As a 510 (c)(3) charitable organization (ID #16-1572964), the International Motor Racing Research Center receives no local, state or federal operating money. To ensure that we stay true to our mission of preserving and sharing the history of motorsports we have established a variety of ways through which individuals and organizations can support the center.
The International Motor Racing Research Center is dependent on donations of financial support. We have established several levels at which support may be given. A motorsports VIP is invited to serve as the appeal campaign chairman each year. Past chairmen include Mario Andretti, Phil Hill, Sir Stirling Moss, Dan Gurney, Sam Posey, Denise McCluggage and Butch Leitzinger. Learn more.
The Center's east and west entryways are paved with engraved bricks and granite blocks. The prices are $200 for a brick and $1,000 for a granite block.
Upwards of 400 bricks and more than 40 blocks now welcome visitors to the Center. Learn more or place your order.
Many families of Center supporters have chosen to honor their loved ones by designating the Center as the recipient of memorials. The Center is extremely grateful for this type of support because it is enhanced by memories of those being honored.
We also actively participate in employers' matching gift programs. Matching gifts may double or triple the value of gifts. The Center encourages our supporters to contact their employer's benefits office to inquire about matching gifts. We truly appreciate the additional time and effort participation requires.
An endowment fund to ensure future operations has been established, to which Center supporters may designate donations. Further information about the endowment is available from the staff.
In addition to the benefits of the gift to the Center, donors receive additional benefits through income tax deductions and savings in estate taxes for themselves and survivors. A personal financial advisor is the best person to analyze those benefits.
The Briggs S. Cunningham Legacy Society provides an opportunity to contribute estate assets to the International Motor Racing Research Center. Gifts may be financial and/or material items such as automobiles, significant collections of photographs, film, books and other motorsports memorabilia subject to acceptance by the Center. Donors will be acknowledged, commensurate with level of contribution, appropriately on display plaques at the Center.
By execution of a formal Pledge Commitment and Agreement making your gift an irrevocable legal obligation of your estate. This method has the advantage to the International Motor Racing Research Center of certainty of both the designation and the amount of your legacy.
By simply adding the International Motor Racing Research Center as a beneficiary of your estate and notifying us in writing that you have done so. The amount of your intended gift may or may not be specified. This alternative is an agreement in principle, but is not legally binding, in that you may modify your estate beneficiary designations at any time.


"Scrapping" redirects here. It is not to be confused with scraping.
"Scrap metal" redirects here. For other uses, see Scrap Metal (disambiguation).
Scrap metal originates both in business and residential environments. Typically a "scrapper" will advertise their services to conveniently remove scrap metal for people who don't need it.
Scrap is often taken to a wrecking yard (also known as a scrapyard, junkyard, or breaker's yard), where it is processed for later melting into new products. A wrecking yard, depending on its location, may allow customers to browse their lot and purchase items before they are sent to the smelters, although many scrap yards that deal in large quantities of scrap usually do not, often selling entire units such as engines or machinery by weight with no regard to their functional status. Customers are typically required to supply all of their own tools and labor to extract parts, and some scrapyards may first require waiving liability for personal injury before entering. Many scrapyards also sell bulk metals (stainless steel, etc.) by weight, often at prices substantially below the retail purchasing costs of similar pieces.
A scrap metal shredder is often used to recycle items containing a variety of other materials in combination with steel. Examples are automobiles and white goods such as refrigerators, stoves, clothes washers, etc. These items are labor-intensive to manually sort things like plastic, copper, aluminum, and brass. By shredding into relatively small pieces, the steel can easily be separated out magnetically. The non-ferrous waste stream requires other techniques to sort.
In contrast to wrecking yards, scrapyards typically sell everything by weight, instead of by item. To the scrapyard, the primary value of the scrap is what the smelter will give them for it, rather than the value of whatever shape the metal may be in. An auto wrecker, on the other hand, would price exactly the same scrap based on what the item does, regardless of what it weighs. Typically, if a wrecker cannot sell something above the value of the metal in it, they would then take it to the scrapyard and sell it by weight. Equipment containing parts of various metals can often be purchased at a price below that of either of the metals, due to saving the scrapyard the labor of separating the metals before shipping them to be recycled.
Scrap prices may vary markedly over time and in different locations. Prices are often negotiated among buyers and sellers directly or indirectly over the Internet. Prices displayed as the market prices are not the prices that recyclers will see at the scrap yards. Other prices are ranges or older and not updated frequently. Some scrap yards' websites have updated scrap prices.
In the US, scrap prices are reported in a handful of publications, including American Metal Market, based on confirmed sales as well as reference sites such as Scrap Metal Prices and Auctions. Non-US domiciled publications, such as The Steel Index, also report on the US scrap price, which has become increasingly important to global export markets. Scrap yards directories are also used by recyclers to find facilities in the US and Canada, allowing users to get in contact with yards.
With resources online for recyclers to look at for scrapping tips, like web sites, blogs, and search engines, scrapping is often referred to as a hands and labor-intensive job. Taking apart and separating metals is important to making more money on scrap, for tips like using a magnet to determine ferrous and non-ferrous materials, that can help recyclers make more money on their metal recycling. When a magnet sticks to the metal, it will be a ferrous material, like steel or iron. This is usually a less expensive item that is recycled but usually is recycled in larger quantities of thousands of pounds. Non-ferrous metals like copper, aluminum, and brass do not stick to a magnet. Some cheaper grades of stainless steel are magnetic, other grades are not. These items are higher priced commodities for metal recycling and are important to separate when recycling them. The prices of non-ferrous metals also tend to fluctuate more than ferrous metals so it is important for recyclers to pay attention to these sources and the overall markets.
Great potential exists in the scrap metal industry for accidents in which a hazardous material present in scrap causes death, injury, or environmental damage. A classic example is radioactivity in scrap; the Goiânia accident and the Mayapuri radiological accident were incidents involving radioactive materials. Toxic materials such as asbestos, and toxic metals such as beryllium, cadmium, and mercury may pose dangers to personnel, as well as contaminating materials intended for metal smelters.
Many specialized tools used in scrapyards are hazardous, such as the alligator shear, which cuts metal using hydraulic force, compactors, and scrap metal shredders.
90% savings in raw materials used.
86% reduction in air pollution.
40% reduction in water use.
76% reduction in water pollution.
97% reduction in mining wastes.
1,115 kg of iron ore.
Aluminium savings of 95% energy.
Copper savings of 85% energy.
Lead savings of 65% energy.
Zinc savings of 60% energy.
The metal recycling industry encompasses a wide range of metals. The more frequently recycled metals are scrap steel, iron (ISS), lead, aluminium, copper, stainless steel and zinc. There are two main categories of metals: ferrous and non-ferrous. Metals which contain iron in them are known as ferrous.
Metals without iron are non-ferrous.
Common non-ferrous metals are copper, brass, aluminum, zinc, magnesium, tin, nickel, and lead.
Usable coins can be deposited in banks. Damaged US coins can be redeemed for money via the Mutilated Coin Redemption Program.
Precious metals are metals with a high market value in any form, such as gold, silver, and platinum group metals.
Exotic metals contain rare elements such as cobalt, mercury, titanium, tungsten, arsenic, beryllium, bismuth, cerium, cadmium, niobium, indium, gallium, germanium, lithium, selenium, tantalum, tellurium, vanadium, and zirconium. Some types of metals are radioactive. These may be "naturally occurring" or formed by nuclear reactions. Metals that have been exposed to radioactive sources may also become radioactive in settings such as medical environments, research laboratories, and nuclear power plants.
OSHA guidelines should be followed when recycling any type of scrap metal to ensure safety.
Ferrous metals are able to be recycled, with steel being one of the most recycled materials in the world. Ferrous metals contain an appreciable percentage of iron and the addition of carbon and other substances creates steel.
In the United States, steel containers, cans, automobiles, appliances, and construction materials contribute the greatest weight of recycled materials. For example, in 2008, more than 97% of structural steel and 106% of automobiles were recycled, comparing the current steel consumption for each industry with the amount of recycled steel being produced (the late 2000s recession and the associated sharp decline in automobile production in the US explains the over-100% calculation). A typical appliance is about 75% steel by weight and automobiles are about 65% steel and iron.
The steel industry has been actively recycling for more than 150 years, in large part because it is economically advantageous to do so. It is cheaper to recycle steel than to mine iron ore and manipulate it through the production process to form new steel. Steel does not lose any of its inherent physical properties during the recycling process, and has drastically reduced energy and material requirements compared with refinement from iron ore. The energy saved by recycling reduces the annual energy consumption of the industry by about 75%, which is enough to power eighteen million homes for one year. According to the International Resource Panel's Metal Stocks in Society report, the per capita stock of steel in use in Australia, Canada, the European Union EU15, Norway, Switzerland, Japan, New Zealand and the US combined is 7,085 kilograms (15,620 lb) (about 860 million people in 2005).
Basic oxygen steelmaking (BOS) uses 25–35% recycled steel to make new steel. BOS steel usually contains lower concentrations of residual elements such as copper, nickel, and molybdenum, and is therefore more malleable than electric arc furnace (EAF) steel, and is often used to make automotive fenders, tin cans, industrial drums, or any product with a large degree of cold working. EAF steelmaking uses almost 100% recycled steel. This steel contains greater concentrations of residual elements that cannot be removed through the application of oxygen and lime. It is used to make structural beams, plates, reinforcing bar, and other products that require little cold working. Downcycling of steel by hard-to-separate impurities such as copper or tin can only be prevented by well-aimed scrap selection or dilution by pure steel. Recycling one metric ton (1,000 kilograms) of steel saves 1.1 metric tons of iron ore, 630 kilograms of coal, and 55 kilograms of limestone.
Turnings – Remains of drilling or shaping steels. Also known as "borings" or "swarf"
Manganese steel – Non-magnetic, hardened steel used in the mining industry, cement mixers, rock crushers, and other high-impact and abrasive environments.
The hulls of ships, with any usable equipment salvaged and removed, can be broken up to provide scrap steel. For a time countries in south Asia carried out most ship breaking, often using manual methods that were hazardous to workers and the environment. International regulations now dictate treatment of old ships as sources of hazardous waste, so ship breaking has returned to ports in more developed countries. In 2013, about 29 million tons of scrap steel was recovered from broken ships. Some of the scrap can be reheated and rolled to make products such as concrete reinforcing bars, or the scrap may be melted to make new steel.
The scrap industry was valued at more than $90 billion in 2012, up from $54 billion in 2009 balance of trade, exporting $28 billion in scrap commodities to 160 countries. Since 2010, the industry has added more than 15,000 jobs, and supports 463,000 workers, both directly and indirectly. In addition, it generates more than $10 billion in revenue for federal, state, and local governments. Scrap recycling also helps reduce greenhouse gas emissions and conserves energy and natural resources. For example, scrap recycling diverts 135 million short tons (121,000,000 long tons; 122,000,000 t) of materials away from landfills. Recycled scrap is a raw material feedstock for nearly 60% of steel made in the US, for almost 50% of the copper and copper alloys produced in the US, for more than 75% of the US paper industry's needs, and for 50% of US aluminum. Recycled scrap helps keep air and water cleaner by removing potentially hazardous materials and keeping them out of landfills.
^ "Benefits of Recycling Scrap Metal". Retrieved 2011-04-04.
^ Hartman, Roy A. (2009). "Recycling". Encarta. Archived from the original on 2008-04-14.
^ "Steel Recycling Rates at a Glance" (PDF). Archived from the original (PDF) on 2010-02-20. Retrieved 2010-02-20.
^ "Recycling steel appliances". Stell Recycling Institute. 2014. Retrieved 2017-04-06.
^ "End-of-life vehicle recycling process". Retrieved 2016-01-18.
^ "Facts About Steel Recycling". Archived from the original on 2009-08-25. Retrieved 2009-07-18.
^ M.A. Reuter; K. Heiskanen; U. Boin; A. Van Schaik; E. Verhoef; Y. Yang; G. Georgalli, eds. (November 2005). "13" (Book). The metrics of material and metal ecology: harmonizing the resource, technology, and environmental cycles. Developments in Mineral Processing. 16. Elsevier. p. 396. ISBN 978-0-444-51137-9.
^ "Information on Recycling Steel Products". WasteCap of Massachusetts. Archived from the original on 2007-10-11. Retrieved 2007-02-28.
^ "Ferrous Scrap Metal". Retrieved 2011-04-24.
Wikimedia Commons has media related to Scrap metal.
Look up scrap in Wiktionary, the free dictionary.


The weekend is almost here and that means it's time to relax with another 'Time Out'. This week's questions are answered by Walaiti Rathore, Managing Director at Licensing Law Consultancy, who in another life could have worked in the film industry.
Check to see what my family have planned for me!
I am fortunate in that a lot of what I enjoy doing is related to what I do at work – perks of working in the leisure and hospitality industry. However, completely outside of that I continue to be a film enthusiast; I still work on scripts for example. I also love watching and following sport – especially cricket.
I love and watch a lot of films, so this is a very difficult question. If I had to pick one, then it would have to be the short film called "Chinese Takeout". For anyone who has seen this, it would probably be their worst film. It is my favourite because it was the first film I produced and directed. It was screened at Broadway cinema as part of the Shots in the Dark film festival held in Nottingham (many years ago) where I received an award for my efforts.
Writing, producing and directing films as well as some acting.
I like the idea of being a "normal" human being with power coming from science and technology which can be constantly updated/improved. Therefore, I would be happy with an Iron Man type suit.
I like a variety of music but with preference for traditional qawwali.
Any memorable moment shared with my mother. Everything else I need I have right here in the present.


text:
Microsoft phasing out its virtual points currency?
Microsoft could be preparing to phase out its virtual points currency by the end of the year.
By Tom Warren@tomwarren Jan 24, 2012, 3:58am EST
Source Inside Mobile Apps
Share All sharing options for: Microsoft phasing out its virtual points currency?
Microsoft Points
Microsoft may be preparing to discontinue its virtual currency system by the end of the year. Inside Mobile Apps reports that the software giant will phase out Microsoft Points in favor of real money based on the local region of Xbox Live accounts. The move may sound surprising, but Microsoft has already started to introduce cash purchases for its full Xbox Live games instead of using the points system.
Microsoft Points allow customers to exchange money for allotments of points to purchase game add-ons, themes, gamertag pictures, and more. The points can also be used in the Zune Marketplace to purchase music and videos. Microsoft has long been criticized for its approach to the virtual points system, with most criticism focusing on the fact that users have to purchase points in increments. 80 Microsoft Points converts to one dollar in the US, and users often have to purchase more than they actually require, resulting in a confusing experience for consumers.
Windows Phone accepts Microsoft Points for Xbox Live avatar items, but game purchases and applications use traditional currency. Despite the broad, but inconsistent, use of Microsoft Points, it makes sense that Microsoft could be looking to phase the system out, although we cannot verify the report. Windows 8, Microsoft's next-generation operating system, is expected to debut later this year and will include Xbox LIVE integration. The Windows Store, used to purchase apps and games on Windows 8, will use cash instead of Microsoft Points, and the timing appears to coincide with Microsoft's broader strategy to simplify and consolidate its Xbox, Windows and Phone platforms, both in user interface and functionality. The company isn't discussing any plans for Microsoft Points, saying it "doesn't comment on rumours or speculation."

redpajama_set_name: RedPajamaCommonCrawl


text:
pidgeonholes
#WKSL
lessons from a distance
The Wizard
by Chris Bedell
I went to bed without dinner one night shortly after my 13th birthday under the pretense of not feeling well when in reality it was about the fact that the feeling that someone had been following me for the last week was unshakable.
The hairs on my back pricked up hours later at the sighting of a faint outline of a man with red eyes hovering in my bedroom.
Screaming didn't even matter since no one came to my aid because the man must have knocked out my parents, as it was the only logical explanation.
Each end of the man's tailcoat spurted at me, breaking into a total of four pieces while they wrapped around my hands and feet.
A fog dragged us through the air and I made a loud thumping sound after landing in a shed. I then shuffled to the exit, attempting to open the door only to discover it remained locked.
The 1,470 tally marks on the walls of the shed from the chalk the Wizard provided screamed out to me. It was as if it were some sort of sick torture since the Wizard wanted me to know how long I had been held captive. There was also a fourth wall, which was scarred with pink, green, and purple tally marks, proving other people had been previously held captive because the Wizard only provided me with blue chalk.
The rays of the descending sunlight poked into the shed, signaling the start of dinner.
A cloud crackled through the air, making a plate of toast and glass of water appear on the table.
I shuffled over to the table, shoving the piece of toast into my mouth in one sitting before gulping down the water in case the Wizard decided to take it away as a punishment.
Perfume wafted through the air, as if the Wizard realized it would make me nostalgic for my Mom.
A shadow slashed by the window hours later since a daily evening check was another part of the routine.
The only problem was this evening was different because flames flowed out of my right arm, making me pierce the night with agonizing shrills. It must have been my punishment for trying to open the door at the end of the maze during my exercise hour in the morning today.
Tears trickled down my face all night at the realization that enough was enough because the threat of being killed no longer mattered. Sneaking into the Wizard's tower and stealing his wand during my exercise hour tomorrow morning was the only option. It was ultimately the right decision because he might have taken naps during that time since vigilance wasn't required. After all, magic was the only way out of the maze since the Wizard needed a way to go between Earth and his world if he wanted more victims.
The door to the shed opened the following morning, as if on cue, which meant it was time for exercise.
I darted out of the shed, scanning the property before making my way to the tower.
Climbing up the wisteria was easier than it first seemed, as the Wizard ended up being in the middle of a nap.
I slid through the window, making sure not to make a noise while grabbing the wand and climbing down the wisteria again after flinching when the Wizard almost woke up.
The maze got smaller and smaller while clipping through it, and I arrived at the door, waiving the wand. The door dissolved and I stepped through it before waving the wand again, causing itto seal as the neon light of an open sign glowed in the distance, revealing that it was time to rejoin civilization.
The train honked, waking me up from my dream before rolling into the train station while slicing through the fog.
Getting off the train proved more difficult than it should have been as a result of having to push through the crowd of people.
."It's good to see you Lucy," said a guy's voice as I stepped off the platform and onto to the ground.
"You too Javier."
The two of us embraced for a quick hug before pulling back a moment later.
A missing poster containing a picture of my face popped out at me in the distance, causing me to scoff. Someone must have forgotten to take the poster down or something, which was odd because that should have been done a little over a year ago.
The wind whistled in the background, pushing forward a poster of another missing girl who also happened to be a blonde like me.
"It's nice you're finally visiting me. I mean you've been home long enough…"
I flipped my hairover my shoulders. "Well, you are my boyfriend."
He grabbed my hand while we trekked towards the parking lot. "I know, but your parents have been kind of strict lately."
"It's because they don't believe my story since they still think I'm the girl that lied about not getting expelled from middle school."
Javier stared at the scar on my right arm. "How did you get that?"
"It's not important."
"Okay. No problem."
A lump lingered in my throat. "You believe me, don't you?"
He rolled his eyes. "Yes, I do. But they never found the man Lucy."
"That's because they aren't looking in the right spot, and his name is the Wizard."
"Is he the one who gave you that scar?" Javier asked.
Javier had one thing going for him; he was more patient than my therapist.
But knowing the truth still mattered though because Javier and the rest of the world would find out, just not today, which was okay because nobody would take away my hope.
As for the wand, it remained hidden in a drawer at my parents' house because the Wizard was smart and must have had emergency magic even if his wand was gone. There was no doubt about it. He lurked in the background somewhere, waiting to prey on his next victim.
Chris Bedell's previous publishing credits include essays on the online magazine Thought Catalog, and 2 short stories on online literary magazines, which include "Surface Tension" on Crab Fat Literary Magazine and "A Little Accident" on Quail Bell Magazine.
weird birds. weird words.
fearless literature 2015 - 2018


However, if you're worried about if it'll work or not, you'll love My ATM Location Guarantee.
Because the trainings are digital however, we only offer the ATM Location Guarantee, we do not offer refunds of any kind.
There is no contract or no minimum for this monthly group coaching program. You may cancel anytime that you'd like and you'll never be billed for any future payment again. Just send an email to support when you're ready to cancel.
In order to obtain a full or partial refund under this policy, physical products should be shipped at your expense to the address listed below, must include your purchase information (such as a copy of your receipt) so that we can make a refund, include the date you attended ATM Business Blueprint LIVE! and proof that you've but what you learned into motion, and any physical products should be in merchantable condition (i.e. we could resell them because they're in good condition). Returned physical products are subject to a re-stocking fee equivalent to 0 % of the purchase price.
E-mail your refund requests to returns[at]atmbusinessblueprint[dot]com. Please include the words Refund Request in the subject line of your e-mail.
This Returns Policy was last updated on 08-08-2017.


1. Am I not free? Am I not an Apostle? Have I not seen Christ Jesus our Lord? Are you not my work in the Lord?
2. And if I am not an Apostle to others, yet still I am to you. For you are the seal of my Apostleship in the Lord.
4. Do we not have the authority to eat and to drink?
5. Do we not have the authority to travel around with a woman who is a sister, just as do the other Apostles, and the brothers of the Lord, and Cephas?
6. Or is it only myself and Barnabas who do not have the authority to act in this way?
7. Who has ever served as a soldier and paid his own stipend? Who plants a vineyard and does not eat from its produce? Who pastures a flock and does not drink from the milk of the flock?
8. Am I saying these things according to man? Or does the law not also say these things?
9. For it is written in the law of Moses: "You shall not bind the mouth of an ox, while it is treading out the grain." Is God here concerned with the oxen?
10. Or is he saying this, indeed, for our sake? These things were written specifically for us, because he who plows, ought to plow in hope, and he who threshes, too, in hope of receiving the produce.
11. If we have sown spiritual things in you, is it important if we harvest from your worldly things?
12. If others are sharers in this authority over you, why are we not more entitled? And yet we have not used this authority. Instead, we bear all things, lest we give any hindrance to the Gospel of Christ.
13. Do you not know that those who work in the holy place eat the things that are for the holy place, and that those who serve at the altar also share with the altar?
14. So, too, has the Lord ordained that those who announce the Gospel should live by the Gospel.
15. Yet I have used none of these things. And I have not written so that these things may be done for me. For it is better for me to die, rather than to let anyone empty out my glory.
16. For if I preach the Gospel, it is not glory for me. For an obligation has been laid upon me. And woe to me, if I do not preach the Gospel.
17. For if I do this willingly, I have a reward. But if I do this reluctantly, a dispensation is granted to me.
18. And what, then, would be my reward? So, when preaching the Gospel, I should give the Gospel without taking, so that I may not misuse my authority in the Gospel.
19. For when I was a free man to all, I made myself the servant of all, so that I might gain all the more.
20. And so, to the Jews, I became like a Jew, so that I might gain the Jews.
21. To those who are under the law, I became as if I were under the law, (though I was not under the law) so that I might gain those who were under the law. To those who were without the law, I became as if I were without the law, (though I was not without the law of God, being in the law of Christ) so that I might gain those who were without the law.
22. To the weak, I became weak, so that I might gain the weak. To all, I became all, so that I might save all.
23. And I do everything for the sake of the Gospel, so that I may become its partner.
24. Do you not know that, of those who run in a race, all of them, certainly, are runners, but only one achieves the prize. Similarly, you must run, so that you may achieve.
25. And one who competes in a contest abstains from all things. And they do this, of course, so that they may achieve a corruptible crown. But we do this, so that we may achieve what is incorruptible.
26. And so I run, but not with uncertainty. And so I fight, but not by flailing in the air.
27. Instead, I chastise my body, so as to redirect it into servitude. Otherwise, I might preach to others, but become myself an outcast.


text:
Detailed description (Plateau Mont-Royal)
The plateau is a large, mostly residential, district that as a lot of interesting neighbourhood restaurants and cafés. The Plateau is a very popular place to live or just hang out, I don't think you can mention Montreal withouth talking about the Plateau (it certainly grabs a lot of media attention). It started out at first as a working class neighbourhood (and like other districts, there were many small villages and cities in that area that merged with Montreal at the turn of the century) but in the 60's and 70's it became the place where many writers, singers and artists lived and worked. For this reason, the Plateau has been the main character in numerous songs, novels and movies, including Michel Tremblay and Yves Beauchemin's work. One of its obvious characteristic is that most appartement buildings are duplexes or triplexes (two or three stories high) and have outside stairs to the second floor (it's architectural trait of a lot of Montreal houses). Although you can find those in many other districts, there are mainly associated with the Plateau. This district is also the most densely populated district in Canada and, luckily, the district with the least number of cars in north america (bicycles are the prefered mode of transportation).
The Plateau is a great place for interesting restaurants and cafés as well as for shopping. It's main commercial streets are St-Denis street north of Roy street, Mont-Royal ave. and St-Laurent blvd. (but you could say that St-Laurent blvd. is a neighbourhood in itself).There are guided tours every sunday morning organized by the district's tourism office starting from the Mont-Royal metro station at around 10:00am (call (514) 848-0099 for more information or visit their web site).
The Plateau can be divided in three sections :
the Mile-End district (in the north-west section of the Plateau) the neighbourhood featured in Mordicai Richler's novels and some fillms including "Leolo" and "Lies My Father Told Me". The Mile-End is technically part of the Plateau but it is quite different. It is more ethnically diverse, with an important hassidic community to the north-east (Jeanne-Mance and Esplanade streets north of Fairmount) and portuguese community (see St-Louis district). It is also influenced by the municipality of Outremont to the west since there are not many stores and no bars there so most people from Outremont shop and hangout in the western section of the Mile-End (some realtors selling condos and houses in this section describe it as "Outremont-adjacent"). The Mile-End's main streets are Bernard street (for cafes like Romolo cafe or le Futenbulle and a new restaurant called Brasileirinho), St-Viateur (for St-Viateur Bagel, Zorba or Arahova's souvlakis, many small restaurants like Découvrir le Sénégal, Le Jardin du Curry, etc and new terrasses and upscale food shops, it would seem that St-Viateur has become very -too? - popular recently), St-Laurent blvd and its cluster of about a dozen restaurants and other stores between Laurier and St-Viateur streets), du Parc ave. with its persian carpet stores and authentic greek restaurants (Milos, Psarotaverna du Symporium, Mythos Ouzeri, Rodos, etc). P.S. the south-west section of the Mile-End district, between Mont-Royal ave. and St-Joseph blvd., was also known as the Annex.
the Plateau itself to to east (a good way to visit that part of the district is by using the bicycle path that goes from the corner of Cherrier and Berri streets then goes east to the Lafontaine park then north along Brébeuf and Christophe-Colomb to Laurier street). There's not much to see north-east of the corner of St-Joseph blvd. and St-Denis street, except for some small farm houses that seem to be the remnants of a small village around Berri street just north of Laurier street. The most interesting section are the streets around or close to the Lafontaine park (Mentana, Marie-Anne, Brébeuf, Chambord, etc). The main commercial streets are St-Denis street (gift shops and decorative object stores, lot's of restaurants and cafes, etc) and Mont-Royal ave. (it used to be a neighbourhood commercial street with everything-for-a-dollar stores and used record stores but it has changed the past few years, there are a lot more restaurants, cafes and upscale food stores).
the St-Jean-Baptiste and St-Louis districts, a "transition" zone between the the latin quarter, the Mile-End (south east section of the Plateau) and the rest of the Plateau. It is quite varied in every sense since it takes from each neighbouring district. The portuguese community has many stores and cultural institutions in the north-western part of St-Jean-Baptiste (see Mile-End). The community's main church is on the corner of St-Urbain and Rachel and the "Parc du Portugal" is on the corner of Marie-Anne and St-Laurent blvd. Many portuguese restaurants and stores can be found around these places. Other well known spots are Duluth street and on St-Laurent blvd., the Balattou (live african and world beat music dance club) and Schwartz's deli (smoked meat).
I could add St-Laurent blvd. as a district in itself because of its distinctive features and importance. While St-Laurent was considered to separate the french community to the east and english community to the west (that line is now quite blurred), St-Laurent blvd. and its surroundings could be considered to be the center of an "international" zone between those two sections.
Surrounding streets :
Sherbrooke to the south
Van Horne and Rosemont blvd. to the north
Frontenac to the east
Hutchison to the west
Metro stations :


text:
\section{Introduction}

With machine learning applications becoming ubiquitous in modern-day life, there exists an increasing concern about the robustness of the deployed models. Since first reported in \citep{szegedy2013intriguing,goodfellow2014explaining,biggio2013evasion}, these \emph{adversarial attacks} are small perturbations of the input, imperceptible to the human eye, which can nonetheless completely fluster otherwise well-performing systems. Because of clear security implications \citep{darpa}, this phenomenon has sparked an increasing amount of work dedicated to devising defense strategies \citep{metzen2017detecting,gu2014towards,madry2017towards} and correspondingly more sophisticated attacks \citep{carlini2017adversarial,athalye2018obfuscated,tramer2020adaptive}, with each group trying to triumph over the other in an arms-race of sorts.

A different line of research attempts to understand adversarial examples from a theoretical standpoint. Some works have focused
on giving robustness certificates, thus providing a guarantee to withstand the attack of an adversary under certain assumptions \citep{cohen2019certified,raghunathan2018certified,wong2017provable}. Other works address questions of learnabiltiy \citep{shafahi2018adversarial,cullina2018pac,bubeck2018adversarial,tsipras2018robustness} or sample complexity \citep{schmidt2018adversarially,yin2018rademacher,tu2019theoretical}, in the hope of better characterizing the increased difficulty of learning hypotheses that are robust to adversarial attacks.
While many of these results are promising, the analysis is often limited to simple models.

Here, we strike a better balance by considering a model that involves learning a representation while at the same time giving a precise generalization bound and a robustness certificate.
In particular, we focus our attention on the adversarial robustness of the supervised sparse coding model \citep{mairal2011task}, or task-driven dictionary learning, consisting of a linear classifier acting on the representation computed via a supervised sparse encoder.
We show an interesting interplay between the expressivity and stability of a (supervised) representation map and a notion of margin in the feature space.
The idea of employing sparse representations as data-driven features for supervised learning goes back to the early days of deep learning \citep{coates2011importance,kavukcuoglu2010fast,zeiler2010deconvolutional,ranzato2007unsupervised}, and has had a significant impact on applications in computer vision and machine learning \citep{wright2010sparse,henaff2011unsupervised,mairal2008discriminative,mairal2007sparse,gu2014projective}. More recently, new connections between deep networks and sparse representations were formalized by~\cite{papyan2018theoretical}, which further helped deriving stability guarantees \citep{papyan2017working}, providing architecture search strategies and analysis \citep{tolooshams2019deep,murdock2020dataless,sulam2019multi}, and other theoretical insights \citep{xin2016maximal,aberdam2019multi,aghasi2020fast,aberdam2020ada,moreau2016understanding}.
While some recent work has leveraged the stability properties of these latent representations to provide robustness guarantees against adversarial attacks \citep{romano2019adversarial}, these rely on rather stringent generative model assumptions that are difficult to be satisfied and verified in practice. In contrast, our assumptions rely on the existence of a positive \emph{gap} in the encoded features, as proposed originally by  \cite{mehta2013sparsity}. This distributional assumption is significantly milder -- it is directly satisfied by making traditional sparse generative model assumptions -- and can be directly quantified from data.

This work makes two main contributions: The first is a bound on the robust risk of hypotheses that achieve a mild encoder gap assumption, where the adversarial corruptions are bounded in $\ell_2$-norm.
Our proof technique follows a standard argument based on a minimal $\epsilon$-cover of the parameter space, dating back to \cite{vapnik1971uniform} and adapted for matrix factorization and dictionary learning problems in \cite{gribonval2015sample}. However, the analysis of the Lipschitz continuity of the adversarial loss with respect to the model parameters is considerably more involved. The increase in the sample complexity is mild with adversarial corruptions of size $\nu$ manifesting as an additional term of order $\mathcal{O}\left((1+\nu)^2/m\right)$ in the bound, where $m$ is the number of samples, and a minimal encoder gap of $\mathcal{O}(\nu)$ is necessary. Much of our results extend directly to other supervised learning problems (e.g. regression).
Our second contribution is a robustness certificate that holds for every hypothesis in the function class for $\ell_2$ perturbations for multiclass classification. In a nutshell, this result guarantees that the label produced by the hypothesis will not change if the encoder gap is \emph{large enough} relative to the energy of the adversary, the classifier margin, and properties of the model (e.g. dictionary incoherence).

\section{Preliminaries and Background} \label{sec:Preliminiaries}
In this section, we first describe our notation and the learning problem, and then proceed to situate our contribution in relation to prior work.

Consider the spaces of inputs, $\mathcal{X}\subseteq {B}_{\mathbb{R}^d}$, i.e. the unit ball in $\mathbb{R}^d$, and labels, $\mathcal{Y}$. Much of our analysis is applicable to a broad class of label spaces, but we will focus on binary and multi-class classification setting in particular. We assume that the data is sampled according to some unknown distribution $P$ over $\mathcal X \times \mathcal Y$.
Let $\mathcal{H} = \{f:\mathcal{X}\to\mathcal{Y}'\}$ denote a hypothesis class mapping inputs into some output space $\mathcal{Y}' \subseteq \mathbb{R}$.
Of particular interest to us are norm-bounded linear predictors, $f(\cdot)=\langle {\mathbf w}, \cdot \rangle$, parametrized by $d$-dimensional vectors ${\mathbf w}\in \mathcal W = \{{\mathbf w} \in \mathbb{R}^d : \|{\mathbf w}\|_2\leq B\}$.

From a learning perspective, we have a considerable understanding of the linear hypothesis class, both in a stochastic non-adversarial setting as well as in an adversarial context~ \citep{charles2019convergence,li2019inductive}. However, from an application standpoint, linear predictors are often too limited, and rarely applied directly on input features. Instead, most state-of-the-art systems involve learning a representation. In general, an \emph{encoder} map $\varphi:\mathcal{X}\to\mathcal{Z} \subseteq \mathbb{R}^p$, parameterized by parameters $\theta$, is composed with a linear function so that $f({\mathbf x}) = \langle {\mathbf w}, \varphi_\theta({\mathbf x}) \rangle$, for ${\mathbf w}\in\mathbb{R}^p$. This description applies to a large variety of popular models, including kernel-methods, multilayer perceptrons and deep convolutional neural networks.
Herein we focus on an encoder given as the solution to a Lasso problem \citep{tibshirani1996regression}. More precisely, we  consider $\varphi_{\mathbf D}({\mathbf x}):\mathbb{R}^d\to \mathbb{R}^p$, defined by
\begin{equation}
    \label{eqn:enc-1}
\varphi_{\mathbf D}({\mathbf x}) \coloneqq \arg\min_{\mathbf z} \frac{1}{2} \|{\mathbf x} - {\mathbf D}{\mathbf z}\|^2_2 +\lambda \|{\mathbf z}\|_1.
\end{equation}
Note that when ${\mathbf D}$ is overcomplete, i.e. $p>d$, this problem is not strongly convex. Nonetheless, we will assume that that solution to Problem~\ref{eqn:enc-1} is unique\footnote{The solution is unique under mild assumptions \citep{tibshirani2013lasso}, and otherwise our results hold for any solution returned by a deterministic solver.}, and study the hypothesis class given by $ \mathcal{H} = \{ f_{{\mathbf D},{\mathbf w}}({\mathbf x}) = \langle {\mathbf w}, \varphi_{\mathbf D}({\mathbf x}) \rangle : {\mathbf w}\in \mathcal{W}, {\mathbf D} \in \mathcal{D} \}$, where $\mathcal{W} = \{{\mathbf w} \in \mathbb{R}^p : \|{\mathbf w}\|_2\leq B\}$, and $\mathcal{D}$ is the oblique manifold of all matrices with unit-norm columns (or \emph{atoms}); i.e. $\mathcal{D} =\{ {\mathbf D} \in \mathbb{R}^{d\times p} : \|{\mathbf D}_i\|_2 = 1 ~ \forall i \in [p] \}$.
While not explicit in our notation, $\enc[{\mathbf D}]{({\mathbf x})}$ depends on the value of $\lambda$. For notational simplicity, we also suppress subscripts $({\mathbf D},{\mathbf w})$ in $f_{{\mathbf D},{\mathbf w}}(\cdot)$ and simply write $f(\cdot)$.

We consider a bounded loss function $\ell: \mathcal{Y}\times  \mathcal{Y}' \to [0,b]$, with Lipschitz constant $L_\ell$.
The goal of learning is to find an $f \in \mathcal{H}$ with  minimal risk, or expected loss, $R(f) = \textstyle\mathbb{E}_{({\mathbf x},y)\sim P} \left[ \ell(y,f({\mathbf x})) \right]$. Given a sample $S=\{({\mathbf x}_i,y_i)\}_{i=1}^m$, drawn i.i.d. from $P$, a popular learning algorithm is empirical risk minimization (ERM) which involves finding $f_{{\mathbf D}, {\mathbf w}}$ that solves the following problem: \vspace{-.1cm}
\begin{equation}
  \min_{{\mathbf D},{\mathbf w}}~ \frac{1}{m} \sum_{i=1}^{m} \ell(y_i,f_{{\mathbf D},{\mathbf w}}({\mathbf x}_i)). \nonumber
\end{equation}\vspace{-.1cm}

\textbf{Adversarial Learning.} In an adversarial setting, we are interested in hypotheses that are robust to adversarial perturbations of inputs. We focus on \emph{evasion attacks}, in which an attack is deployed at test time (while the training samples are not tampered with). As a result, a more appropriate loss that incorporates the robustness to such contamination is the robust loss \citep{madry2017towards},  $\tilde{\ell}_\nu(y,f({\mathbf x})) \coloneqq \max_{{\mathbf v}\in\Delta_\nu} ~ \ell(y,f({\mathbf x}+{\mathbf v}))$,
where $\Delta$ is some subset of $\mathbb{R}^d$ that restricts the {power} of the adversary. Herein we focus on $\ell_2$ norm-bounded corruptions, $\Delta _\nu = \{{\mathbf v} \in \mathbb{R}^d ~ : ~ \|{\mathbf v}\|_2\leq \nu \}$, and denote by $\tilde{R}_S(f) = \frac{1}{m} \sum_{i=1}^{m} \tilde{\ell}_\nu(y_i,f({\mathbf x}_i))$ the empirical robust risk of $f$ and  $\tilde{R}(f) = \mathbb{E}_{({\mathbf x},y)\sim P}[ \tilde{\ell}_\nu(y,f({\mathbf x}))]$ its population robust risk w.r.t. distribution $P$.

\textbf{Main Assumptions.} We make two general assumptions throughout this work.
First, we assume that the dictionaries in $\mathcal{D}$ are $s$-incoherent, i.e, they satisfy a restricted isometry property (RIP). More precisely, for any $s$-sparse vector, ${\mathbf z} \in \mathbb R^p$ with $\|{\mathbf z}\|_0 = s$, there exists a minimal constant $\eta_s<1$ so that ${\mathbf D}$ is close to an isometry, i.e.  $(1-\eta_s)\|{\mathbf z}\|^2_2 \leq \|{\mathbf D}{\mathbf z}\|^2_2 \leq (1 + \eta_s) \|{\mathbf z}\|^2_2$. Broad classes of matrices are known to satisfy this property (e.g. sub-Gaussian matrices \citep{foucart2017mathematical}), although empirically computing this constant for a fixed (deterministic) matrix is generally intractable. Nonetheless, this quantity can be upper bounded by the correlation between columns of ${\mathbf D}$, either via mutual coherence \citep{donoho2003optimally} or the Babel function \citep{tropp2003improved}, both easily computed in practice.

Second, we assume that the map $\enc[{\mathbf D}]$ induces a positive \emph{encoder gap} on the computed features. Given a sample ${\mathbf x}\in\mathcal{X}$ and its encoding, $\enc[{\mathbf D}]{({\mathbf x})}$, we denote by $\Lambda^{p-s}$ the set of atoms of cardinality $(p-s)$, i.e., $\Lambda^{p-s} = \{ \mathcal{I} \subseteq \{1,\dots,p\} : |\mathcal{I}|=p-s \}$. The encoder gap $\tau_s(\cdot)$ induced by $\enc[{\mathbf D}]$ on any sample ${\mathbf x}$ is defined \citep{mehta2013sparsity} as
\begin{equation}
   \tau_s({\mathbf x}) \coloneqq \max_{\mathcal{I}\in\Lambda^{p-s}} \min_{i \in \mathcal{I}} ~\left( \lambda - | \langle {\mathbf D}_i , {\mathbf x} - {\mathbf D}\enc[{\mathbf D}]({\mathbf x}) \rangle | \right).  \nonumber
\end{equation}
{An equivalent and conceptually simpler definition for $\tau_s({\mathbf x})$ is the $(s+1)^{th}$ smallest entry in the  vector $\lambda\mathbf{1} - | \langle {\mathbf D} , {\mathbf x} - {\mathbf D}\enc[{\mathbf D}]({\mathbf x}) \rangle |$. Intuitively, this quantity can be viewed as a measure of maximal energy along any dictionary atom that is not in the support of an input vector.}
More precisely, recall from the optimality conditions of Problem \eqref{eqn:enc-1} that $|{\mathbf D}_i^T({\mathbf x}-{\mathbf D}\enc[{\mathbf D}]{({\mathbf x})})|=\lambda$ if $[\enc[{\mathbf D}]{({\mathbf x})}]_i\neq 0$, and $|{\mathbf D}_i^T({\mathbf x}-{\mathbf D}\enc[{\mathbf D}]{({\mathbf x})})|\leq\lambda$ otherwise. Therefore, if $\tau_s$ is large, this indicates that there exist a set $\mathcal{I}$ of $(p-s)$ atoms that are \emph{far} from entering the support of $\enc[{\mathbf D}]{({\mathbf x})}$.
If $\enc[{\mathbf D}]({\mathbf x})$ has exactly $k$ non-zero entries, we may choose some $s>k$ to obtain $\tau_s({\mathbf x})$. In general, $\tau_s(\cdot)$ depends on the energy of the residual, ${\mathbf x}-{\mathbf D}\enc[{\mathbf D}]{({\mathbf x})}$, the correlation between the atoms, the parameter $\lambda$, and the cardinality $s$.
In a nutshell, if a dictionary ${\mathbf D}$ provides a quickly decaying approximation error as a function of the cardinality $s$, then a positive encoder gap exists for some $s$.

{We consider dictionaries that induce a positive encoder gap in every input sample from a dataset, and define the minimum such margin as $\tau_s^*:=\min_{i\in[m]} \tau_s({\mathbf x}_i) > 0.$}
Such a positive encoder exist for quite general distributions, such as $s$-sparse and approximately sparse signals. However, this definition is more general and it will allow us to avoid making any other stronger distributional assumptions. We now illustrate such the encoder gap with both analytic and numerical examples\footnote{Code to reproduce all of our experiments is made available at \href{https://github.com/Sulam-Group/Adversarial-Robust-Supervised-Sparse-Coding}{our github repository.}}.

\textbf{Approximate $k$-sparse signals} Consider signals ${\mathbf x}$ obtained as ${\mathbf x} = {\mathbf D}{\mathbf z} + {\mathbf v}$, where ${\mathbf D}\in\mathcal D$, $\|{\mathbf v}\|_2\leq\nu$ and ${\mathbf z}$ is sampled from a distribution of sparse vectors with up to $k$ non-zeros, with $k < \frac{1}{3}\left(1+\frac{1}{\mu({\mathbf D})}\right)$, where $\mu({\mathbf D}) = \max_{i\neq j}\langle {\mathbf D}_i , {\mathbf D}_j \rangle$ is the mutual coherence of ${\mathbf D}$. Then, for a particular choice of $\lambda$, we have that $\tau_s({\mathbf x}) > \lambda - \frac{15\mu\nu}{2}, \forall s>k $. This can be shown using standard results in \citep{tropp2006just}; we defer the proof to the Appendix \ref{supp:EncoderGapExample}.
Different values of $\lambda$ provide different values of $\tau_s({\mathbf x})$. To illustrate this trade-off, we generate synthetic approximately $k$-sparse signals ($k=15$) from a dictionary with 120 atoms in 100 dimensions and contaminate them with Gaussian noise. We then numerically compute the value of $\tau^*_s$ as a function of $s$ for different values of $\lambda$, and present the results in \autoref{fig:encoder_gap_synthetic}.

\textbf{Image data} We now demonstrate that a positive encoder exist for natural images as well. In \autoref{fig:encoder_gap_mnist} we similarly depict the value of $\tau_s(\cdot)$, as a function of $s$, for an encoder computed on MNIST digits and {CIFAR images} (from a validation set) with learned dictionaries (further details in \autoref{Sec:Experiments}).

{In summary, the encoder gap is a measure of the ability of a dictionary to sparsely represent data, and one can induce a larger encoder gap by increasing the regularization parameter or the cardinality $s$.
As we will shortly see, this will provide us with a a controllable knob in our generalization bound.}

\begin{figure}
\centering
\subcaptionbox{\label{fig:encoder_gap_synthetic}}
{\includegraphics[width = .32\textwidth,trim = 30 0 30 20]{figures/encoder_gap_synthetic.pdf}}
\subcaptionbox{\label{fig:encoder_gap_mnist}}
{\includegraphics[width = .32\textwidth,trim = 30 0 30 20]{figures/encoder_gap_mnist_trans.pdf}
}
\subcaptionbox{\label{fig:encoder_gap_cifar}}
{\includegraphics[width = .32\textwidth,trim = 30 0 30 20]{figures/encoder_gap_cifar_supervised.pdf}
}
\caption{ Encoder gap, $\tau^*_s$, for synthetic approximately sparse signals (a) MNIST digits (b) and CIFAR10 images (c). \vspace{-.3cm}}
\label{fig:Encoder_Gap}
\end{figure}

\vspace{-5pt}
\section{Prior Work} \label{sec:prior_work}
Many results exist on the approximation power and stability of Lasso (see \citep{foucart2017mathematical}), which most commonly rely on assuming data is (approximately) $k$-sparse under a given dictionary. As explained in the previous section, we instead follow an analysis inspired by \cite{mehta2013sparsity}, which relies on the encoder gap. \cite{mehta2013sparsity} leverage encoder gap to derive a generalization bound for the supervised sparse coding model in a stochastic  (non-adversarial) setting. Their result, which follows a symmetrization technique~\citep{mendelson2004importance}, scales as $\tilde{\mathcal{O}} (\sqrt{(dp + \log(1/\delta))/m}$, and requires  a minimal number of samples that is $\mathcal O (1/(\tau_s\lambda))$. In contrast, we study an generalization in the adversarially robust setting, detailed above. Our analysis is based on an $\epsilon$-cover of the parameter space and on analyzing a local-Lipschitz property of the adversarial loss. The proof of our generalization bound is simpler, and shows a mild deterioration of the upper bound on the generalization gap due to adversarial corruption.

Our work is also inspired by the line of work initiated by~\cite{papyan2017convolutional} who regard the representations computed by neural networks as approximations for those computed by a Lasso encoder across different layers. In fact, a first analysis of adversarial robustness for such a model is presented by \cite{romano2019adversarial}; however, they make strong generative model assumptions and thus their results are not applicable to real-data practical scenarios. Our robustness certificate mirrors the analysis from the former work, though leveraging a more general and new stability bound (\autoref{lemma:stability_of_rep_adversarial}) relying instead on the existence of positive encoder gap. In a related work, and in the context of neural networks, \cite{cisse2017parseval} propose a regularization term inspired by Parseval frames, with the empirical motivation of improving adversarial robustness. Their regularization term can in fact be related to minimizing the (average) mutual coherence of the dictionaries, which naturally arises as a control for the generalization gap in our analysis.

Lastly, several works have employed sparsity as a beneficial property in adversarial learning \citep{marzi2018sparsity,demontis2016security}, with little or no theoretical analysis, or in different frameworks (e.g. sparse weights in deep networks \citep{guo2018sparse,balda2019adversarial}, or on different domains \citep{bafna2018thwarting}). Our setting is markedly different from that of \cite{chen2013robust} who study adversarial robustness of Lasso as a sparse predictor directly on input features. In contrast, the model we study here employs Lasso as an encoder with a data-dependent dictionary, on which a linear hypothesis is applied. A few works have recently begun to analyze the effect of learned representations in an adversarial learning setting \citep{ilyas2019adversarial,allen2020feature}. Adding to that line of work, our analysis demonstrates that benefits can be provided by exploiting a trade-off between expressivity and stability of the computed representations, and the classifier margin.

\vspace{-10pt}
\section{Generalization bound for robust risk}
\label{sec:GenBound}
In this section, {we present a bound on the robust risk for models satisfying a positive encoder gap}. Recall that given a $b$-bounded loss $\ell$ with Lipschitz constant $L_\ell$,  $\tilde{R}_S(f) = \frac{1}{m} \sum_{i=1}^{m} \tilde{\ell}_\nu(y_i,f({\mathbf x}_i))$ is the empirical robust risk, and $\tilde{R}(f) = \mathbb{E}_{({\mathbf x},y)\sim P} \big[ \tilde{\ell}_\nu(y,f({\mathbf x})) \big]$ is the population robust risk w.r.t. distribution $P$. Adversarial perturbations are bounded in $\ell_2$ norm by $\nu$. {Our main result below guarantees that if a hypothesis $f_{{\mathbf D},{\mathbf w}}$ is found with a sufficiently large encoder gap, and a large enough training set, its generalization gap is bounded as $\tilde{\mathcal O}\Big(b\sqrt{\frac{ (d+1)p}{m}}\Big)$, where $\tilde{\mathcal O}$ ignores  poly-logarithmic factors.}

\begin{theorem}
\label{thm:generalization} Let $\mathcal W = \{{\mathbf w} \in \mathbb R^p : \|{\mathbf w}\|_2 \leq B \}$,  and $\mathcal{D}$ be the set of column-normalized dictionaries with $p$ columns and with RIP at most $\eta^*_s$. Let $\mathcal{H} = \{ f_{{\mathbf D},{\mathbf w}}({\mathbf x}) = \langle {\mathbf w}, \varphi_{\mathbf D}({\mathbf x}) \rangle : {\mathbf w}\in \mathcal{W}, {\mathbf D} \in \mathcal{D} \}$. Denote $\tau^*_s$ the minimal encoder gap over the $m$ samples. Then, with probability at least $1-\delta$ over the draw of the $m$ samples, the generalization gap for any hypothesis $f\in\mathcal H$ that achieves an encoder gap on the samples of $\tau_s^*>2\nu$, satisfies
\begin{multline}
      \left| \tilde{R}_S(f)\! -\! \tilde{R}(f) \right| \!\leq \frac{b}{\sqrt{m}} \left(  (d+1) p \log\left(\frac{3m}{2\lambda (1-\eta^*_s)}\right) + p\log(B) + \log{\frac{4}{\delta}} \right)^{\frac{1}{2}} \\ + b\sqrt{\frac{2\log(m/2)+2\log(2/\delta)}{m}}  + 12\frac{(1+\nu)^2 L_\ell B \sqrt{s} }{m} , \nonumber
\end{multline}
as long as $m > \frac{\lambda(1-\eta_s)}{(\tau_s^* - 2\nu)^2} K_\lambda$,
where $K_\lambda = \left(  2 \left( 1+\frac{1+\nu}{2\lambda} \right) + \frac{5 (1+\nu)}{\sqrt{\lambda}}  \right)^2$.
\end{theorem}

A few remarks are in order. First, note that adversarial generalization incurs a polynomial dependence on the adversarial perturbation $\nu$. This is mild, especially since it only affects the fast $\mathcal O(1/m)$ term. Second, the bound requires a minimal number of samples. Such a requirement is intrinsic to the stability of Lasso (see \autoref{lemma:advers_D_stable} below) and it exists also in the non-adversarial setting \citep{mehta2013sparsity}. In the adversarial case, this requirement becomes more demanding, as reflected by the term $(\tau^*_s-2\nu$) in the denominator. Moreover, a minimal encoder gap $\tau_s^*>2\nu$ is needed as well.

\autoref{thm:generalization} suggests an interesting trade-off. One can obtain a large $\tau^*_s$ by increasing $\lambda$ and $s$ -- as demonstrated in in \autoref{fig:Encoder_Gap}. But increasing $\lambda$ may come at an expense of hurting the empirical error, while increasing $s$ makes the term $1-\eta_s$ smaller. {Therefore, if one obtains a model with small training error, along with large $\tau^*_s$ over the training samples for an appropriate choice of $\lambda$ and $s$ while ensuring that $\eta_s$ is bounded away from $1$,
then $f_{{\mathbf D},{\mathbf w}}$ is guaranteed to generalize well. Furthermore, note that the excess error depends mildly (poly logarithmically) on $\lambda$ and $\eta_s$.}

Our proof technique is based on a minimal $\epsilon$-cover of the parameter space, and the full proof is included in the
Appendix~\ref{supp:Generalization}. Special care is needed to ensure that the encoder gap of the dictionary holds for a sample drawn from the population, as we can only measure this gap on the provided $m$ samples. To address this, we split the data equally into a training set and a development set: the former is used to learn the dictionary, and the latter to provide a high probability bound on the event that $\tau_s({\mathbf x})>\tau^*_s$. This is to ensure that the random samples of the encoder margin are i.i.d. for measure concentration. Ideally, we would like to utilize the entire dataset for learning the predictor; we leave that for future work.

While most of the techniques we use are standard~\footnote{See \citep{seibert2019sample} for a comprehensive review on these tools in matrix factorization problems.}, the Lipschitz continuity of the robust loss function requires a more delicate analysis. For that, we have the following result.

\begin{lemma}[Parameter adversarial stability] \label{lemma:advers_D_stable}
Let ${\mathbf D},{\mathbf D}' \in \mathcal D$. If $\|{\mathbf D}-{\mathbf D}'\|_2 \leq \epsilon\leq 2 \lambda/(1+\nu)^2$,  then
    \begin{equation} \label{eq:adver_perturb_stability}
       \max_{{\mathbf v}\in\Delta} \|\enc[{\mathbf D}]({\mathbf x}+{\mathbf v}) -  \enc[{\mathbf D}']({\mathbf x}+{\mathbf v}) \|_2 \leq \gamma (1+\nu)^2 \epsilon,
    \end{equation}
    with $\gamma = \frac{3}{2} \frac{\sqrt{s}}{\lambda(1-\eta_s)}$, as long as $\tau_s({\mathbf x}) \geq 2 \nu +  \sqrt{\epsilon}\left( \sqrt{\frac{25}{\lambda}}(1+\nu) + 2 \left( \frac{(1+\nu)}{\lambda} + 1\right) \right).$
\end{lemma}
\autoref{lemma:advers_D_stable} is central to our proof, as it provides a bound on difference between the features computed by the encoder under model deviations. Note that the condition on the minimal encoder gap, $\tau_s({\mathbf x})$, puts an upper bound on the distance between models ${\mathbf D}$ and ${\mathbf D}'$. This in turn results in the condition imposed on the minimal samples in \autoref{thm:generalization}. It is worth stressing that the lower bound on $\tau_s({\mathbf x})$ is on the \emph{unperturbed} encoder gap -- that which can be evaluated on the samples from the dataset, without the need of the adversary. We defer the proof of this Lemma to
Appendix~\ref{supp:ParameterAdversarialStability}.

\section{Robustness Certificate}
\label{sec:certificates}

Next, we turn to address an equally important question about robust adversarial learning, that of giving a formal certification of robustness. Formally, we would like to guarantee that the output of the trained model, $f_{{\mathbf D},{\mathbf w}}({\mathbf x})$, does not change for norm-bounded adversarial perturbations of a certain size.
Our second main result provides such a certificate for the supervised sparse coding model.

Here, we consider a multiclass classification setting with $y\in\{1,\dots,K\}$; simplified results for binary classification are included in Appendix \ref{supp:certificate}. The hypothesis class is parameterized as $f_{{\mathbf D},{\mathbf W}}({\mathbf x}) = {\mathbf W}^T\enc[{\mathbf D}]({\mathbf x})$, with ${\mathbf W} = [{\mathbf W}_1, {\mathbf W}_2, \ldots, {\mathbf W}_K] \in\mathbb{R}^{p\times K}$. The multiclass margin is defined as follows:
\[ \rho_{\mathbf x} = {\mathbf W}^T_{y_i} \enc[{\mathbf D}]({\mathbf x}) - \max_{j\neq y_i} {\mathbf W}^T_{j} \enc[{\mathbf D}]({\mathbf x}). \]
We show the following result.
\begin{theorem}[Robustness certificate for multiclass supervised sparse coding] \label{thm:multiclass_certificate}
Let $\rho_x > 0$ be the multiclass classifier margin of $f_{{\mathbf D},{\mathbf w}}({\mathbf x})$ composed of an encoder with a gap of $\tau_s({\mathbf x})$ and a dictionary, ${\mathbf D}$, with RIP constant $\eta_s<1$. Let $c_{\mathbf W} := \max_{i\neq j}\|{\mathbf W}_i-{\mathbf W}_j\|_2$. Then,
    \begin{equation}
        \arg\max_{j\in[K]} ~[{\mathbf W}^T \enc[{\mathbf D}]({\mathbf x})]_j\ = \arg\max_{j\in[K]}~ [{\mathbf W}^T \enc[{\mathbf D}]({\mathbf x}+{\mathbf v})]_j,\quad \forall~ {\mathbf v}:\|{\mathbf v}\|_2\leq \nu,
    \end{equation}
    so long as $\nu \leq \min\{ \tau_s({\mathbf x})/2 , \rho_{\mathbf x} \sqrt{1-\eta_s} / c_{\mathbf W} \}.$
\end{theorem}

\autoref{thm:multiclass_certificate} clearly captures the potential contamination on two flanks: robustness can no longer be guaranteed as soon as the energy of the perturbation is enough to either significantly modify the computed representation \emph{or} to induce a perturbation larger than the classifier margin on the feature space.
Proof of \autoref{thm:multiclass_certificate}, detailed in Appendix \ref{supp:certificate}, relies on the following lemma showing that under an encoder gap assumption, the computed features are moderately affected despite adversarial corruptions of the input vector.
\begin{lemma}[Stability of representations under adversarial perturbations] \label{lemma:stability_of_rep_adversarial}
Let ${\mathbf D}$ be a dictionary with RIP constant $\eta_s$. Then, for any ${\mathbf x} \in \mathcal X$ and its additive perturbation ${\mathbf x}+{\mathbf v}$, for any $\|{\mathbf v}\|_2\leq \nu$, if $\tau_s({\mathbf x}) >  2 \nu$, then we have that
\begin{equation}
    \| \enc[{\mathbf D}]{({\mathbf x})} - \enc[{\mathbf D}]{({\mathbf x}+{\mathbf v})} \|_2 \leq \frac{\nu}{\sqrt{1-\eta_s}}.
\end{equation}
\end{lemma}
An extensive set of results exist for the stability of the solution provided by Lasso relying generative model assumptions \citep{foucart2017mathematical,elad2010sparse}. The novelty of Lemma~\ref{lemma:stability_of_rep_adversarial} is in replacing such an assumption with the existence of a positive encoder gap on $\enc[{\mathbf D}]({\mathbf x})$.

Going back to \autoref{thm:multiclass_certificate}, note that the upper bound on $\nu$ depends on the RIP constant $\eta_s$, which is not computable for a given (deterministic) matrix ${\mathbf D}$. Yet, this result can be naturally relaxed by upper bounding $\eta_s$ with measures of correlation between the atoms, such as the mutual coherence. This quantity provides a measure of the worst correlation between two atoms in the dictionary ${\mathbf D}$, and it is defined as $\mu({\mathbf D}) = \max_{i\neq j} | \langle {\mathbf D}_i , {\mathbf D}_j \rangle|$ (for ${\mathbf D}$ with normalized columns). For general (overcomplete and full rank) dictionaries, clearly $0<\mu({\mathbf D})\leq 1$.

While conceptually simple, results that use $\mu({\mathbf D})$ tend to be too conservative. Tighter bounds on $\eta_s$ can be provided by the Babel function\footnote{Let $\Lambda$ denote subsets (supports) of $\{1,2,\dots,p\}$. Then, the Babel function is defined as \mbox{$\mu_{(s)} = \max_{\Lambda:|\Lambda|=s} \max_{j\notin \Lambda} \sum_{i\in\Lambda} |\langle {\mathbf D}_i , {\mathbf D}_j \rangle|$.}}, $\mu_{(s)}$, which quantifies the maximum correlation between an atom and \emph{any other} collection of $s$ atoms in ${\mathbf D}$. It can be shown \citep[Chapter 2]{tropp2003improved,elad2010sparse} that $\eta_s\leq \mu_{(s-1)}\leq(s-1)\mu({\mathbf D})$. Therefore, we have the following:
\begin{corollary} \label{cor:certificate_babel_multiclass}
Under the same assumptions as those in \autoref{thm:multiclass_certificate},
\begin{equation}
       \arg\max_{j\in[K]} ~[{\mathbf W}^T \enc[{\mathbf D}]({\mathbf x})]_j\ = \arg\max_{j\in[K]}~ [{\mathbf W}^T \enc[{\mathbf D}]({\mathbf x}+{\mathbf v})]_j,\quad \forall~ {\mathbf v}:\|{\mathbf v}\|_2\leq \nu
    \end{equation}
    so long as $\nu \leq \min\{ \tau_s({\mathbf x})/2 , \rho_{\mathbf x} \sqrt{1-\mu_{(s-1)}}/c_{\mathbf W} \}$.
\end{corollary}
Although
the condition on $\nu$ in the corollary above is stricter (and the bound looser), the quantities involved can easily be computed numerically leading to practical useful bounds, as we see~next.

\vspace*{-5pt}
\section{Experiments}
\label{Sec:Experiments}

In this section, we illustrate the robustness certificate guarantees
both in synthetic and real data, as well as the trade-offs between constants in our sample complexity result. First, we construct samples from a separable binary distribution of $k$-sparse signals. To this end, we employ a dictionary with 120 atoms in 100 dimensions with a mutual coherence of 0.054. Sparse representations ${\mathbf z}$ are constructed by first drawing their support (with cardinality $k$) uniformly at random, and drawing its non-zero entries from a uniform distribution away from zero. Samples are obtained as ${\mathbf x} = {\mathbf D}{\mathbf z}$, and normalized to unit norm. We finally enforce separability by drawing ${\mathbf w}$ at random from the unit ball, determining the labels as $y=\text{sign}( {\mathbf w}^T\enc[{\mathbf D}]{({\mathbf x})})$, and discarding samples with a margin $\rho$ smaller than a pre-specified amount (0.05 in this case). Because of the separable construction, the accuracy of the resulting classifier is 1.

We then attack the obtained model employing the projected gradient descent method \citep{madry2017towards}, and analyze the degradation in accuracy as a function of the energy budget $\nu$. We compare this empirical performance with the bound in  \autoref{cor:certificate_babel_multiclass}: given the obtained margin, $\rho$, and the dictionary's $\mu_{s}$, we can compute the maximal certified radius for a sample ${\mathbf x}$ as
\begin{equation} \label{eq:computing_bound}
\nu({\mathbf x}) = \max_s \min\{ \tau_s({\mathbf x})/2 , \rho_{\mathbf x} \sqrt{1-\mu_{(s-1)}}/c_{\mathbf W}\}.
\end{equation}
{For a given dataset, we can compute the minimal certified radius over the samples, $\nu^* = \min_{i\in[n]} \nu({\mathbf x}_i)$.} This is the bound depicted in the vertical line in \autoref{fig:synthetic_separable}. As can be seen, despite being somewhat loose, the attacks do not change the label of the samples, thus preserving the accuracy.

In non-separable distributions, one may study how the accuracy depends on the \emph{soft margin} of the classifier. In this way, one can determine a target margin that results in, say, 75\% accuracy on a validation set. One can obtain a corresponding certified radius of $\nu^*$ as before, which will guarantee that the accuracy will not drop below 75\% as long as $\nu<\nu^*$. This is illustrated in \autoref{fig:synthetic_non_separable}.

An alternative way of employing our results from \autoref{sec:certificates} is by studying the \emph{certified accuracy} achieved by the resulting hypothesis.
The certified accuracy quantifies the percentage of samples in a test set that are classified correctly while being \emph{certifiable}. In our context, this implies that a sample ${\mathbf x}$ achieves a margin of $\rho_{\mathbf x}$, for which a certified radius of $\nu^*$ can be obtained with \autoref{eq:computing_bound}. In this way, one may study how the certified accuracy decreases with increasing $\nu^*$.

This analysis lets us compare our bounds with those of other popular certification techniques, such as randomized smoothing \citep{cohen2019certified}. Randomized smoothing provides high probability robustness guarantees against $\ell_2$ attacks for \emph{any} classifier by composing them with a Gaussian distribution (though other distributions have been recently explored as well for other $l_p$ norms \citep{salman2020black}). In a nutshell, the larger the variance of the Gaussian, the larger the certifiable radius becomes, albeit at the expense of a drop in accuracy.

We use the MNIST dataset for this analysis. We train a model with 256 atoms by minimizing the following regularized empirical risk using stochastic gradient descent (employing Adam \citep{kingma2014adam}; the implementation details are deferred to Appendix \ref{supp:numerical})
\begin{equation}
    \min_{{\mathbf W},{\mathbf D}}~ \frac{1}{m} \sum_{i=1}^m \ell\left(y_i,\langle {\mathbf W},\enc[{\mathbf D}]{({\mathbf x}_i)} \rangle\right) + \alpha \|{\mathbf I} - {\mathbf D}^T{\mathbf D}\|^2_F + \beta \|{\mathbf W}\|^2_F,
\end{equation}
where $\ell$ is the cross entropy loss.
Recall that $\enc[{\mathbf D}]({\mathbf x})$ depends on $\lambda$, and we train two different models with two values for this parameter ($\lambda=0.2$ and $\lambda=0.3$).

\begin{figure}
\centering
\subcaptionbox{\label{fig:synthetic_separable}}
{\hspace*{-2pt}\includegraphics[width = .25\textwidth,trim = 0 0 0 0 0]{figures/synthetic_demo_sep_trans_m.pdf}}
\subcaptionbox{\label{fig:synthetic_non_separable}}
{\hspace*{-12pt}\includegraphics[width = .25\textwidth,trim = 0 0 0 0]{figures/synthetic_demo_non_sep_trans.pdf}}
\subcaptionbox{\label{fig:mnist_comparison_a}}
{\hspace*{3pt}\includegraphics[width = .25\textwidth,trim = 0 -5 20 30]{figures/CertAcc_L02_trans.pdf}}
\subcaptionbox{\label{fig:mnist_comparison_b}}
{\hspace*{3pt}\includegraphics[width = .25\textwidth,trim = 0 -5 20 30]{figures/CertAcc_L03_transb.pdf}}
\caption{Numerical demonstrations of our results. (a) synthetic separable distribution. (b) synthetic non-separable distribution. (c-d) certified accuracy on MNIST with $\lambda = 0.2$ and $\lambda = 0.3$, respectively, comparing with Randomized Smoothing with different variance levels.\vspace{-8pt}}
\label{fig:Certificates}
\end{figure}
\autoref{fig:mnist_comparison_a} and  \ref{fig:mnist_comparison_b} illustrate the certified accuracy on 200 test samples obtained by different degrees of randomized smoothing and by our result. While the certified accuracy resulting from our bound is comparable to that by randomized smoothing, the latter provides a certificate by \emph{defending} (i.e. composing it with a Gaussian distribution). In other words, different \emph{smoothed} models have to be constructed to provide different levels of certified accuracy. In contrast, our model is not defended or modified in any way, and the certificate relies solely on our careful characterization of the function class.
Since randomized smoothing makes no assumptions about the model, the bounds provided by this strategy rely on the estimation of the output probabilities. This results in a heavy computational burden to provide a high-probability result (a failure probability of 0.01\% was used for these experiments). In contrast, our bound is deterministic and trivial to compute.

Lastly, comparing the results in \autoref{fig:mnist_comparison_a} (where $\lambda = 0.2$) and \autoref{fig:mnist_comparison_b} (where $\lambda = 0.3$), we see the trade-off that we alluded to in \autoref{sec:GenBound}: larger values of $\lambda$ allow for larger encoder gaps, resulting in overall larger possible certified radius. In fact, $\lambda$ determines a hard bound on the possible achieved certified radius, given by $\lambda/2$, as per \autoref{eq:computing_bound}. This, however, comes at the expense of reducing the complexity of the representations computed by the encoder $\enc[{\mathbf D}]{({\mathbf x})}$, which impacts  the risk attained.

\vspace{-5pt}
\section{Conclusion}
\label{sec:Conclusion}

In this paper we study the adversarial robustness of the supervised sparse coding model from two main perspectives: {we provide a bound for the robust risk of any hypothesis that achieves a minimum encoder gap over the samples, as well as a robustness certificate for the resulting end-to-end classifier}. Our results describe guarantees relying on the interplay between the computed representations, or features, and the classifier margin.

While the model studied is still relatively simple, we envision several ways in which our analysis can be extended to more complex models. First, high dimensional data with shift-invariant properties (such as images) often benefit from convolutional features. Our results do hold for convolutional dictionaries, but the conditions on the mutual coherence may become prohibitive in this setting. An analogous definition of encoder gap in terms of convolutional sparsity \citep{papyan2017working} may provide a solution to this limitation. Furthermore, this analysis could also be extended to sparse models with multiple layers, as in \citep{papyan2017convolutional,sulam2019multi}. {On the other hand, while our result does not provide a uniform learning bound over the hypothesis class, we have found empirically that regularized ERM does indeed return hypotheses satisfying non-trivial encoder gaps. The theoretical underpinning of this phenomenon needs further research.}
More generally, even though this work focuses on sparse encoders, we believe similar principles could be generalized to other forms of representations in a supervised learning setting, providing a framework for the principled analysis of adversarial robustness of machine learning models.

\section*{Broader Impact}
This work contributes to the theoretical understanding of the limitations and achievable robustness guarantees for supervised learning models. Our results can therefore provide tools that could be deployed in sensitive settings where these types of guarantees are a priority. On a broader note, this work advocates for the precise analysis and characterization of the data-driven features computed by modern machine learning models, and we hope our results facilitate their generalization to other more complex models.

\section*{Acknowledgements} {
This research was supported, in part, by DARPA GARD award HR00112020004, NSF BIGDATA award IIS-1546482, NSF CAREER award IIS-1943251 and NSF TRIPODS award CCF-1934979. Jeremias Sulam kindly thanks Aviad Aberdam for motivating and inspiring discussions. Raman Arora acknowledges support from the
Simons Institute as part of the program on the Foundations of Deep Learning and the
Institute for Advanced Study (IAS), Princeton, NJ, as part of the special year on Optimization, Statistics, and Theoretical Machine Learning.
}
\medskip
\bibliographystyle{plainnat}

redpajama_set_name: RedPajamaArXiv


of the most efficient systems for water delivery using flexilble layflat hose.
Helping industries overcome the challenge of limited water supply.
Our complete range is made locally right here in Australia.
ADIA (Australian Drilling Industry Association) & AWA (Australian Water Association).
Crusader Hose is the leading Australian manufacturer and supplier of layflat hoses suitable for high volume and high-pressure pumping applications. Layflat hose is a type of hose that is retrievable or used for permanent fluid discharge. Applications range from rapid emergency response, cooling systems on- or off-shore, to mine dewatering and fuel pumping. Used for decades in various sectors like firefighting, agriculture, construction and industry, layflat hoses from Crusader Hose offer a flexible, mobile, easy to handle alternative to rigid piping, providing cost savings and a solution that's easy to transport.
At Crusader, we produce hoses built to last for transferring most liquids – from drinking water to industrial or waste water, to fuel and chemicals, in the volume and pressure needed.
Offering high reliability and quality, our industrial hoses outperform less durable ones. With a large range of industrial hose solutions, Crusader Hose is a leading industrial hose manufacturer and industrial hose supplier.
World-leading Australian Manufacturers of the most efficient systems for water delivery using flexible layflat hose.
Explore featured case studies of our layflat hose used across a wide spectrum of industrial and government projects.


Join us on Saturday February 13 for an evening of entertainment celebrating Valentine's Day!
Good Medicine is a duo hailing form North Fork , CA. Debbie Hutchins plays 6 and 12 string acoustic electric and electric guitars and sings, Frank Sipes writes the songs, sings and plays bass. Both have impressive musical credentials. They call their style of music "Folkadelic" drawing from folk and blues to country and tinged with rock, jazz and funk.
We are very pleased to present a special reading by Lee Underwood, from his recently released anthology Diamondfire.
Poet, musician, author, and journalist Lee Underwood toured and recorded with singer/songwriter Tim Buckley in the '60s and '70s, playing lead guitar. While living in Los Angeles, Underwood served as West Coast editor for Down Beat magazine from 1975-1981. He is also the author of Blue Melody: Tim Buckley Remembered. As well, he has recorded two solo piano albums, Gathering Light and Phantom Light.
Spring into Summer - Recent works by Jonathan Bock, Jerry Bosworth, Michael Costa, Michael Frye, Franka Gabler, Jeff Grandy, David Hoffman and Nancy Robbins - Fine Art Photography and Digital Art.
An exhibit that considers the roads we choose and the choices we make.
Saturday December 1st at 7 PM - $5 admission.
An evening of Entertainment inspired by the Pathways art exhibit.
December 1st from 7 to 9 PM. call 559-658-8844 for reservations.
Autumn 2012 Fine Art Photography - featuring works by Jon Bock, Jerry Bosworth, Michael Costa, Franka Gabler, Jeff Grandy, David Hoffman, Terry Robinson, Nancy Robbins, and Heidi Vetter.
Sierra Art Hop takes place on the second Saturday of every month, from 11 AM to 5 PM.
Most participating venues present featured artists, hold demonstrations and other activities, and offer light refreshments.
Current participants include Artifacts Ethnic and Tribal Gallery, Nancy Robbins Photography, The Mind Shop, Stellar Gallery, Timberline Gallery, Williams Gallery West, and Willow Bridge Books in Oakhurst.
Featured artists demonstrate their work throughout the day, discuss techniques, and visit with clients and customers. Come join us!
Stellar Gallery Combines Fine Art with Asian Style.
Stellar Gallery carries a unique collection of hand-picked antique Chinese Country furniture. The style combines simple, yet elegant lines with a sense of practicality and vigor. The gallery's collection includes trunks, tables, benches, cabinets, and assorted accent pieces.


text:
Special Area of Conservation
A Special Area of Conservation (SAC) is defined in the European Union's Habitats Directive (92/43/EEC), also known as the Directive on the Conservation of Natural Habitats and of Wild Fauna and Flora. They are to protect the 220 habitats and approximately 1000 species listed in annex I and II of the directive which are considered to be of European interest following criteria given in the directive. They must be chosen from the Sites of Community Importance by the State Members and designated SAC by an act assuring the conservation measures of the natural habitat.[1]
SACs complement Special Protection Areas and together form a network of protected sites across the European Union called Natura 2000. This, in turn, is part of the Emerald network of Areas of Special Conservation Interest (ASCIs) under the Berne Convention.
1 Assessment methodology in the United Kingdom
1.1 Stage one
1.2 Stage two
Assessment methodology in the United KingdomEdit
Prior to being designated as a Special Area of Conservation (SAC), sites are assessed under a two-stage process set out by the Joint Nature Conservation Committee. The value of the proposed site is considered in relation to the whole national resource of each habitat type and of each species.
Stage oneEdit
Firstly, assessment of the relative importance of sites containing examples of the individual habitat types. Four criteria are used:
the degree of representativity;
the area;
the degree of conservation of habitat structure and functions;
restoration possibilities and a global assessment of conservation value (i.e. an overall assessment).
Secondly, species assessment evaluates population size and density, the degree of conservation of the features of the habitat that are important for the species and restoration possibilities, the degree of isolation of the population in relation to the species' natural range and a global assessment of conservation value.[2]
Stage twoEdit
This stage is often informally referred to as 'moderation'. The criteria used in Stage 2 are intended to be used to assess the sites at the level of the nine biogeographical regions and the EU as a whole. The Stage 2 criteria may be summarised as:
relative value of the site at national level;
relationship of the site to migration routes or its role as part of an ecosystem on both sides of one or more Community frontiers;
total area of the site;
number of habitat types and species present;
global ecological value of the site at the level of the biogeographical region and/or EU as a whole.
List of Special Areas of Conservation in the Republic of Ireland
List of Special Areas of Conservation in Northern Ireland
List of Special Areas of Conservation in England
List of Special Areas of Conservation in Scotland
List of Special Areas of Conservation in Wales
Conservation designations
Index of conservation articles
Area of Outstanding Natural Beauty (AONB)
Environmentally Sensitive Area (ESA)
Heritage Coast
Scheduled monument
Site of Special Scientific Interest (SSSI)
^ SAC definition in the Directive
^ Cors Caron - Special Area of Conservation - JNCC SAC - Habitats Directive
A full list of UK SACs on the Joint Nature Conservation Committee website
Interactive map of designations in Great Britain, including SAC boundaries (select 'Nature Reserves' in UK Map Layers).
Retrieved from "https://en.wikipedia.org/w/index.php?title=Special_Area_of_Conservation&oldid=913067452"


Green coffee bean capsules give a significant boost to weight loss when taken daily. The initial study that has actually acquired everybody so excited monitored 16 participants over a duration of 22 weeks, and also found that each had actually surprisingly shed on average concerning 17 pounds.
Much more persuading proof was when Dr Oz himself conducted his very own study making use of 100 women from a television audience. The females which got green coffee supplements lost twice as much as those which received an inactive medicine.
The adhering to offers all you have to learn about Dr Oz green coffee beans, clarifying the rumors and questions many individuals in Maracaibo Venezuela are grappling with.
Dr Oz Green coffee beans have chlorogenic acid. The acid is essential to its numerous weight loss benefits given that it signals your liver to burn the stored fatty tissue FIRST. Additionally, it slows processing of sugars into sugar, so there's less in your blood producing far more fatty tissue burning. Excess sugar doesn't turn to fat deposits later on. By doing this there are prompt benefits along with lasting ones.
This is specifically what happened with some participants in a lately conducted research on the green coffee bean extracts by Scranton College under Joe Vinson. Lookings for were disclosed during the 243rd National Meeting of the long time running American Chemical Society (ACS).
When coffee is roasted, the result is just what lots of people in Maracaibo Venezuela appreciate throughout numerous times of the day. However, when freshly picked, pure green coffee is just unroasted coffee- coffee in its purest type- which might be the answer to all weight associated issues out there. Green coffee extract has actually been located to significantly minimize fatty tissue absorption in addition to fuel metabolic rates in the liver, both which are extremely important in weight decrease.
Green coffee contains a natural chemical called chlorogenic acid. This chemical decreases the launch of sweets in the circulatory system, a procedure which also aids individuals in Maracaibo Venezuela in weight loss. Since chlorogenic acid increases thermogenesis (a process when body heat is created), fatty tissue is burned much faster for power intake. Furthermore, given that green coffee consists of much more anti-oxidants than roasting coffee, taking green coffee supplements can likewise assist in reducing the manufacturing of fat deposits cells in the physical body.
Chlorogenic acid is prevalent mainly in green coffee grains, nonetheless it is ruined during the cooking process that is made use of to make coffee cocktails we're all knowledgeable about. It's important that a procedure utilized consequently grains into conveniently taken in capsules doesn't use warmth that ruins the chemical framework therefore drastically influencing quantities of chlorogenic acid that is present in an end product.
Some bogus merchants in Maracaibo Venezuela use terms such as Pure Green Coffee Bean Extracts, but the term Pure suggests nothing right here. There's NO POLICY guaranteeing just what Pure means.
If you are reading any sort of green coffee reviews and all they keep doing is rave regarding the amount of of a miracle it is, so take care. Excellent suppliers of supplements make sure to inform you the possible downsides or side effects. They'll additionally tell you the precise ingredients.
The key is ensuring you're obtaining a true top quality extract and also nothing fulled of unnecessary elements which can in fact be damaging. CHECK OUT THE LABEL! The fraudsters can't push a label. Dr. Oz will certainly advise a product with:.
In a recent study that existed at the ACS's spring national conference which was in San Diego, sixteen over weight adults took, reduced dosages of green coffee bean extracts, higher amounts of green coffee supplements, as well as a placebo. Although the research was tiny, results were striking. It ended that Subjects taking complete amounts of the green coffee extracts shed about concerning 17.5 pounds.
Studies have proposed chlorogenic acid decreases the absorption of fat during meals intake in addition to activating metabolic process of extra fats.
Green Coffee contains several parts. Each of them might have the prospective medicinal impacts. The trick to genuine long term weight loss is one crucial active compound that's called Chlorogenic Acid discovered in green coffee grains.
Green Coffee Beans will prevent body fat absorption. They additionally turn on fatty tissue metabolism which is done in the liver. The Chlorogenic acids located in the Green Coffee Beans can be called an all-natural phytochemical found in an extremely large array of plants, all with high focus of Green Coffee Beans.
Green Coffee Extract is been found to hinder launch of sugar into a blood stream, specifically after meals. This appears to really help individuals in Maracaibo Venezuela drop weight much a lot faster. Chlorogenic acid could additionally generate body system fat loss with increasing the body heat produced. This way it causes promotion of thermogenesis, natural burning of body fat to produce electricity.
Green Coffee Extract is also believed to minimize generation of fatty tissue cells because of its superior antioxidant results.
Green Coffee Extract is highly thought by nutritional experts and health club fitness instructors in Maracaibo Venezuela that a trick to weight loss is healthy diet and also workout. Fortunately below is an unbelievable extremely food that provides results! This super meals, the green coffee grain, has been creating major media talk.
Green Coffee Grain extract truly is a reliable weight loss supplement. Nonetheless the popularity of this brand-new offer in Maracaibo Venezuela integrated with the simplicities involved with web retail business imply there exist several fraud firms that are functioning to make use of this remarkable interest in using Green Coffee beans for weight loss.
It is vital currently to get an item that is legit and real. The popularity of this green coffee extract is fanned by its reputation to deliver results.
Additionally, to encourage even more reliable weight loss, make sure to work out whenever feasible. Integrating this with the green coffee grains will lead to outcomes that will shock you.
Where to Buy Weight Loss Pill in Jelgava Latvia?


Helps to bring your desires to you. This is intended for relationship issues, but will also help you access what you need in other areas. NOTE: this is not a guarantee of requited love! We do not condone imposing one's will on others! Wear this when you're hoping to attract someone/something.


Our site BLS-House.com is a platform featuring the largest possible range of Italian furniture from hundreds of manufacturers. The Buyer can select a Seller on the website. We have no legal relationship with the furniture sellers and are not responsible for their actions. The Sellers are independent legal entities. Please carefully read the general terms and conditions listed below. To cooperate with the website BLS-House.com you have to agree with these terms. If you do not agree with these terms, please do not use the services of the website BLS-House.com. Varga D.O.O. operating the website BLS-House.com reserves the right to change the terms of the online store at any time without preliminary notification. The notification about the changes in general conditions will be available for users on the website BLS-House.com.
Every BLS-House.com visitor becomes the User or the Customer when registering or purchasing in the online store.
Please note that due to photo quality and differences in monitor settings, the colors shown in the photos may not fully reflect true colors. All the photos are shown for reference and cannot serve as the standard of true products colors.
The products are made of natural materials and materials imitating wood textures, so they can have differences in hues or patterns even within the same order. Possible differences in hues and patterns of furniture, wood knots, annual rings at the front part of cabinet furniture, as well as possible cuts, scrapes and bumps on the leather parts are not defects and shall not be considered for reclamation.
Leather products manufacturing process includes various structural nuances due to leather natural origin, such as folds, scars, bites, brands, etc.
Both individuals and legal persons can be customers of BLS-House.com website Individuals are protected by the Consumer Protection Law.
The Customer places an order online on the website www.BLS-House.com (hereinafter referred to as the "Website") or by e-mail.
When placing the order on the ebsite the Customer states his name and surname, gender, e-mail address, telephone number and delivery address. Customer nickname is not possible. The Customer is responsible for the accuracy of his personal data when ordering.
The description of the goods presented on the website is for information only. The Customer can contact customer support for more details. Each order has an identification number. All goods and related services must be paid at the price fixed by the Seller.
Order procedure is the same for both individuals and legal entities.
a) Reliable information about the company (name, address, registration number).
b) The actual address for prompt and effective communication (telephone, e-mail).
c) Basic descriptions of the goods and services (including after-sales services and warranty).
g) Availability of products (each product or service on the webpage should be available within a reasonable time).
d) Delivery terms and services provided (method, place and time of delivery).
e) All prices must be plain and accurate and must indicate if taxes and shipping costs are included.
g) Payment and delivery methods.
h) Time for cancellation of the agreement, cancellation terms and costs to the Customer in case of return of goods.
i) Customer actions in case of a claim, including information about the contact person or department responsible for customer relations.


text:
Places to Visit around Oklahoma City, Oklahoma
Red Earth Museum
6 Santa Fe Plaza, Oklahoma City, OK, 73102, United States
The Elsing Museum comprises an extensively rare collection of some of the world's most amazing gems, minerals, natural art, Indian artifacts, oriental artifacts, and much more. A visitor's first glimpse of the museum makes it clear that many years of dedication and...
Oklahoma City Museum of Art
415 Couch Drive, Oklahoma City, OK, 73102, United States
The museum has outgrown its first building and has two annexes. There is also a genealogical library in the main museum.
620 N Harvey Avenue, Oklahoma City, OK, 73101, United States
Stanton County Museum plays a vital role in the community by collecting, preserving, sharing, and interpreting the county's history and heritage through exhibits, programs, and research files in a manner that is open and accessible to county residents and visitors.
International Gymnastics Hall of Fame
120 N Robinson, Oklahoma City, OK, 73102, United States
The Coronado-Quivira Museum in Lyons is one of the best small museums in Kansas. Separate displays focus on early inhabitants, Spanish explorers, the Sante Fe Trail, and the coming of homesteaders and permanent settlers.
Untitled (ArtSpace)
1 NE 3rd Street, Oklahoma City, OK, 73104, United States
Visitors are encouraged to experience for themselves the bleak, "Dust Bowl" years from the seats of a Model A Ford, then move on through the booming Forties and Fifties from the seat of a 1948 Willys Overland Jeep. The happy days of the 1960s and on can be...
0.5 miles Art Gallery
[Artspace] at Untitled
A non-profit art center in downtown Oklahoma City presenting exhibitions and programs on contemporary art in a variety of mediums.
0.6 miles Botanic Garden
Gaylord-Pickens Oklahoma Heritage Museum
1400 Classen Drive, Oklahoma City, OK, 73106, United States
Pioneer museum with antique farm machinery, autos, log cabin and jail from the 1880s, war memorabilia, rural school, printshop, church and opera house.
Henry Overholser Mansion
405 NW 15 Street, Oklahoma City, OK, 73103, United States
In addition to items related to the Butterfield Trail, the museum's collection includes clothing, tools, and furniture used by early plains settlers. We also have an extensive list of books, periodicals, and photographs dating to the mid nineteenth century, as well...
Great museums in Oklahoma City
Fun in Midwest City
School education trip in Moore
What to do with kids in Edmond
Day out with the kids around Norman
School trip in Shawnee
Fun family activities in Stillwater
Best cheap things to do around Enid
Going out with the kids in Lawton
Best places to visit in Ponca City
Museums in Tulsa