-metilpiridin-5-karboksilat + -{NADPH + H}-+ + O2  2-(acetamidometilin)-3-(hidroksimetil)sukcinat + -{NADP}-+

Ovaj enzim je flavoprotein (FAD).

Reference

Literatura

Spoljašnje veze

ЕЦ 1.14.12
Флавопротеински ензими

redpajama_set_name: RedPajamaWikipedia


text:
Hundreds of Clarke County 10th-graders experience arts at UGA during trips
by mills-julia
What began as a vision for a Clarke Central High School social studies teacher became reality Nov. 10 and 11 as hundreds of Clarke County 10th-graders visited UGA to explore the arts.
The students, from Clarke Central one day and Cedar Shoals High School the next, came to campus as part of Experience UGA, a partnership between the university and the Clarke County School District. The program is designed to introduce students in grades pre-K-12 to the myriad academic opportunities at UGA to foster interest in higher education. The 10th-grade trips were scheduled to coincide with Spotlight on the Arts, an annual event designed to draw attention to UGA's visual literary and performing arts programs.
Clarke Central social studies teacher Ashley Goodrich, a UGA alumna who is getting her doctorate in educational theory and practice, came up with the idea for an arts-based field trip after several years of taking her students to the Georgia Museum of Art to study history through connections in art.
While on campus the 10th-graders visited the theatre and film studies department, the Hugh Hodgson School of Music, the museum, the Lamar Dodd School of Art, the dance department and the College of Education.
Students filled the seats in Ramsey Hall to hear the Red Priest, a world-renowned baroque quartet, play a collection of classical music.
In the theatre and film studies department, assistant professor Emily Sahakian's service-learning students introduced the high school students to theater as a socially engaged art form, not just for entertainment. They played games that helped students reflect on issues such as racism and bullying, building community and problem solving.
"Games are a wonderful way in theater to engage with a community and to create a dialogue with the audience," said Wyatt Geist, a master's degree candidate in performance. "It's also a great way to display ideas and emotions."
Experience UGA is led by the Office of Service-Learning, a unit that reports jointly to the UGA Vice President for Public Service and Outreach and the Vice President for Instruction. Launched last year, the partnership aims to bring every Clarke County student to campus for an annual field trip and opportunity to experience learning on a college campus, explore college options and interact with UGA students. More than 3,000 Clarke County students came to UGA as part of the program last year. UGA will host about 10,000 students this year and expects to bring all 13,000 pre-K-12 students to campus next year and in the following years. Twelve schools and colleges, as well as eight other university units, will host trips for Experience UGA this year.
During the November field trips, students were introduced to dance as a way to be part of a community. Under the direction of associate professor Rebecca Enghauser, they learned about studio work and careers in dance. From their chairs they practiced movements and worked with partners, which taught them about listening skills and leadership.
At the art school, the students learned about light graffiti—art created by moving light and a camera's long exposure—and 3-D printing. Jeremy Blair, an assistant professor of art education, helped them make short movies using green screen technology, props and costumes. They toured the building and saw a photography darkroom, a drawing and painting studio and a critique space.
At the museum, students discussed ways of responding to artwork through tours of the permanent collection and temporary exhibits led by students in education curator Carissa DiCindio's service-learning course "Engaging Art Museum Audiences as Student Docents." Current exhibitions on Italian fashion designer Emilio Pucci and the Elephant Six Collective provided an opportunity for students to also make sketches inspired by fashion and album covers on display.
For many, the Experience UGA arts field trip was a way to better understand art and its relationship to their lives, teachers said.
"A lot of our students have never had exposure to the arts before," said Mary Claire Nixon, a dance teacher at Cedar Shoals High School. "Giving them this opportunity is the most important thing; we hope to create lifelong lovers of art and learning."
Columns Inside UGA Office of Service-Learning
Fallows passes away at age 60
Professor's new textbook equips students to solve…
University Woman's Club plans Feb. 8 meeting
School of Law wins regional round of the National Moot…


text:
The Infinite Wonders of Creation est le  solo de Luca Turilli, célèbre guitariste du groupe de metal italien Rhapsody of Fire.

Cet album est remarquable par la prédominance des chœurs, donnant à l'ensemble un effet "mystique". Il est vendu avec deux disques, un premier qui est celui énuméré ci-dessous, et un autre qui est un aperçu de l'album Lost Horizons de Luca Turilli's Dreamquest

Titres
 Secrets of a Forgotten Age (3:12)
 Mother Nature (4:39)
 Angels of a Winter Dawn (4:16)
 Altitudes (4:37)
 The Miracle of Life (4:24)
 Silver Moon (5:37)
 Cosmic Revelation (4:48)
 Pyramids and Stargates (6:07)
 Mystic and Divine (4:21)
 The Infinite Wonders of Creation (8:40)
 Altitudes (Piano Version) (3:20)

Album musical sorti en 2006
Album de Luca Turilli


text:
Malmö FF competed in Division 2 Sydsvenska Serien for the 1926–27 season.

Players

Squad stats

|}

Club

Other information

References

Malmö FF seasons
Association football clubs 1926–27 season


With the release of 2016's Life With Lobsters, an album consisting of glitchy, summery indie dance pop, the Los Angeles-based collective NVDES fronted by founding member and primary songwriter Josh Ocean received over 10 million streams across all digital platforms, landed on Spotify's Global Viral Chart — and as a result, also received phrase from The Fader, Nylon and others. Building on a growing profile, the project's latest single "Turning Heads" off their forthcoming La NVDITÉ EP will further cement Ocean's reputation for crafting scuzzy, breakneck dance punk along the lines of Sound of Silver-era LCD Soundsystem, Radio 4 and others, complete with angular guitar chords, a propulsive bass line, boom-bap beats and a rousingly anthemic hook. And while clearly being summery, the track also manages to be incredibly dance floor friendly.


A pudgy, mulleted southern boy clad in black, Bill Hicks wasn't the most obvious guy to follow in the comedic footsteps of Jewish New Yorker Lenny Bruce, but that's exactly what he did.
Bill taketh on the Bible and Bush. He speaketh the truth. Not for the kiddies.


National human rights institutions (NHRIs) in the form of ombuds institutions and classic ombuds are examined critically to explore the extent to which they strengthen gender equality in public administration and their own operations, improve women's access to justice, and protect and promote women's rights. Ombuds effectiveness depends not only on their engagement with international and domestic human rights instruments, ranging from the UN Paris Principles on NHRIs to international women's rights law, but also on extralegal factors. Gender-based violence, gender discrimination and violations of women's economic, social and cultural rights are the most common issues addressed by all ombuds institutions in this study.
Professor Linda C. Reif, CN Professor of International Trade, University of Windsor (LLB), University of Cambridge (LLM), has been a member of the Faculty of Law, University of Alberta since 1987. Professor Reif has published extensively on national human rights institutions, ombuds institutions, children's rights institutions, international human rights law and international business and trade. Her work appears in Canadian, US, European and Asia-Pacific publications, including her book, The Ombudsman, Good Governance and the International Human Rights System (Brill/Martinus Nijhoff), with a second edition scheduled for publication in 2019, co-authorship of International Law: Chiefly as Interpreted and Applied in Canada (Emond Montgomery, 9th edition in press), and numerous book chapters and law review articles, including in the Harvard Human Rights Journal and Human Rights Law Review.


text:
\section{Introduction}
\noindent \textbf{Background.}
Neural network pruning has garnered significant recent interest \citep{lth, rethink_pruning, conv_filt_prune}, as obtaining high-performing sub-networks from larger, dense networks enables a reduction in the computational and memory overhead of neural network applications \citep{han2015deep, han2015learning}.
Many popular pruning techniques are based upon empirical heuristics that work well in practice \citep{stable_lth, thinet, he_pruning}.
Generally, these methodologies introduce some notion of ``importance'' for network parameters (or groups of parameters) and eliminate parameters with negligible importance.

The empirical success of pruning methodologies inspired the development of pruning algorithms with theoretical guarantees \citep{sipping_nn, liebenwein2019provable, mussay2019data, baykal2018data, whats_hidden}.
Among such work, greedy forward selection (GFS) \citep{provable_subnetworks, log_pruning}---inspired by the Frank-Wolfe algorithm \citep{frank1956algorithm}---differentiated itself as a methodology that performs well in practice and provides theoretical guarantees.
However, GFS is inefficient in comparison to popular pruning heuristics.

\noindent
\textbf{This Work.}
We leverage greedy selection \citep{cosamp, khanna2018iht} to develop a structured pruning algorithm that is provable, practical, and efficient.
This algorithm, called \textbf{i}terative, \textbf{Spa}rse \textbf{S}tructured \textbf{P}runing (i-SpaSP), iteratively estimates the most important parameter groups\footnote{``Parameter groups'' refers to the minimum structure used for pruning (e.g., neurons or filters).} within each network layer, then thresholds this set of parameter groups based on a pre-defined pruning ratio---a similar procedure to the CoSAMP algorithm for sparse signal recovery \citep{cosamp}.
Theoretically, we show for two and multi-layer networks that $i)$ the output residual between pruned and dense networks decays polynomially with respect to the size of the pruned network and $ii)$ the order of this polynomial increases as the dense network's hidden representations become more sparse.\footnote{To the best of the authors' knowledge, we are the first to provide theoretical analysis showing that the quality of pruning depends upon the sparsity of representations within the dense network.}
In experiments, we show that i-SpaSP is capable of discovering high-performing sub-networks across numerous different models (i.e., two-layer networks, ResNet34, MobileNetV2, and BERT) and datasets (i.e., MNIST, ImageNet, and XNLI).
i-SpaSP is simple to implement and significantly improves upon the runtime of GFS variants.

\section{Preliminaries} \label{S:prelim}
\noindent
\textbf{Notation.}
Vectors and scalars are denoted with lower-case letters.
Matrices and certain constants are denoted with upper-case letters.
Sets are denoted with upper-case, calligraphic letters (e.g., $\mathcal{G}$) with set complements $\mathcal{G}^c$.
We denote $[n] = \{0, 1, \dots, n\}$.
For $x \in \mathbb{R}^N$, $\|x\|_p$ is the $\ell_p$ vector norm.
$x_s$ is the $s$ largest-valued components of $x$, where $|x| \geq s$.
$\supp(x)$ returns the support of $x$.
For index set $\mathcal{G}$, $x|_{\mathcal{G}}$ is the vector with non-zeros at the indices in $\mathcal{G}$.
For $X \in \mathbb{R}^{m \times n}$, $\|X\|_F$ and $X^\top$ represent the Frobenius norm and transpose of $X$.
The $i$-th row and $j$-th column of $X$ are given by $X_{i, :}$ and $X_{:, j}$, respectively.
$\rsupp(X)$ returns the row support of $X$ (i.e., indices of non-zero rows).
For index set $\mathcal{G}$, $X_{\mathcal{G}, :}$ and $X_{:, \mathcal{G}}$ represent row and column sub-matrices, respectively, that contain rows or columns with indices in $\mathcal{G}$.
$\mu(X): \mathbb{R}^{m \times n} \longrightarrow \mathbb{R}^m$ sums over columns of a matrix (i.e., $\mu(X) = \sum_{i} X_{:, i}$), while $\texttt{vec}(X): \mathbb{R}^{m \times n} \longrightarrow \mathbb{R}^{mn}$ stacks columns of a matrix.
$X \in \mathbb{R}^{m \times n}$ is $p$-row-compressible with magnitude $R \in \mathbb{R}^{\geq 0}$ if $|\mu(X)|_{(i)} \leq \frac{R}{i^{\frac{1}{p}}},~~\forall i \in [m],$ where $|\cdot|_{(i)}$ denotes the $i$-th sorted vector component (in magnitude).
Lower $p$ values indicate a nearly row-sparse matrix and vice versa.

\medskip
\noindent
\textbf{Network Architecture.}
Our analysis primarily considers two-layer, feed forward networks\footnote{Though \eqref{eq:2layer_nn} has no bias, a bias term could be implicitly added as an extra element within the input and weight matrices.}:
\begin{align}
    f(X, \mathcal{W}) = W^{(1)}\cdot \sigma(W^{(0)} \cdot X) \label{eq:2layer_nn}
\end{align}
The network's input, hidden, and output dimensions are given by $d_{in}$, $d_{hid}$, and $d_{out}$.
$X \in \mathbb{R}^{d_{in} \times B}$ stores the full input dataset with $B$ examples.
$W^{(0)} \in \mathbb{R}^{d_{hid} \times d_{in}}$ and $W^{(1)} \in \mathbb{R}^{d_{out} \times d_{hid}}$ denote the network's weight matrices.
$\sigma(\cdot)$ denotes the ReLU activation function and $H = \sigma(W^{0}\cdot X)$ stores the network hidden representations across the dataset.
We also extend our analysis to multi-layer networks with similar structure; see Appendix \ref{A:multi_layer} for more details.

\section{Related Work} \label{S:related_work}
\noindent \textbf{Pruning.}
Neural network pruning strategies can be roughly separated into structured \citep{eie, conv_filt_prune, net_slimming, provable_subnetworks, log_pruning} and unstructured \citep{rigl, how_lth_wins, lth, deep_comp} variants.
Structured pruning, as considered in this work, prunes parameter groups instead of individual weights, allowing speedups to be achieved without sparse computation \citep{conv_filt_prune}.
Empirical heuristics for structured pruning include removing parameter groups with low $\ell_1$ norm \citep{conv_filt_prune, net_slimming}, measuring the gradient-based sensitivity of parameter groups \citep{sipping_nn, preserve_grad_flow, discrim_prune}, preserving network output \citep{he_pruning, thinet, nisp}, and more \citep{suau2020filter, chin2019legr, huang2018data, molchanov2016pruning}.
Pruning typically follows a three-step process of pre-training, pruning, and fine-tuning \citep{conv_filt_prune, rethink_pruning}, where pre-training is typically the most expensive component \citep{early_bert, early_bird}.

\medskip
\noindent \textbf{Provable Pruning.}
Empirical pruning research inspired the development of theoretical foundations for network pruning, including sensitivity-based analysis \citep{sipping_nn, liebenwein2019provable}, coreset methodologies \citep{mussay2019data, baykal2018data}, random network pruning analysis \citep{pruning_is_all_you_need, log_pruning_is_all_you_need, subset_lth, whats_hidden}, and generalization analysis \citep{zhang2021lottery}.
GFS---analyzed for two-layer \citep{provable_subnetworks, wolfe2021provably} and multi-layer \citep{log_pruning} networks---was one of the first pruning methodologies to provide both strong empirical performance and theoretical guarantees.

\medskip
\noindent \textbf{Greedy Selection.}
Greedy selection efficiently discovers approximate solutions to combinatorial optimization problems \citep{frank1956algorithm}.
Many algorithms and frameworks for greedy selection exist; e.g., Frank-Wolfe \citep{frank1956algorithm}, sub-modular optimization \citep{submod_opt}, CoSAMP \citep{cosamp}, and iterative hard thresholding \citep{khanna2018iht}.
Frank-Wolfe has been used within GFS and to train deep neural networks \citep{bach_fw, frankwolfe_nn}, thus forming a connection between greedy selection and deep learning.
Furthering this connection, we leverage CoSAMP \citep{cosamp} to formulate our proposed methodology.

\section{Methodology} \label{S:method}

i-SpaSP is formulated for two-layer networks in Algorithm \ref{A:cosamp_prune}, where the pruned model size $s$ and total iterations $T$ are fixed.
$\mathcal{S}$ stores active neurons in the pruned model, which is refined over iterations.

\begin{wrapfigure}{L}{0.49\textwidth}
\vspace{-0.3cm}
\begin{minipage}{0.49\textwidth}
\begin{algorithm}[H]
\SetAlgoLined
\textbf{Parameters:}~$T$,~$s$; $\mathcal{S}$ := $\emptyset$; $t$ := 0
\vspace{0.1cm} \\
\textcolor{myblue2}{\# compute hidden representation} \\
$H = \sigma(W^{(0)} \cdot X)$\\
$h = \mu(H)$\\\vspace{-.8cm}\\

\textcolor{myblue2}{\# compute dense network output} \\
$U = W^{(1)} \cdot H$ \\
$V = U$\\\vspace{-.3cm}\\
\While{t < T}{
$t = t + 1$\\\vspace{-.3cm}\\
\textcolor{myblue2}{\# \textbf{Step I:} Estimating Importance} \\
$Y = (W^{(1)})^\top \cdot V$\\
$y = \mu(Y)$\\
$\Omega = \supp(y_{2s})$ \\\vspace{-.3cm}\\
\textcolor{myblue2}{\# \textbf{Step II:} Merging and Pruning} \\
$\Omega^\star = \Omega \cup \mathcal{S}$ \\
$b = h|_{\Omega^\star}$ \\
$\mathcal{S} = \supp(b_s)$\\\vspace{-.3cm}\\
\textcolor{myblue2}{\# \textbf{Step III:} Computing New Residual} \\
$V = U - W^{(1)}_{:, \mathcal{S}} \cdot H_\mathcal{S, :}$
}
\textcolor{myblue2}{\# return pruned model with neurons in $\mathcal{S}$}\\
\texttt{return} $\{W^{(0)}_{\mathcal{S}, :}, W^{(1)}_{:, \mathcal{S}}\}$
\caption{i-SpaSP for Two-Layer Networks}
\label{A:cosamp_prune}
\end{algorithm}
\end{minipage}
\vspace{-1.4cm}
\end{wrapfigure}

\medskip
\noindent \textbf{Why does this work?}
Each iteration of i-SpaSP follows a three-step procedure in Algorithm \ref{A:cosamp_prune}:
\begin{itemize}[leftmargin=1.6cm]
 \item [\textbf{Step I:}] Compute neuron ``importance'' $Y$ given the current residual matrix $V$. \vspace{-0.2cm}
\end{itemize}

\vspace{-.3cm}
\begin{itemize}[leftmargin=1.65cm]
 \item[\textbf{Step II:}] Identify $s$ neurons within the combined set of important and active neurons (i.e., $\Omega \cup \mathcal{S}$) with the largest-valued hidden representations.
\end{itemize}

\vspace{-.6cm}
\begin{itemize}[leftmargin=1.7cm]
 \item[\textbf{Step III:}] Update $V$ with respect to the new pruned model estimate.
\end{itemize}

\noindent
We now provide intuition regarding the purpose of each individual step within i-SpaSP.

\medskip
\noindent
\textbf{Estimating Importance.}
$Y_{ij}$ is the importance of hidden neuron $i$ with respect to dataset example $j$.
We can characterize the discrepancy between pruned and dense network output $U'$ and $U$ as:
\begin{align}
    \mathcal{L}(U, U') = \frac{1}{2}\|W^{(1)}\cdot H - U' \|_F^2. \label{eq:prune_obj}
\end{align}
Considering $U'$ fixed, $\nabla_H \mathcal{L}(U, U') = (W^{(1)})^\top \cdot V$.
As such, if $Y_{ij}$ is a large, positive (negative) value, decreasing (increasing) $H_{ij}$ will decrease the value of $\mathcal{L}$ locally.
Then, because $H$ is non-negative and cannot be modified via pruning, one can realize that the best methodology of minimizing \eqref{eq:prune_obj} is including neurons with large, positive importance values within $\mathcal{S}$, as in Algorithm \ref{A:cosamp_prune}.

\medskip
\noindent
\textbf{Merging and Pruning.}
The $2s$ most-important neurons---based on $\mu(Y)$ components---are selected and merged with $\mathcal{S}$, allowing a larger set of neurons (i.e., more than $s$) to be explored.
From here, $s$ neurons with the largest-valued components in $\mu(H)$ are sub-selected from this combined set to form the next pruned model estimate.
Because hidden representation values are not considered in importance estimation, performing this two-step merging and pruning process ensures neurons within $\mathcal{S}$ have both large hidden activation and importance values, which together indicate a meaningful impact on network output.

\medskip
\noindent
\textbf{Computing the New Residual.}
The next pruned model estimate is used to re-compute $V$, which can be intuitively seen as updating $U'$ in \eqref{eq:prune_obj}.
As such, importance in Algorithm \ref{A:cosamp_prune} is based on the current pruned and dense network residual and \eqref{eq:prune_obj} is minimized over successive iterations.

\subsection{Implementation} \label{S:implementation}

\medskip
\noindent \textbf{Automatic Differentiation.}
Because $Y = \nabla_H \mathcal{L}(U, U')$, importance within i-SpaSP can be computed efficiently using automatic differentiation \citep{paszke2017automatic, tensorflow}; see Algorithm \ref{A:autodiff} for a PyTorch-style example.
Automatic differentiation simplifies importance estimation and allows it to be run on a GPU, making the implementation efficient and parallelizable.
Because the remainder of the pruning process only leverages basic sorting and set operations, the overall implementation of i-SpaSP is both simple and efficient.

\medskip
\noindent \textbf{Other Architectures.}
Algorithm \ref{A:autodiff} can be easily generalized to more complex network modules (i.e., beyond feed-forward layers) using automatic differentiation.
Notably, convolutional filters or attention heads can be pruned using the importance estimation from Algorithm \ref{A:autodiff} if $\texttt{sum}(\cdot)$ is performed over both batch and spatial dimensions.
Furthermore, i-SpaSP can be used to prune multi-layer networks by greedily pruning each layer of the network from beginning to end.

\medskip
\noindent \textbf{Large-Scale Datasets.}
Algorithm \ref{A:cosamp_prune} assumes the entire dataset is stored within $X$.
For large-scale experiments, such an approach is not tractable.
As such, we redefine $X$ within experiements to contain a subset or mini-batch of data from the full dataset, allowing the pruning process to be performing in an approximate---but computationally tractable---manner.
To improve the quality of this approximation, a new mini-batch is sampled during each i-SpaSP iteration, but the size of such mini-batches becomes a hyperparameter of the pruning process.\footnote{Both GFS \citep{provable_subnetworks} and multi-layer GFS \citep{log_pruning} adopt a similar mini-batch approach during pruning.}

\begin{minipage}{0.5\textwidth}
\begin{tcolorbox}
    \begin{footnotesize}
    $H$\texttt{.requires\_grad} $:=$ \texttt{True} \\
    \texttt{with torch.no\_grad():} \\
    \text{~~~~~~~~}\texttt{prune\_out} $:=$ \texttt{prune\_layer}$(H_{\mathcal{S}, :})$ \\
    \texttt{dense\_out} $:=$ \texttt{dense\_layer}$(H)$ \\
    \texttt{obj} $:=$ \texttt{sum} $\left(\frac{1}{2}(\texttt{dense\_out} - \texttt{prune\_out})^2\right)$\\
    \texttt{obj.backward()}\\
    \texttt{importance} $:= \texttt{sum}(H\texttt{.grad}, \texttt{dim}=0)$\\
    \texttt{return~importance}
    \end{footnotesize}
\end{tcolorbox}
\captionof{algorithm}{i-SpaSP importance computation via automatic differentiation.}
\label{A:autodiff}
\end{minipage}\hspace{0.5cm}
\begin{minipage}{0.45\textwidth}
    \centering
    \includegraphics[width=0.95\textwidth]{cs_time_compare.pdf}
    \vspace{-0.4cm}
    \captionof{figure}{Runtime of pruning ResNet34 blocks with i-SpaSP and GFS variants to 20\% (i.e., plain) or 40\% (i.e., dotted) of filters.}
    \label{fig:cs_time}
\end{minipage}

\subsection{Computational Complexity and Runtime Comparisons} \label{S:comp_complex}
Denote the complexity of matrix-matrix multiplication as $\xi$.
The complexity of pruning a network layer with $T$ iterations of i-SpaSP is $\mathcal{O}(T\xi + T d_{hid}\log(d_{hid}))$.
GFS has a complexity of $\mathcal{O}(s \xi d_{hid})$, as it adds a single neuron to the (initially empty) network layer at each iteration by exhaustively searching for the neuron that minimizes training loss.
Though later GFS variants achieve complexity of $\mathcal{O}(s \xi)$ \citep{log_pruning}, i-SpaSP is more efficient in practice because the forward pass (i.e. $\mathcal{O}(\xi)$) dominates the pruning procedure and $T \ll s$ (e.g., $T=20$ in Section \ref{S:experiments}).

As a practical runtime comparison, we adopt a ResNet34 model \citep{resnet} and measure wall-clock pruning time\footnote{We prune the 2nd (64 channels) and 15th (512 channels) convolutional blocks. We choose blocks in different network regions to view the impact of channel and spatial dimension on pruning efficiency.} with i-SpaSP and GFS.
We use the public implementation of GFS \citep{gfs_github} and test both stochastic and vanilla variants\footnote{Vanilla GFS exhaustively searches neurons within each iteration, while the stochastic variant randomly selects 50 neurons to search per iteration.}.
i-SpaSP uses settings from Section \ref{S:experiments}, and selected ResNet blocks are pruned to ratios of 20\% or 40\% of original filters; see Figure \ref{fig:cs_time}.
i-SpaSP significantly improves upon the runtime of GFS variants; e.g., i-SpaSP prunes Block 15 in roughly 10 seconds, while GFS takes over 1000 seconds in the best case.
Furthermore, unlike GFS, the runtime of i-SpaSP is not sensitive to the size of the pruned network (i.e., wall-clock time is similar for ratios of 20\% and 40\%), though i-SpaSP does prune later network layers faster than earlier layers.

\section{Theoretical Results} \label{S:theory}
Proofs are deferred to Appendix \ref{A:proofs}.
The dense network is pruned from $d_{hid}$ to $s$ neurons via i-SpaSP.
We assume that $W^{(1)}$ satisfies the restricted isometry property (RIP) \citep{rip}:

\begin{assumption}[Restricted Isometry Property (RIP) \citep{rip}] \label{assm:rip}
    Denote the $r$-th restricted isometry constant as $\delta_r$ and assume $\delta_r \leq 0.1$ for $r = 4s$.\footnote{This is a numerical assumption adopted from \citep{cosamp}, which holds for Gaussian matrices of size $\mathbb{R}^{m \times n}$ when $m \geq \mathcal{O}\left(r\log(\frac{n}{r})\right)$.} Then, $W^{(1)}$ satisfies the RIP with constant $\delta_r$ if  $ (1 - \delta_r)\|x\|_2^2 \leq \|W^{(1)} \cdot x\|_2^2 \leq (1 + \delta_r) \|x\|_2^2~$ for all $\|x\|_0 \leq r$.
\end{assumption}

No assumption is made upon $W^{(0)}$.
We hypothesize that Assumption \ref{assm:rip} is mild due to properties like semi or quarter-circle laws that bound the eigenvalues symmetric, random matrices within some range \citep{edelman2013random}, but we leave the formal verification of this assumption as future work.
We define $H = \sigma(W^{(0)}\cdot X) \in \mathbb{R}^{d_{hid} \times B}$, which can be theoretically reformulated as $H = Z + E$ for $s$-row-sparse $Z$ and arbitrary $E$.
From here, we can show the following about the residual between pruned and dense network hidden representations after $t$ iterations of Algorithm \ref{A:cosamp_prune}.

\begin{lemma} \label{L:hidden_res}
If Assumption \ref{assm:rip} holds, the pruned approximation to $H$ after $t$ iterations of Algorithm \ref{A:cosamp_prune}, $H_{\mathcal{S}_t, :}$, is $s$-row-sparse and satisfies the following inequality:
\begin{align*}
    \|\mu(H - H_{\mathcal{S}_t, :})\|_2 \leq (0.444)^t \|\mu(H)\|_2 + \left(14 + \frac{7}{\sqrt{s}}\right)\|\mu(E)\|_1
\end{align*}
\end{lemma}

\noindent
Going further, Lemma \ref{L:hidden_res} can be used to bound the residual between pruned and dense network output.

\begin{theorem} \label{T:vfrob}
    Let $U = W^{(1)} \cdot H$ and $U' = W_{:, \mathcal{S}_t}^{(1)} \cdot H_{\mathcal{S}_t, :}$ denote pruned and dense network output, respectively. $V_t = U - U'$ stores the residual between pruned and dense network output over the entire dataset. If Assumption \ref{assm:rip} holds, we have the following at iteration $t$ of Algorithm \ref{A:cosamp_prune}:
    \begin{align*}
        \|V_t\|_F \leq \|W^{(1)}\|_F \cdot \left ( (0.444)^t \|\mu(H)\|_2 + \left ( 14 + \frac{7}{\sqrt{s}}\right)\|\mu(E)\|_1 \right)
    \end{align*}
\end{theorem}

Because $\|\mu(H)\|_2$ decays linearly in Theorem \ref{T:vfrob}, $\|\mu(E)\|_1$ dominates the above expression for large $t$.
By assuming $H$ is row-compressible, we can derive a bound on $\|\mu(E)\|_1$.

\begin{lemma} \label{L:noise_bound}
Assume $H$ is $p$-row-compressible with magnitude $R$, where $H = Z + E$ for $s$-row-sparse $Z$ and arbitrary $E$.
Then,
$\|\mu(E)\|_1 \leq R\cdot \frac{s^{1 - \frac{1}{p}}}{\frac{1}{p} - 1}.$
\end{lemma}

\noindent
Lemma \ref{L:noise_bound} can then be combined with Theorem \ref{T:vfrob} to bound the error due to pruning via i-SpaSP.

\begin{theorem} \label{T:final_bound}
    Assume Algorithm \ref{A:cosamp_prune} is run for a sufficiently large number of iterations $t$.
    If Assumption \ref{assm:rip} holds and $H$ is $p$-row-compressible with factor $R$, the output residual between the dense network and the pruned network discovered via i-SpaSP can be bounded as follows:
    \begin{align*}
        \|V_t\|_F \leq \mathcal{O}\left ( \frac{s^{\frac{1}{2} - \frac{1}{p}} p (2\sqrt{s} + 1)}{1 - p} \right).
    \end{align*}
    \begin{proof}
    This follows directly from substituting Lemma \ref{L:noise_bound} into Theorem \ref{T:vfrob}, assuming $t$ is large enough such that $(0.444)^t \|W^{(1)}\|_F \|\mu(H)\|_2 \approx 0$, and factoring out constants in the resulting expression.
    \end{proof}
\end{theorem}

Theorem \ref{T:final_bound} indicates that the quality of the pruned network is dependent upon $s$ and $p$.
Intuitively, one would expect that lower values of $p$ (implying sparser $H$) would make pruning easier, as neurons corresponding to zero rows in $H$ could be eliminated without consequence.
This trend is observed exactly within Theorem \ref{T:final_bound}; e.g., for $p = \left \{\frac{3}{4}, \frac{1}{2}, \frac{1}{4} \right \}$ we have $\|V_t\|_F \leq \{\mathcal{O} (s^{-\frac{1}{3}}), \mathcal{O} (s^{-1} ), \mathcal{O}(s^{-3} )  \}$, respectively.
\textit{To the best of the authors' knowledge, our work is the first to theoretically characterize pruning error with respect to sparsity properties of network hidden representations.}
This bound can also be extended to similarly-structured (see Appendix \ref{A:multi_layer}), multi-layer networks:

\begin{theorem} \label{T:multi_layer}
Consider an $L$-hidden-layer network with weight matrices $\{W^{(0)}, \dots, W^{(L)} \}$ and hidden representations $\{H^{(1)}, \dots H^{(L)} \}$.
The hidden representations of each layer are assumed to have dimension $d$ for simplicity.
We define $H^{(\ell)} = \sigma (W^{(\ell)} \cdot H^{(\ell - 1)})$, where $H^{(1)} = \sigma(W^{(0)} \cdot X)$ and $H^{(L)} = W^{(L)} \cdot H^{(L - 1)}$.
We assume all weight matrices other than $W^{(0)}$ obey Assumption \ref{assm:rip} and all hidden representations other than $H^{(L)}$ are $p$-row-compressible.
i-SpaSP is applied greedily to prune each network layer, in layer order, from $d$ to $s$ hidden neurons.
Given sufficient iterations $t$, the residual between pruned and dense multi-layer network output behaves as:
\begin{align}
\|V^{(L)}_t\|_F \leq \mathcal{O} \left ( \sum_{i=1}^L  \textcolor{mygreen}{\left (14 + \frac{7}{\sqrt{s}}\right)^{L - i + 1}}  \textcolor{myblue2}{\left (\|W^{(L)}\|_F   \prod_{j=1}^{L - i} \|\texttt{vec}(W^{(j)})\|_1 \right)} \textcolor{myred}{\left (\frac{ d^{\frac{L-i}{2}} s^{1 - \frac{1}{p}}}{ \frac{1}{p} - 1} \right)} \right ) \label{eq:multi_layer}
\end{align}
\end{theorem}

Pruning error in \eqref{eq:multi_layer} is summed over each network layer.
Considering layer $i$, the green factor is inherited from Theorem \ref{T:vfrob} with an added exponent due to a recursion over network layers after $i$.
Similarly, the blue factor accounts for propagation of error through weight matrices after layer $i$, \emph{revealing that green and blue factors account for propagation of error through network layers}.
The red portion of \eqref{eq:multi_layer}, which captures the convergence properties of multi-layer pruning, comes from Lemma \ref{L:noise_bound}, where an extra factor $d^{\frac{L - i}{2}}$ arises as an artifact of the proof.\footnote{This artifact arises because $\ell_1$ norms must be replaced with $\ell_2$ norms in certain areas to form a recursion over network layers. This artifact can likely be removed by leveraging sparse matrix analysis for expander graphs \citep{bah2018construction}, but we leave this as future work to simplify the analysis.}
Because $d$ is fixed multiple of $s$ determined by the pruning ratio, this factor disappears when $p$ is small, leading the expression to converge.
For example, if $p=\frac{1}{4}$, the red expression in \eqref{eq:multi_layer} behaves asymptotically as $\{\mathcal{O}(s^{-2.5}), \mathcal{O}(s^{-2}), \mathcal{O}(s^{-1.5}), \dots \}$ for layers at the end of the network moving backwards.

\section{Experiments} \label{S:experiments}

\begin{wrapfigure}{r}{0.6\textwidth}
    \vspace{-0.6cm}
    \centering
    \includegraphics[width=1\linewidth]{synthetic_compress.pdf}
    \caption{i-SpaSP pruning experiments for two-layer networks of different sizes and $p$ ratios. \emph{The error decay rate increases with decreasing $p$, confirming that more compressible hidden representations aid in the pruning process.}}
    \label{fig:synthetic} \vspace{-0.4cm}
\end{wrapfigure}

Within this section, we provide empirical analysis of i-SpaSP.
We first present synthetic results using two-layer neural networks to numerically verify Theorem \ref{T:final_bound}.
Then, we perform experiments with two-layer networks on MNIST \citep{mnist}, convolutional neural networks (CNNs) on ImageNet (ILSVRC2012), and multi-lingual BERT (mBERT) \citep{devlin2018bert} on the cross-lingual NLI corpus (XNLI) \citep{conneau2018xnli}.
For all experiments, we adopt best practices from previous work \citep{conv_filt_prune} to determine pruning ratios within the dense network, often performing less pruning on sensitive layers; see Appendix \ref{A:exp_details} for details.
As baselines, we adopt both greedy selection methodologies \citep{provable_subnetworks, log_pruning} and several common, heuristic methods.
We find that, in addition to improving upon the pruning efficiency of GFS (i.e., see Section \ref{S:comp_complex} for runtime comparisons on ResNet pruning experiments), i-SpaSP performs comparably to baselines in all cases, demonstrating that it is both performant and efficient.

\subsection{Synthetic Experiments} \label{S:synthetic_exp}
To numerically verify Theorem \ref{T:final_bound}, we construct synthetic $H$ matrices with different row-compressibility ratios $p$, denoted as $\mathcal{H} = \{H^{(i)}\}_{i=1}^K$.
For each $p \in \{0.3, 0.5, 0.7, 0.9\}$, we randomly generate three unique entries within $\mathcal{H}$ (i.e., $K=12$) and present average performance across them all.
We prune a randomly-initialized $W^{(1)}$ from size $d_{hid} \times d_{out}$ to sizes $s \times d_{out}$ for $s \in [d_{hid}]$. (i.e., each setting of $s$ is a separate experiment) using $T = 20$.\footnote{We find that the active set of neurons selected by i-SpaSP becomes stable (i.e., few neurons are modified) after $20$ iterations or less in all synthetic experiments.}.
We test $d_{hid} \in \{100, 200\}$, and the same $W^{(1)}$ matrix is used for experiments with equal hidden dimension; see Appendix \ref{A:synthetic} for more details.

Results are displayed in Figure \ref{fig:synthetic}, where  $\|V\|_F^2$ is shown to decay polynomially with respect to the number of neurons within the pruned network $s$.
Furthermore, as predicted by Theorem \ref{T:final_bound}, the decay rate of $\|V\|^2_F$ increases as $p$ decreases, revealing that higher levels of sparsity within the dense network's hidden representations improve pruning performance and speed with respect to $s$.
This trend holds for all hidden dimensions that were considered.

\subsection{Two-Layer Networks} \label{S:two_layer}
\begin{wrapfigure}{r}{0.6\textwidth}
    \vspace{-0.4cm}
    \centering
    \includegraphics[width=1\linewidth]{mnist_prune.pdf}
    \caption{Performance of networks pruned with different greedy algorithms on MNIST before (left) and after (right) fine-tuning. Although GFS performs well prior to fine-tuning, \emph{i-SpaSP always yields the top-performing network after fine-tuning.}}
    \label{fig:mnist_prune} \vspace{-0.2cm}
\end{wrapfigure}

We perform pruning experiments with two-layer networks on MNIST \citep{mnist}.
All MNIST images are flattened and no data augmentation is used.
The dense network has $10 \times 10^{3}$ hidden neurons and is pre-trained before pruning.
We prune the dense network using i-SpaSP, GFS, and Top-K, which we use as a naive greedy selection baseline.\footnote{Top-k selects $k$ neurons with the largest-magnitude hidden representations in a mini-batch of data. It is a naive baseline for greedy selection that is not used in previous work to the best of our knowledge.}
After pruning, the network is fine-tuned using stochastic gradient descent (SGD) with momentum.
Performance is reported as an average across three separate trails; see Appendix \ref{A:two_layer} for more details.

As shown in Figure \ref{fig:mnist_prune}-right, i-SpaSP outperforms other pruning methodologies after fine-tuning in all experimental settings.
Networks obtained with GFS perform well without fine-tuning (i.e., Figure \ref{fig:mnist_prune}-left) because neurons are selected to minimize loss during the pruning process.
Because i-SpaSP selects neurons based upon the importance criteria described in Section \ref{S:method}, the pruning process does not directly minimize training loss, thus leading to poorer performance prior to fine-tuning (i.e., Top-K exhibits similar behavior).
Nonetheless, i-SpaSP, in addition to improving upon the pruning efficiency of GFS, discovers a set of neurons that more closely recovers dense network output, as revealed by its superior performance after fine-tuning.

\subsection{Deep Convolutional Networks} \label{S:deep_net}
We perform structured filter pruning of ResNet34 \citep{resnet} and MobileNetV2 \citep{mobilenetv2} with i-SpaSP on ImageNet (ILSVRC 2012).
Beginning with public, pre-trained models \citep{paszke2017automatic}, we use i-SpaSP to prune chosen convolutional blocks within each network, then fine-tune the pruned model using SGD with momentum.
After pruning each block, we perform a small amount of fine-tuning; see Appendix \ref{A:deep_net} for more details.
Numerous heuristic and greedy selection-based algorithms are adopted as baselines; see Table \ref{tab:imagenet}.

\begin{wraptable}{r}{0.6\textwidth}
    \centering
    \begin{footnotesize}
    \begin{tabular}{c|c|cc}
    \toprule
          & Pruning Method & Top-1 Acc. & FLOPS \\
         \midrule
         \multirow{12}{*}{\rotatebox{90}{ResNet34}} & Full Model \cite{resnet} & 73.4 & 3.68G\\
         \cmidrule{2-4}
         & Filter Pruning \cite{conv_filt_prune} & 72.1 & 2.79G\\
         & Rethinking Pruning \cite{rethink_pruning} & 72.0 & 2.79G \\
         & More is Less \cite{dong2017more} & 73.0 & 2.75G \\
         & i-SpaSP & 73.5 & 2.69G\\
         & GFS \cite{provable_subnetworks} & 73.5 & 2.64G \\
         & Multi-Layer GFS \cite{log_pruning} & 73.5 & 2.20G\\
         \cmidrule{2-4}
         & SFP \cite{he2018soft} & 71.8 & 2.17G \\
         & FPGM \cite{he2019filter} & 72.5 & 2.16G \\
         & i-SpaSP & 72.5 & 2.13G\\
         & GFS \cite{provable_subnetworks} & 72.9 & 2.07G \\
         & Multi-Layer GFS \cite{log_pruning} & 73.3 & 1.90G \\
         \midrule
         \multirow{12}{*}{\rotatebox{90}{MobileNetV2}} & Full Model \cite{mobilenetv2} & 72.0 & 314M \\
         \cmidrule{2-4}
         & i-SpaSP & 71.6 & 260M\\
         & GFS \cite{provable_subnetworks} & 71.9 & 258M\\
         & Multi-Layer GFS \cite{log_pruning} & 72.2 & 245M\\
         & i-SpaSP & 71.3 & 242M\\
         \cmidrule{2-4}
         & LEGR \cite{chin2019legr} & 71.4 & 224M\\
         & Uniform & 70.0 & 220M \\
         & AMC \cite{he2018amc} & 70.8 & 220M\\
         & i-SpaSP & 70.7 & 220M \\
         & GFS \cite{provable_subnetworks} & 71.6 & 220M \\
         & Multi-Layer GFS \cite{log_pruning} & 71.7 & 218M \\
         & Meta Pruning \cite{liu2019metapruning} & 71.2 & 217M\\
    \bottomrule
    \end{tabular}
    \caption{Test accuracy of ResNet34 and MobileNetV2 models pruned to different FLOP levels with various pruning algorithms on ImageNet.}
        \label{tab:imagenet} \vspace{-0.5cm}
    \end{footnotesize}
\end{wraptable}

\medskip
\noindent \textbf{ResNet34.}
We prune ResNet34 to 2.69 and 2.13 GFlops with i-SpaSP.
As shown in Table \ref{tab:imagenet}, i-SpaSP yields comparable performance to GFS variants at similar FLOP levels; e.g., i-SpaSP matches 75.5\% test accuracy of GFS with the 2.69 GFlop model and performs within 1\% of GFS variants for the 2.13 GFlop model.
However, the multi-layer variant of GFS \citep{log_pruning} does discover sub-networks with fewer FLOPS and similar performance in both cases.
i-SpaSP improves upon or matches the performance of all heuristic methods at similar FLOP levels.

\medskip
\noindent \textbf{MobileNetV2.}
We prune MobileNetV2 to 260, 242, and 220 MFlops with i-SpaSP.
In all cases, sub-networks discovered with i-SpaSP achieve performance within 1\% of those obtained via both GFS variants and heuristic methods.
Although MobileNetV2 performance is relatively lower in comparison to ResNet34, i-SpaSP is still capable of pruning the more difficult network to various different FLOP levels and performs comparably to baselines in all cases.

\medskip
\noindent \textbf{Discussion.} Within Table \ref{tab:imagenet}, the performance of i-SpaSP is never more than 1\% below that of a similar-FLOP model obtained with a baseline pruning methodology, including both greedy selection and heuristic-based methods.
As such, similar to GFS, i-SpaSP can be seen as a theoretically-grounded pruning methodology that is practically useful, even in large-scale experiments.
Such pruning methodologies that are both theoretically and practically relevant are few.
In comparison to GFS variants, i-SpaSP significantly improves pruning efficiency; see Section \ref{S:comp_complex}.
Thus, i-SpaSP can be seen as a viable alternative to GFS---both theoretically and practically---that may be preferable when runtime is a major concern.

\subsection{Transformer Networks} \label{S:transformers}

\begin{wraptable}{r}{0.5\textwidth}
    \centering
    \begin{footnotesize}
    \begin{tabular}{c|cc}
    \toprule
          Method & Pruning Ratio & Top-1 Acc. \\
          \midrule
          Full Model & - & 71.02\\
          \midrule \multirow{2}{*}{Uniform} & 25\% & 62.73 \\
          & 40\% & 66.43 \\
          \midrule \multirow{2}{*}{\cite{michel2019sixteen}} & 25\% & 62.97 \\
          & 40\% & 67.31 \\
          \midrule \multirow{2}{*}{i-SpaSP} & 25\% & 63.70\\
          & 40\% & 68.11\\
    \bottomrule
    \end{tabular}
    \caption{Test accuracy of mBERT models pruned and fine-tuned on the XNLI dataset.}
        \label{tab:xnli} \vspace{-0.4cm}
    \end{footnotesize}
\end{wraptable}

We perform structured pruning of attention heads using the mBERT model \citep{devlin2018bert} on the XNLI dataset \citep{conneau2018xnli}, which contains textual entailment annotations across 15 different languages.
We begin with a pre-trained mBERT model and fine-tune it on XNLI prior to pruning.
Then, structured pruning is performed on the attention heads of each layer using i-SpaSP, uniform pruning (i.e., randomly removing a fixed ratio of attention heads), and a sensitivity-based masking approach \citep{michel2019sixteen} (i.e., a state-of-the-art heuristic approach for structured attention head pruning for transformers).
After pruning models to a fixed ratio of 25\% or 40\% of original attention heads, we fine-tune the pruned networks and record their performance on the XNLI test set; see Appendix \ref{A:transformers} for further details.

As shown in Table \ref{tab:xnli}, models pruned with i-SpaSP outperform those pruned to the same ratio with either uniform or heuristic pruning methods in all cases.
The ability of i-SpaSP to effectively prune transformers demonstrates that the methodology can be applied to the structured pruning of numerous different architectures without significant implementation changes (i.e., using the automatic differentiation approach described in Section \ref{S:implementation}).
Furthermore, i-SpaSP even outperforms a state-of-the-art heuristic approach for the structured pruning of transformer attention heads, thus again highlighting the strong empirical performance of i-SpaSP in comparison to heuristic pruning methods that lack theoretical guarantees.

\section{Conclusion}
We propose i-SpaSP, a pruning methodology for neural networks inspired by sparse signal recovery.
Our methodology comes with theoretical guarantees that indicate, for both two and multi-layer networks, the quality of a pruned network decays polynomially with respect to its size.
We connect this theoretical analysis to properties of the dense network, showing that pruning performance improves as the dense network's hidden representations become more sparse.
Practically, i-SpaSP performs comparably to numerous baseline pruning methodologies in large-scale experiments and drastically improves upon the computationally efficiency of the most common provable pruning methodologies.
As such, i-SpaSP is a practical, provable, and efficient algorithm that we hope will enable a better understanding of neural network pruning both in theory and practice.
In future work, we wish to extend i-SpaSP to cover cases in which network pruning and training are combined into a single process, such as utilizing regularization-based approaches to induce sparsity during pre-training or updating network weights to improve sub-network performance as pruning occurs.

\clearpage

\acks{Compute resources used for this work were provided by Alegion Inc. This work is supported by NSF FET:Small no. 1907936, NSF MLWiNS CNS no. 2003137 (in collaboration with Intel), NSF CMMI no. 2037545, NSF CAREER award no. 2145629, and Rice InterDisciplinary Excellence Award (IDEA).}

redpajama_set_name: RedPajamaArXiv


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Weather Due to the unpredictable nature of weather, we reserve the right to cancel, postpone, shorten or alter any flight due to weather. Weather includes, but is not limited to; rain, thunderstorms, strong winds, gusty winds, reduced visibility due to rain, fog, smoke or haze and high temperatures. Our highest priority is the safety of our passengers. Please contact us to discuss further or to arrange other plans should your intended flight be affected by weather.
Helicopter Serviceability Our helicopters are maintained to the highest possible standards by a CASA approved Maintenance Facility. There are instances where our helicopters are required for scheduled or unscheduled servicing and may not be available on your intended day of travel.
Controlled Airspace From the Geelong Waterfont our helicopters operate in controlled airspace. Therefore we require a clearance from Air Traffic Control before we can take off and land. Although rare, Air Traffic Control may differ the flight path from what has being described or there may be delays.
Minimum fare is equivalent to two adult passengers per flight. Gift certificates are available for a single passenger, however on the day of flight the gift certificate holder will need to either pay the difference, join another passenger or bring someone else along to make up the difference.
Child's Fare is valid for any child aged up to and including 15 years old and is only applicable to the Corio Bay Fun Fly from the Geelong Waterfront.
Gift Certificates are valid for a period of 6 months from date of issue. For a Gift Certificate to be redeemed, it must be presented on the day of the flight in digital or paper form. Gift certificates are not redeemable for cash. Gift certificates can be upgraded. If a gift certificate has been fraudulently changed or altered it will be deemed to be invalid.
Special Offers Geelong Helicopters has from time to time a number of special offers. Please note that only one special offer can be used at any one time. Special offers which include additional items to the helicopter flight (i.e. souvenir caps, discount vouchers) must be collected on the day of flight from the departure point.
Passenger Safety All passengers will be given a safety briefing before their flight. Geelong Helicopters reserves the right to cancel or terminate any flight if a passenger is deemed a threat to the safety of the aircraft or other passengers and crew.
Departure Location If a flight is unavailable to be flown from its usual departure point, it may be moved to an alternate location. Passengers will be notified of any changes prior to their flight. If customer requires a flight to be flown from a particular location, a minimum number of flights may be required otherwise a helicopter relocation fee will apply.
Unauthorised landings Any landing by a Geelong Helicopters aircraft on public or private property where the client has provided unauthorised or fraudulent landing permission will be charged a fee of $2,500 (no exceptions), plus any fines or penalties which are associated with the unauthorised landing. This amount is payable by the client unless the unauthorised landing is due to an emergency or deteriorating weather conditions.
Single passenger price Geelong Helicopters may offer a single passenger ticket for certain helicopter flights. These are available by appointment only, between 11am and 2pm on Saturday and Sunday and are subject to the above terms and conditions.
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