text:
Finding the best deals for cheap tickets and discount
In the past, if you wanted to find cheap airline tickets and discount airfares you had to contact a travel agent and explain where and when you wanted to go. They would do some research and eventually get back to you with a deal.
Locations To Take To 8 Awesome Adventure Sports In Your Caribbean Vacation
Should your notion of an excellent getaway requires adventure, adrenaline, and taking a break from ordinary, then the Caribbean islands are calling your name. Even though the region's pristine white sand shores and turquoise seas often call in your thoughts images of …
vacation – BingNews
IIDs in California
Ignition Interlock Devices can be enforced by California courts in a DUI case. These devices are installed in cars driven by people who have been convicted of driving under the influence in California. An Ignition Interlock Device works like a breathalyzer and is connected to the vehicle's dashboard and linked to the ignition system. This system is used to prevent and deter drivers from driving under the influence of alcohol.
Ignition Interlock Devices (also known as IIDs) are breath alcohol detection systems used to prevent/deter selected drivers from operating motor vehicles under the influence of alcohol. Before a driver can start the engine, he or she must blow into the device, which then analyzes the ethanol alcohol content of the breath using an electrochemical fuel cell to detect concentration. If the device reads that concentration is greater than a preset limit (usually between 0.02% and 0.04% BAC), it triggers the ignition system to lock and prevent the car from starting. If the fuel cell system does not detect a strong alcohol trace, the ignition will trigger without a problem.
To deter people from cheating the IID system, manufacturers employ something called a rolling retest. At random intervals after the ignition has been started, the device will require the driver to retest by blowing into the breathalyzer again. (This prevents drunken drivers from having friends take the test for them). If the driver fails the retest or refuses to take the retest, the device records this information and triggers the car horn to beep and the lights to flash. The only way to silence the horn and get the lights to stop flashing is to turn the car off and to retake (and pass) another breathalyzer test. (For obvious safety reasons, the IID never actually shuts down the car engine.)
Ignition Interlock Devices must be maintained and calibrated periodically. Drivers who have been court ordered to install these devices must pay for installation and maintenance. IIDs also record and log noteworthy events. If for instance, a driver attempts to remove the IID device or to circumvent it somehow or to impair fuel cell operation, authorities will be alerted and will set punishments accordingly.
California law empowers the courts to impose that IIDs be installed in cars driven by individuals who have been convicted of a DUI in California. These stipulations set out in Vehicle Code Section 14601.2, give the California courts the power to require that said individuals drive with IIDs for up to three years.
These intrusive devices can make day-to-day living exceptionally difficult. Particularly in California, where it is nearly impossible to work or do errands without a working motor vehicle. The sentencing practices vary widely depending upon the specific court your case is pending in and variables like the circumstances behind your DUI and the county you are facing charges in can have a substantial impact on whether or not you will be required to have an IID installed on your car.
Ignition Interlock Devices (also known as IIDs) are breath alcohol detection systems used to prevent/deter selected drivers from operating motor vehicles under the influence of alcohol. Essentially, IIDs work like breathalyzers, but they are connected to the dashboard and linked electrically to the ignition system. Before a driver can start the engine, he or she must blow into the device, which then analyzes the ethanol alcohol content of the breath using an electrochemical fuel cell to detect concentration. If the device reads that concentration is greater than a preset limit (usually between 0.02% and 0.04% BAC), it triggers the ignition system to lock and prevent the car from starting. If the fuel cell system does not detect a strong alcohol trace, the ignition will trigger without a problem.
Our law firm has offices from San Diego to Ventura and our experienced California DUI defense attorneys know the procedures in the local court where your case is pending. The firm of Wallin & Klarich boasts a tremendous track record helping DUI defendants. Contact us today to get your questions answered and to have a case evaluation.
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Why should your organization think about the option of an unlimited getaway system? It is a fantastic concern for HR pros because it begins with a strong mental response; individuals will automatically genuinely believe that it is both an excellent concept or an awful concept.
May California Weather
California Weather in May sees the beginning of the hot time of year in the desert areas. No part of California is particularly prone to rain or humid conditions in May. Overall, May California weather is very pleasant, the desert is not hot yet, and there are no areas with major rainfall.
Here are the May Weather conditions in some of the major areas of California:
The highest recorded temperature in May for Bakersfield for the past 68 years was 107 degrees Fahrenheit, while the regular high temperature is 83.8. The lowest documented temperature for Bakersfield for the past 68 years in May was 37, while the average low temperature is 56.8.
Bakersfield typically receives an average of 0.24 inches of precipitation during the month.
In the last 58 years the highest measured May temperature in Bishop was 102 degrees Fahrenheit, while the daily mean high temperature is 81.2. The lowest documented temperature in May for Bishop for the past 58 years was 25, and the normal low daily temperature is 43.7.
Bishop normally receives an average of 0.26 inches of rain during May and averages 0.1 inches of snow.
In the last 95 years the highest measured temperature in Eureka California for May was 84 degrees Fahrenheit, while the normal high daily temperature is 59.6. The lowest measured temperature in Eureka for the last 95 years in May was 35, and the average low daily temperature is 47.6.
Eureka receives an average of 1.62 inches of precipitation during May.
The highest recorded temperature in May for Fresno California for the past 56 years was 107 degrees Fahrenheit, while the regular high temperature is 82.7. The lowest documented temperature for Fresno for the past 56 years in May was 36, and the normal low daily temperature is 54.9.
Fresno typically receives an average of 0.39 inches of precipitation during the month.
The hottest day in Long Beach California in May for the past 53 years was 104 degrees Fahrenheit, and the mean high is 74&. The lowest measured temperature in Long Beach for the last 53 years in May was 40, and the mean daily low in the month is 57.8.
Long Beach normally receives an average of 0.23 inches of rain during May.
The highest recorded temperature for Los Angeles California for the past 70 years in May was 97 degrees Fahrenheit, and the typical daily high is 69.3. The lowest measured temperature in Los Angeles for the last 70 years in May was 43, while the regular low temperature for a given day during the month is 56.9.
Los Angeles usually receives an average of 0.24 inches of rain during May.
The record high May temperature in Mount Shasta California for the last 42 years was 94 degrees Fahrenheit, while the daily mean high temperature is 67.3. In the last 42 years the highest reported temperature in Mount Shasta for May was 21, while the mean low daily temperature during the month is 39.
Mount Shasta typically receives an average of 1.87 inches of rain during the month and normally receives an average of 0.8 inches of snow.
In the last 19 years the highest measured temperature in Redding California for May was 104 degrees Fahrenheit, and the average high is 80.7. The lowest reported temperature in May for Redding for the past 19 years was 34, and the regular daily low is 51.6.
Redding usually receives an average of 1.66 inches of precipitation during the month.
In the last 55 years the highest measured temperature in Sacramento California for May was 105 degrees Fahrenheit, while the normal high daily temperature is 80. The lowest reported temperature in May for Sacramento for the past 55 years was 36, and the regular daily low is 50.9.
Sacramento normally receives an average of 0.53 inches of rain during May.
In the last 65 years the highest measured temperature in San Diego California for May was 96 degrees Fahrenheit, and the typical daily high is 69.3. In the last 65 years the highest reported temperature in San Diego for May was 48, and the typical low is 59.8.
San Diego receives an average of 0.2 inches of precipitation during May.
In the last 78 years the highest measured temperature in San Francisco
California for May was 97 degrees Fahrenheit, and the mean high is 66.8&. In the last 78 years the highest reported temperature in San Francisco for May was 36, and the standard low recorded daily temperature is 50.5.
San Francisco usually receives an average of 0.38 inches of precipitation during May.
In the last 69 years the highest measured May temperature in Santa Barbara was 92 degrees Fahrenheit, while the typical high daily temperature is 71.2. The lowest reported temperature in May for Santa Barbara for the past 69 years was 38, while the normal low measured temperature is 50.5.
Santa Barbara usually receives an average of 0.23 inches of rain during May.
The record high May temperature in Santa Maria California for the last 63 years was 100 degrees Fahrenheit, while the typical high daily temperature is 68.6. The lowest reported temperature in May for Santa Maria for the past 63 years was 31, and the standard low recorded daily temperature is 46.9.
Santa Maria typically receives an average of 0.32 inches of precipitation during the month.
In the last 46 years the highest measured May temperature in Stockton was 107 degrees Fahrenheit, while the standard high temperature is 81.3. The lowest measured temperature in Stockton for the last 46 years in May was 38, while the regular low temperature for a given day during the month is 52.1.
Stockton usually receives an average of 0.5 inches of rain during the month.
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This Travel Business Areas Rich Asian Families Within Their Fantasy Holiday Houses
While luxurious vacations with the household are often something to look toward, planning the particulars is time-consuming. Choosing the perfect location to remain, how to handle it, where you can consume – the list goes on. It's perhaps not a vacation whenever you're stressed …
The "California" is not sold in (California) USA/NA although demand is high, the name "California' was previously used by Volkswagen for the Westfalia-built campervans since 1988. Westfalia was the Official Manufacturer of motorhomes for VW. When the Westfalia company was bought by DaimlerChrysler in 2001, VW decided to build and design their own campervan range, based on the legendary Transporter range.
The California launched in August, 2003. In 2004 two design studies were presented at the Caravan Salon in Dsseldorf: California Beach and California Tramper. VWCV expanded the range in 2005 with the California (Multivan/Kombi) Beach, to bridge the gap between the Multivan/Kombi people carriers and the California.
At the 2007 Caravan Salon in Dsseldorf, VWCV launched the new California Beach with pop top bed, as seen in the higher specified versions of the California range.
VW California (Multivan/Kombi) Beach specs & features
Australian spec Kombi Beach, known as California/Multivan Beach in Germany
Acceleration Slip Regulator (ASR) – a traction control system
power front windows
heated doors and mirrors
CD player with 6 speakers
alloy wheels with full size spares
dimmable dashboard illumination
storage nets
The "Good Night Package" comes with curtains, and the ability to fold the seats into a bed for two.
Indoor/outdoor features
A folding table integrated within the sliding door is able to be used outside and inside with two folding chairs.
Combined with the rear seats, the front seats swivel 180 degrees, so you can have a meeting/meal/break with the rear seat passengers.
California Beach now has the option of the elevating "pop top" aluminium roof, with bed space of 120x200cm in Germany.
VW California specs & features
California Comfortline in Germany
The California comes with an electro-hydraulically elevating aluminium roof with bed space of 120x200cm.
The rear seating in the California can be arranged to make lower level double bed measuring 200x114cm.
The kitchen comprises a sink, a two burner gas cooktop powered by a 2.8 kg gas cylinder, a 42 litre compressor fridge, two kitchen storage cupboards with drawers integrated in the design are also standard.
A standard fitment twin water tanks both hold 30 litres of water, one for drinking water, the other for foul/waste water.
There is an optional 250 cm awning which mounts above the sliding door, and a foldaway table and two chairs are hidden in the tailgate hatch.
satellite navigation system
3 Zone Climatic airconditioner
alcantara and leather upholstery
6 disc CD changer
single seat for living area
parking sensors for front and rear
electric sliding door
privacy glass for rear
removable towbar
4motion four-wheel drive system
^ California Beach Poptop picture thread at VWwatercooled forum
^ a b c Volkswagen Kombi Beach
^ California Beach @ Caravan Salon in Dusseldorf – VWwatercooled Australia
^ a b c d Volkswagen Driver, November 2006 page 27-30 -UK Publication
VW Germany's California Website
VW UK's California Website
VW Australia's Kombi Beach Website
VW Australia's California Website
Brick-Yard UK – Multivan & Transporter forums
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Australian VWCV forum
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I am an expert from China Hardware Suppliers, usually analyzes all kind of industries situation, such as compact pci connector , scart control box.
More California Articles
Obama vacation Residence Up for rental for $4,500 per night
1st Family's holiday residence is up for rental. Veuer's Nick Cardona tells us how much you need to pay up so that you can vacation like the Obama's in Hawaii.
Miss California
For pageant lovers the recently concluded Miss USA was a delight. The show's opening was spunky and fun as they arrived at a winner on the night. What was different about this pageant is that no 'world peace' answer would be have been given or even accepted. The questions were current political discussions that you would typically hear on news stations. How did these enter pageants?
In the final answer segment gay judge Perez Hilton asked Miss California Carrie Prejean her personal opinions on gay marriage in America. With an answer that would disgust persons who support gay marriage Miss California went on to say 'no offense' but she believes that a marriage should be between a man and a woman. With whatever answer she would have provided in answering the question it would isolate a party to the question. With Miss California giving an unpopular answer in today's world where more and more persons are being liberal on the matter accepting gay marriage it would have come to her detriment.
Hours later, Perez Hilton on his blog video taped his response referring to Miss Prejean to derogatory names which were disrespectful and crude. With this, it ignited an old discussion of the acceptance of gay marriage in the American society. With persons taking sides it also started a witch hunt to find skeletons in Miss Prejean's closet.
Being first runner up to the Miss USA title for 2009, this answer skyrocketed Miss California in the media with her making grand appearances on news networks across America. I have been watching this story, I can't recall her making a CNN appearance but she was definitely on MSNBC and FOX time and time again. This kept the discussion current and developed in a huge fight with the Miss California pageant holders.
Persons who follow pageants also blame Miss Prejean for answering the way she did. Of course pageant girls are known for their perfect answers and concise responses. The argument against Miss California is that she knew Perez Hilton was gay, the Miss California pageant holder is, a huge part of her state supports gay marriage, and two other judges were openly gay, so why this stance on the matter if you really want to win the crown knowing that you would isolate some persons from yourself.
With questions ranging from the war in Iraq and free health care, this was a telling night. The questions asked of Miss Prejean also had her stop in her tracks. If anyone should analyze her answer she was liberal in the beginning saying we live in a land where we can choose. However, in recent interviews she says at that point, God intervened and said to her what do you really believe? At that point her answer took a 360 degree turn and her reference that a marriage should be between a man and a woman was made.
Should she really be blamed though? I don't think so. She was asked a question and she gave an honest answer. I believe if she had flip flopped from side to side then she wouldn't have answered her question. Which brings me to the point can we handle the truth. Sometimes we ask questions and we are not going to like the responses we are given but we need to respect an individual's opinion on a certain matter and Perez Hilton could not handle the heat.
Days after, revealing images appeared on the internet exposing Miss California to some harsh critics. In entering pageants, contestants cannot pose nude in images and these should be disclosed before you continue. Miss California did state in her contract that she did not pose nude. However, photographer's images revealed that some parts of her body (breasts) were exposed. But these were taken in and between takes when she was not aware of. This matter was taken to Donald Trump the pageant holder which concluded that she will continue to be Miss California's representative.
The Bigger Issue
The question of gay marriage is topical and ethical some would say. From an ethics point of view by the right's perspective, it is a right to marry whoever you want to marry. From a utilitarian's view if gay marriage will benefit the majority then it should be awarded. However, this political issue just won't go away. Obama, Clinton, Palin etc. believes that a marriage should be between a man and a woman. Bruce Golding the Prime Minister of Jamaica believes the same. He said in a recent interview that he would not award a gay person to his cabinet. The issue of gay marriage is a touching one because gays are demanding the right to marriage in more places.
This issue will always have two sides and it is time for politicians and civilians to sit down and really think about this issue. It will only hurt if we begin to hate those who oppose a position we stand on. Miss California was punished for exercising her freedom of speech right; I do hope that others will not be hateful because someone does not believe the way you do.
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The US Virgin Islands will probably pay You to simply take a secondary in 2017!
Get paid to just take a secondary? Matter us in! I am already a fan, cannot show this again
For the enjoyable weather that Mother Nature gives to California, this beautiful and popular state welcomed an endless amount of individuals and families travelling to each year. Just as with any other state, the specific time of the year that your important business trip is planned for, or the specific time that you would like to plan to take each of your family members, will greatly determine the type of weather that you will experience while view the various cities in the beautiful state.
If you are caught up in the plans of making a trip to a coastal area of sunny Southern California, then you can expect to be able to take advantage of much warmer temperatures than those that can generally be found in many of the Northern areas that are further away from the coastline. Perhaps one of the most beneficial advantages that individuals and numerous families enjoy about Southern areas that are by the coast, is the fact that when the months of the winter arrive there is not that much of a difference in the temperatures that are generally enjoyed during the months of the summer.
Many of the inland areas offer weather than can be found in southern California to include hot desert areas that are extremely dry. There are several areas that are located to the east of Los Angeles that are densely populated as well as being dry and hot. The inland areas are also known for a much different variation in the difference between the temperatures that are most often felt between the winter months and the summer months.
Death Valley is the popular named area of California that is well known for the extremely hot temperatures that can be found in the area. In fact, the hot temperatures reach such high levels that it is widely known as the hottest place that can be found in the United States.
If your plans will be taking you to the northern or central areas of the state, you can expect to experience more rainfall than what you would experience in the other areas. Of course this is also dependent on the ranges covered in mountains that are in these areas.
One unique factor that a large number of residents and families of the central parts of this state rely a great deal upon is the fact that the weather is so moderate in this specific area that they are actually able to take advantage of having two seasons that can be used for growing a variety of items, instead of just one season like many other areas have. Because of this enjoyed fact, this particular area is considered to be one of the best areas for land that can be found around the country that can be used for farming purposes.
It is true that many individuals enjoy the nice hot weather that is found in this state, but it is also a fact that causes numerous fires every single year. The most prevalent disaster for this state that is caused by nature is fire.
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This will be the last post for at least a week because of the Rotax engine classes that I'll be attending in Virginia this coming week. This is good timing because the DOG Aviation purchasing department just placed the order for the RV-12 powerplant kit which is the final kit to complete the DOG Aviation RV-12.
During the instillation of the left F-1265 flaperon pushrod it was discovered my chosen location for attaching the static air line alongside the seat rib has created interference with the left F-1265 pushrod. At the time this location was chosen for the static air line, it did not seem likely the left F-1265 pushrod would get close enough to the static air line to touch it … but it does. It just barely touches … however any contact is unacceptable.
After installing the left F-1265 flaperon pushrod, it was discovered there are combinations of stick and flap handle positions that create contact with the static air line. My finger is pointing to the area of contact.
If the lightening holes in the seat rib did not have flanges, doubt there would be any interference. The solution is not going to be a big deal … it will just require pulling back the static air line and routing it on the backside of the rib using the exact same routing. The static air line will need to pass through the lightening hole near the flap handle block, follow the same path on the backside of the seat rib it is on and get tied down using the same holes for the wire ties. At the forward end of the seat rib it will need to pass through the large lightening hole and through the wire grommet it is currently in.
Moving the static air line behind the rib will solve the clearance issue and as can be seen in this photo, there will be no issues bringing the static air line back through the large lightening hole in the seat rib and forward through the wire grommet.


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Pržno (in polacco Prżno) è un comune della Repubblica Ceca facente parte del distretto di Frýdek-Místek, nella regione della Moravia-Slesia.

Altri progetti

Collegamenti esterni

Pržno

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Service Loaner vehicle - Available for Sale...kilometers may not be exactly as listed, contact for more details.
Service loaner demo, kilometers may not be exactly as shown. Vehicle has body side moulding and a hood deflector included.


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Jump to: Navigation
From a pale blue dot to distant frozen world
11 Dec 2015 in Astronomy / Op-ed / Science on Astronomy, Science
A little over 25 years ago, on July 6th 1990, the Voyager I space probe turned around and snapped one last picture of planet Earth and the solar system it was about to leave behind. More than 6 billion kilometers from the sun planet earth was only a pale blue dot, barely a pixel large.
This is the "Pale Blue Dot" photograph of the Earth taken by the Voyager 1 spacecraft on July 6, 1990. The Earth is the relatively bright speck of light about halfway across the uppermost sunbeam.[/caption]
Look again at that dot. That's here. That's home. That's us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every "superstar," every "supreme leader," every saint and sinner in the history of our species lived there-on a mote of dust suspended in a sunbeam. - Carl Sagan, Pale Blue Dot
Yet, today, at roughly the same distance from the sun, the New Horizons mission delivered the sharpest photo of Pluto at the far edge of the solar system. The picture shows details the size of less than a city block. Again this is a testament to human ingenuity and drive to explore, but once more emphasizes that we are only "a mote of dust suspended in a sunbeam". Humbling and rather amazing, if you think about it.
As an earth system scientist and ecologist I model ecosystem processes, such as vegetation growth or drought / disturbance resistance, using various -retrospective- proxy measurements in a model data fusion approach, including among others remote sensing, dendrochronology and recovered historical data records.
Throughout my career I worked interdisciplinary borrowing heavily from fields, outside of ecology, such as image vision processing (computer science), remote sensing and engineering to assist in either field measurements and/or model driven analysis.
–Koen Hufkens
Dark nights at Harvard Forest 18 Jul 2015
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text:
As the sun started to go down, and the baby chicklets had to be herded back to their nursery, it was time for The Halloween Poetry Slam…
Everyone lined up to read their poems. Excitement was high and the poetry was wonderful. There was a lot of peeping and wing flapping throughout, and everyone had a very good time. Melanie, read her poems last, because she was the star.
Below are a smattering of poems that were read on stage.
by Rita
why humans are always fighting
everyone at The Coop
including our guests
come in all colors
shapes and sizes
we all love each other
what's wrong with humans
by Jimi
Halloween can be scary
but it's okay if you want to
close your eyes sometimes
because when you open them later
the scary stuff is usually gone
and if it's not
you can always go and stand
next to your sister
HEALING BUNNIES
by Margo
I love the Healing Bunnies
because they make everyone feel better
and they are warm and fuzzy
in a way that chicklets aren't
they smell good too
and they let us play with their ears
if they aren't too busy
I like to hang out in the kitchen
before the parties
because there are always so many
special treats being made
when I get bigger I want to be a chef
and have a baking show of my own
I'll use a lot of corn
everyone will like that
being a dog
at The Coop
is fun
I get a lot of pets
and sometimes
I give the chicklets rides
they like to sleep on me
and I don't mind
because they are like
feathery balls of love
by Trudy
I'm going to be a ballerina
either that
or a journalist
I haven't really decided yet
my mom said
I have to go to school first
but I'm not sure why
And the grand finale…two poems by MELANIE!!!!
by Melanie
Jeffery is a precocious pinto bean
Agnes makes a terrific tambourine
Ivan amazes as a time machine
Little Lily is an adorable tangerine
Watch them bouncing on the trampoline!
I bet it's the first time you've ever seen
A rhyming costume party for Halloween!
is that a mouse squeaking?
is that the floor boards creaking?
do you hear the leaves that crackle?
as the witches darkly cackle?
is that the hoot of an owl?
and is that the wind that howls?
it might be scary, but don't you scream
it's just the sounds of Halloween!
We all hoe you enjoyed the readings. Thank you.
chicken poetry, Chicklets Halloween
Chicklet Poetry Slam
From: Candy…all old white men destroying the world…together…the bottom picture shows us the clowns we are dealing with now.
007 dies at 90…farewell Sean Connery
Too Many Miniskirts, Not Enough Serial Killers — Musings from a Tangled Mind
My favorite time of year is when the cool crispness of fall settles around us. The leaves turn it up a notch with vibrant oranges and brownish reds, dancing towards the ground like feathers in the wind. Cozied up in sweaters and scarves, drinking spiked apple cider is a seasonal bliss we all should have […]
empower our daughters — Coloradopoet
I have two daughters. I want them to have equality in their impact on the world. Culturally women have been taught to make people feel more comfortable and at ease. We can have trouble standing out, speaking up, being ambitious and strong. We undermine our thoughts and ideas with weak language. Have you noticed how […]
Frankenstein; Or, The New Prometheus – Christine E. Ray — Brave & Reckless
stitched carefully together from odd parts of other women you preferred me to be you recklessly called forth roaring thunder flashes of lightening that wracked through my body electric agony ignited a spark of independent life I rose from the table reborn something new something other you gazed upon me your creation not with love […]
Clarissa…Halloween
Clarissa's best friend is a duck named Flo. To honor their friendship, Clarissa's Halloween costume reflects their close bond. Flo quacked for several minutes, when she saw her friend, and they were both so happy they went and had ice cream and missed the first two stage acts. Still, the ice cream was delicious and they played with the rhinos' on the way to the party barn. They are going to take a Gondola ride tomorrow morning, although Flo may be in the water for part of the time.
Clarissa and Flo
One of the treats tables…
Beth brought Fairy Lights and Holly, who is responsible for the treats tables, made her well loved Corn Bread and so many other goodies, as well as bowls of candy corn. The chicklets and their guests love her baking and treats. Holly also brought her dog, Tide, and cat, Bagheera, as well. Tide was a bit shy at first, but the chicklets sat quietly next to where he was hiding, until he felt safe enough to come out. Bagheera was fine right from the start and promised not to catch and eat anyone. There is hay, carrots, apples and all kinds of treats for everyone. Treat tables are always set up everywhere, since there are so many different kinds of guests. Some of the food has to be low to the ground, since a lot of the chicklets and guests can't reach the tables. But everyone manages and they always help each other. Holly make sure everyone has enough to eat and she keeps the tables neat and tidy. She smiles a lot and she knows how to take care of hungry chicklets and their friends.
Chicklets Halloween, food
Chicklets Halloween Party
"So, big day today," he said.
"What do you mean?" she asked, unpacking her tote bag.
"Isn't it Halloween?"
"We're kind of skipping it this year because of the virus."
"You mean the trump virus, or the Covid virus?"
"Good one," she said. The Covid virus."
"He said doctors are liars, saying that people who die don't have Covid but they get more money if they say they die from the virus. So, all the brave women and men who risk their lives everyday are just a bunch of greedy liars to the republicans. I wonder if he is counting the doctors and nurses who died, trying to save people?"
"I can't talk about this," she said, sitting down on the stairs.
"I heard about what he did in Kenya."
"He's so evil, I don't know why he doesn't catch on fire and burn to ash right in front of all of us," she whispered.
"Lots of girls are going to die."
"And not one single male who made them pregnant."
"It's always like that," he said, eating the last of the sardines.
"He pulled all money out of that country if they even talked about birth control, or abortion."
"Suicide will go up."
"And dead women, from back alley abortions." she said. "He's responsible for so many deaths. He's so ugly and hateful."
"Without a doubt. He picks on the poorest people who have no alternatives and absolutely no resources. The women get cancer checks and HIV help from those clinics, and now that's gone as well, all tied to the money he stopped letting them have. He's hatred on two orange legs and he takes children out of the arms of their parents, won't let them have medical treatment, even in other countries. He's the third anti-christ, after Napoleon and his hero, Hitler."
"I can't argue with that," she sighed, "since it's true."
"Hard to argue with reality and the truth," said the cat, licking his paws, washing his face. "He never makes a single thing better, only horrible."
"And yet…people follow him."
She sat down and put her face in her hands.
"What?" he said.
"Let's hope things change on Tuesday."
"What will you do if he doesn't lose? I don't think Canada will let all of you in. I mean a lot of us could probably cross the border, if we wanted to, but everyone loves cats," he said. "Americans aren't liked very much nowadays, even by each other."
"Another truth."
"Truths are all over the place, if you look," he said, rolling over. "But what's truth for one isn't truth for another."
"What happened?" she said. "Where did all the hateful people come from?"
"They were always here, but one of their own finally got into a position of power and it's like a hell mouth opened up and they just all spewed out."
"Is that supposed to be a Halloween image?"
"If you think it is," he said, eating his treats. "You're all acting like a bunch of sheep being herded into the corner of a field."
"Oh, and what do YOU suggest we do?"
"You should have a plan for Wednesday, either way. But you don't, at least not that I can see."
"We're big and bulky."
"Excuses, excuses," he said. "You guys are full of them."
"It's called brainwashing and conditioning."
"No, it's called inability to form a workable plan with enough people. There is such as thing as an internet, you know."
"I think it's more about guns and his ability to kill people in the streets."
"Well, there is that," he said, grooming his back leg. "People will die, one way or the other."
"Winter is pushing its way in."
"Jingle, jingle," said the cat.
She nodded. "Jingle, jingle, all alone. Not a lot of parties this year."
"Oh, boo hoo. You're positively ancient and you've had a million parties."
"Excuse me?" she asked, grinning. "Positively ancient?"
"Got you thinking about something else, didn't I? Besides, all of us will come to your house for a big cat Catmas Party."
"Oh, what fun," she said happily. "I'd love that."
"I already told everyone."
"He did," said Jinx, starting to eat his food. "He told everyone."
"Decorate the tree with live mice, will you?"
"NO, I will NOT decorate the tree with live mice and you know it," she said.
"He's just teasing you," said Jinx. "He likes to tease you."
"I brought more of your favorite treats."
"Thank you. They're delicious."
"I love you Jinx."
"I love you too."
"Oh, gag," said the cat, his tail thrashing back and forth.
"I love you too," she said, making kissing noises all over him, playing with his ears.
He laid as flat as he could. "What is your PROBLEM," he hissed.
She kissed him some more.
"You're just so cute," she said.
"I don't see it,"said Jinx. "I mean he's a good looking cat, but cute?"
"Very funny," said the cat. "You can buy us gifts at Catmas."
"Wow, thank you," she said, snickering.
"He doesn't mean individual gifts, maybe a few blankets for the Queens and kittens, water dishes, things like that," said Jinx.
"I get those for you all the time," she said.
"Sometimes people steal them."
"The homeless take them and mean kids do too."
"All of you come and live with me," she said, immediately. "Tonight."
"You don't have enough room for all of us," said the cat.
"I do. I wasn't going to tell you until next week but I bought the building I live in. You know it's a two-flat and I'm turning the first floor into a cat live in home."
"Yes, really. So you can all stay with me."
"I'll tell the others and let you know. Are there cat flaps?"
Jinx jumped onto her lap and pressed against her. "Thank you."
She held her breath and wiped at her eyes.
"What's wrong?" asked Jinx.
She shrugged. "The world is so mean," she said, softly, "that when someone is kind…it hurts."
Sammy Super Hero…
Sammy opened the Halloween festivities with a talk about superheroes. He loves his Halloween costume. He runs through The Coop, his cape flowing behind him. Everyone thinks he looks great and they have been asking him to help out with things, since he looks so strong.
Chicklet Sammy


text:
Tattva, termine sanscrito (in scrittura devanāgarī, तत्त्व) che significa "stato vero o reale", "verità", "realtà" "principio reale", o anche "verità". In accordo con varie scuole di filosofia indiane, un tattva è un elemento o aspetto della realtà concepito come una emanazione della realtà assoluta nel processo di manifestazione dal sottile al grossolano. Sebbene il numero dei tattva vari a seconda della scuola filosofica essi, nel loro insieme, sono supposti formare la base di ogni nostra esperienza.

La filosofia Sāṃkhya usa un sistema di 25 tattva mentre lo Shivaismo ne riconosce 36.

<<Tattva (n): l'"essere di ciò", essenza, la natura di tale (ente descritto); "quiddità", realtà, verità; principio; "categoria", principio elementare; i venticinque princìpi, o categorie, enumerati dal Darśana Sāṃkhya (o i ventisei del Darśana Yoga); i trentasei principi dello Śivaismo. Nel significato di "verità sì oppone a mata (verità soggettiva, opinione personale, credenza individuale) in quanto esprime una "verità oggettiva" inconfutablle, di Carattere Universale.>>

Note

Bibliografia
  Subramuniyaswami, Sivaya. Yogas Forgotten Foundation. New York: Himalayan Academy Publications, 2004. 0945497911.
  Aa, Vv. What Is Hinduism?. New York: Himalayan Academy Publications, 2007. 978-1-934145-00-5.

Altri progetti

Collegamenti esterni

Discipline spirituali
Concetti induisti


16 From now on, therefore, we regard no one from a human point of view; even though we once knew Christ from a human point of view, we know him no longer in that way. 17 So if anyone is in Christ, there is a new creation: everything old has passed away; see, everything has become new! 18 All this is from God, who reconciled us to himself through Christ, and has given us the ministry of reconciliation; 19 that is, in Christ God was reconciling the world to himself, not counting their trespasses against them, and entrusting the message of reconciliation to us.
Jesus Christ has changed things. He has changed us. He has changed how we understand God. He has changed how we look at one another. He has changed how we look at the world. These changes began with God, Who "was reconciling the world to Himself" (v. 19). God has initiated the change. It is God Who has made this all possible. It is God Who wants us to view things from a new perspective, from a viewpoint of Divine mercy and grace.
In the new year we often celebrate new beginnings. We have hope as we imagine ourselves enjoying sustained resolutions. Some of our efforts will succeed, others will fail. That's always the case with human efforts. I wish you well in your endeavors to craft a "new-you". But what of God's great resolution? It most certainly will not fail. And it all has to do with mercy and grace.
Mercy is when we do not receive the punishment that we deserve. In terms of God, we do not receive the penalties for the sins we commit; we are pardoned because God is merciful. In grace we receive wonderful blessings that we don't deserve (unmerited favor). In salvation both God's grace and mercy are at work. Sinners (that's all of us, folks) are not condemned, but are forgiven. To the soldiers who nailed Jesus to the cross, Christ said, "Father, forgive them, for they know not what they do." They didn't even ask for it, they hadn't repented, but God forgave them.
Not only are sinners forgiven (mercy), but they are invited to become children of God (new life in Christ), what a remarkable blessing- that's grace.
God's love is beyond even incredible forgiveness; God's love offers incredible blessings – mercy and grace.


text:
\section{Introduction}
\input{1Intro}

\section{Maximum-entropy model of diffeomorphisms}\label{sec::maxentropy}
\input{2MaxEntropyModel}

\newpage
\section{Measuring the relative stability to diffeomorphisms}\label{sec::compression-p}
\input{3DiffeoSensitivity}

\newpage
\section{Relative stability to diffeomorphisms indicates performance}
\label{sec::performance}
\input{4Performance}

\section{Stability toward diffeomorphisms {\it vs.} noise}
\label{sec::typical_grad}
\input{5DiffeoVSGauss}

\section{A minimal model for learning invariants}\label{sec:stripe}
\input{6StripeModel}

\leo{
\section{Discussion}
A common belief  is that stability to random noise (small $G_f$) and to diffeomorphisms (small $D_f$) are desirable properties of neural nets. Its underlying assumption is that the true data label mildly depends on such transformations when they are small. Our observations suggest an alternative view:
\begin{enumerate}
    \item Figs.\ref{fig:GDR_p},\ref{fig:te_Gf}: better predictors are more sensitive to small perturbations in input space.
    \item As a consequence, the notion that predictors are especially insensitive to diffeomorphisms is not captured by stability alone, but rather by the relative stability $R_f=\nicefrac{D_f}{G_f}$.
    \item We propose the following interpretation of Fig.\ref{fig:te_Rf}: to perform well, the predictor must build large gradients in input space near the decision boundary -- leading to a large $G_f$ overall. Networks that are relatively insensitive to diffeomorphisms (small $R_f$) can discover with less data that strong gradients must be there and generalize them to larger regions of input space, improving performance and increasing $G_f$.
\end{enumerate}
This last point can be illustrated in the simple model of Section \ref{sec:stripe}, see Fig.\ref{fig:stripe}-left panel. Imagine two data points of different labels falling close to the -- e.g. -- left true decision boundary. These two points can be far from each other if their orthogonal coordinates differ. Yet, if $R_f=0$ (now defined in Eq.\ref{eq:D_f_stripe}), then the output does not depend on the orthogonal coordinates, and it will need to build a strong gradient -- in input space -- along the parallel coordinate to fit these two data. This strong gradient will exist throughout that entire decision boundary, improving performance but also increasing $G_f$. Instead, if $R_f=1$, fitting these two data will not lead to a strong gradient, since they can be far from each other in input space. Beyond this intuition, in this model decreasing $R_f$ can quantitatively be shown to increase performance, see \cite{paccolat2020isotropic}.
}
\section{Conclusion}

We have introduced a novel empirical framework to characterize how deep nets become invariant to diffeomorphisms. It is jointly based on a maximum-entropy distribution for diffeomorphisms, and on the realization that  stability of these transformations relative to generic ones $R_f$ strongly correlates to performance, instead of just the diffeomorphisms stability considered in the past.

The ensemble of smooth deformations we introduced may have interesting applications. It could serve as a complement to traditional data-augmentation techniques (whose effect on relative stability is discussed in  Fig.\ref{fig:aug_R} of the Appendix). \leo{A similar idea is present in \cite{hauberg_dreaming_2016, shen_anatomical_2020} but our deformations have the advantage of being easier to sample and data agnostic. Moreover, the ensemble} could be used to build adversarial attacks along smooth transformations, in the spirit of \cite{kanbak_geometric_2018, engstrom_exploring_2019, alaifari_adef_2018}. It would be interesting to test if networks robust to such attacks are more stable in relative terms, and how such robustness affects their performance.

\leo{Finally, the tight correlation
between relative stability $R_f$ and test error $\epsilon_t$ suggests that if a predictor displays a given $R_f$, its performance may be bounded from below. The relationships we observe $\epsilon_t(R_f)$ may then be indicative of this bound, which would be a fundamental property of a given data set. Can it be
predicted in terms of simpler properties of the data? Introducing simplified models of data with controlled stability to diffeomorphisms beyond the toy model of Section \ref{sec:stripe} would be useful to investigate this key question.}

\section*{Acknowledgements}
We thank Alberto Bietti, Joan Bruna, Francesco Cagnetta, Pascal Frossard, Jonas Paccolat, Antonio Sclocchi and Umberto M. Tomasini for helpful discussions. This work was supported by a grant from the Simons Foundation (\#454953 Matthieu Wyart).

\subsection{Our Contributions}

\begin{itemize}

\item We introduce a {\it maximum entropy distribution} of diffeomorphisms, that allow us to generate typical diffeomorphisms of controlled norm. Their amplitude is governed by a "temperature" parameter $T$.

\item We define the {\it relative stability to diffeomorphisms index} $R_f$ that  characterizes the square magnitude of the variation of the output function $f$ with respect to the input when it is transformed along a diffeomorphism, relatively to that of a random transformation of the same amplitude.  It is averaged on the test set as well as on the ensemble of diffeomorphisms considered.

\item We find that at initialization, $R_f$ is close to unity for various data sets and architectures, indicating that initially the output is as sensitive to smooth deformations as it is to random perturbations of the image.

\item  Our central result is that after training, $R_f$ correlates very strongly with the test error $\epsilon_t$: during training, $R_f$ is reduced by several decades in current State Of The Art (SOTA) architectures
on four benchmark datasets including MNIST \cite{lecun_gradient-based_1998}, FashionMNIST \cite{xiao_fashion-mnist_2017}, CIFAR-10 \cite{krizhevsky_learning_2009} and ImageNet  \cite{deng_imagenet_2009}. For more primitive architectures (whose test error is higher) such as fully connected nets or simple CNNs, $R_f$ remains of order unity. For CIFAR10 we study 15 known architectures and find empirically that $\epsilon_t\approx 0.2 \sqrt{R_f}$.

\item $R_f$ decreases with the size of the training set $P$. We compare it to an inverse power $1/P$ expected in simple models of invariant learning \cite{paccolat2020compressing}.

\end{itemize}

The library implementing diffeomorphisms on images is available online at
\href{https://github.com/pcsl-epfl/diffeomorphism}{github.com/pcsl-epfl/diffeomorphism}.\vspace*{-0.16cm}

The code for training neural nets can be found at
\href{https://github.com/leonardopetrini/diffeo-sota}{github.com/leonardopetrini/diffeo-sota} and the corresponding pre-trained models at \href{https://doi.org/10.5281/zenodo.5589870}{doi.org/10.5281/zenodo.5589870}.\vspace*{-0.2cm}

\subsection{Definition of maximum entropy model}
We consider the case where the input vector $x$ is an image. It can be thought as a function $x(s)$ describing intensity in position $s=(u,v)\in[0,1]^2 $, where $u$ and $v$ are the horizontal and vertical coordinates. To simplify notations we consider a single channel, in which case $x(s)$ is a scalar (but our analysis holds for colored images as well).

    We denote by $\tau x$ the image deformed by $\tau$, i.e. $[\tau x](s)=x(s-\tau(s))$.

$\tau(s)$ is a vector field of components $(\tau_u(s),\tau_v(s))$.
{The deformation amplitude is measured by the norm}
\begin{equation}
\label{norm}
\| \nabla \tau\|^2=\int_{[0,1]^2}( (\nabla \tau_u)^2 + (\nabla \tau_v)^2 )dudv.
\end{equation}

To test the stability of deep nets toward diffeomorphisms, we seek to build {\it typical} diffeomorphisms of controlled norm $\| \nabla \tau\|$.
We thus consider the distribution over diffeomorphisms
that maximizes the entropy  with a norm constraint.
 It can be solved by introducing a Lagrange multiplier $T$ and by decomposing these fields on their Fourier components, see e.g. \cite{kardar2007statistical} or Appendix \ref{app:max_entropy}. In this canonical ensemble, one finds that $\tau_u$ and $\tau_v$ are independent with identical statistics.
 For  the picture frame not to be deformed, we  impose  fixed boundary conditions: $ \tau=0$ if $u=0,1$ or $v=0,1$. One then obtains:
\begin{equation}
\label{fou}
\tau_u=\sum_{i,j\in \mathbb{N}^+} C_{ij} \sin(i \pi u) \sin(j \pi v)
\end{equation}
where the $C_{ij}$ are Gaussian variables of zero mean and variance  $\langle C_{ij}^2\rangle=T/(i^2+j^2)$.
If the  picture is made of $n\times n$ pixels, the result is identical except that the sum runs on $0<i,j\leq n$.
For large $n$, the  norm then reads $\| \nabla \tau\|^2= (\pi^2/2)\, n^2 T$, and is dominated by high spatial frequency modes. It is useful to add another parameter $c$ to cut-off the effect of high spatial frequencies, which can be simply done by constraining the sum in Eq.\ref{fou} to $i^2+j^2\leq c^2$, one then has $\|\nabla \tau\|^2= (\pi^3/8)\, c^2 T$.

Once $\tau$ is generated, pixels are displaced to random positions. A new pixelated image can then be obtained using standard interpolation methods. We use two interpolations, Gaussian and bi-linear\footnote{Throughout the paper, if not specified otherwise, bi-linear interpolation is employed.}, as described in Appendix \ref{app:interp}. As we shall see below, this choice does not affect our result as long as the diffeomorphism induced a displacement of order of the pixel size, or larger.  Examples are shown in Fig.\ref{fig:imgs_phase_diag} as a function of $T$ and $c$.

\begin{figure}[hbt]
  \begin{minipage}[c]{0.73\textwidth}
  \hspace*{-.1cm}\includegraphics[width=\textwidth]{figures/imagenet_phase_diag.pdf}
  \end{minipage}\hspace*{-.05cm}
  \begin{minipage}[c]{0.27\textwidth}

    \caption{Samples of max-entropy diffeomorphisms for different temperatures $T$ and high-frequency cut-offs $c$ for an ImageNet data-point of resolution $320 \times 320$. The green region corresponds to { well behaving} diffeomorphisms (see Section~\ref{sec::delta_phase_diag}). The dashed line corresponds to  $\delta = 1$. The colored points on the line are those we focus our study in Section \ref{sec::compression-p}.}
    \label{fig:imgs_phase_diag}
  \end{minipage}
\end{figure}

\vspace*{-0.1cm}\subsection{Phase diagram of acceptable diffeomorphisms}
\label{sec::delta_phase_diag}
Diffeomorphisms are bijective, which is not the case for our transformations if $T$ is too large. When this condition breaks down, a single domain of the picture can break into several pieces, as apparent in Fig.\ref{fig:imgs_phase_diag}. It can be expressed as a condition on $\nabla \tau$ that must be satisfied in every point in space \cite{lowe2004distinctive}, as recalled in Appendix \ref{app:allowed_diffeo}. This is satisfied locally with high probability if $\| \tau\|^2 \ll 1$, corresponding to $T\ll (8/\pi^3)/c^2$. In Appendix, we extract empirically a curve of similar form in the $(T,c)$ plane at which a diffeomorphism is  obtained {with probability at least $\nicefrac{1}{2}$ }. For much smaller $T$, diffeomorphisms are obtained almost surely.

Finally, for diffeomorphisms to have noticeable consequences, their associated displacement must be of the order of magnitude of the pixel size.
Defining $\delta^2$ as the average square norm of the pixel displacement at the center of the image in the unit of pixel size, it is straightforward to obtain from Eq.\ref{fou} that asymptotically for large $c$ \leo{(cf. Appendix \ref{app:typical_pix_dis_deform_magn} for the derivation)},
\begin{equation}
    \delta^2 = \frac{\pi}{4} n^2 T \ln(c) .
    \label{de}
\end{equation}

The line $\delta=1/2$ is indicated in Fig.\ref{fig:imgs_phase_diag}, {using empirical measurements that add pre-asymptotic terms to Eq.\ref{de}}. Overall, the green region corresponds to transformations that (i) are diffeomorphisms with high probability  and (ii) produce significant displacements at least of the order of the pixel size.

\subsection{The Relative Compression Ratio, $\mathcal{R}$}

\section{Stability toward diffeomorphisms and noise}

The relative stability to diffeomorphisms $R_f$ can be written as
$R_f=\nicefrac{D_f}{G_f}$
where $G_f$ characterizes the stability with respect to additive noise and $D_f$ the stability toward diffeomorphisms:
\begin{eqnarray}
    G_f =  \frac{\langle \|f(x + \eta) - f(x)\|^2\rangle_{x, \eta}}{\langle \|f(x) - f(z)\|^2\rangle_{x, z}}, \qquad\quad

   D_f = \frac{\langle \|f(\tau x) - f(x)\|^2\rangle_{x, \tau}}{\langle \|f(x) - f(z)\|^2\rangle_{x, z}}.
    \label{eq:D_f}
\end{eqnarray}
Here, we chose to normalize these stabilities with the variation of $f$ over the test  set (to which both $x$ and $z$ belong), and $\eta$ is a random noise whose magnitude is prescribed as above. Stability toward additive noise has been studied previously in fully connected architectures \cite{novak2018sensitivity} and for CNNs as a function of spatial frequency in \cite{tsuzuku2019structural,yin2019fourier}.

The decrease of $R_f$ with growing training set size $P$ could thus be due to an increase in the stability toward diffeomorphisms (i.e.  $D_f$ decreasing with $P$) or a decrease of stability toward noise ($G_f$ increasing with $P$). To test these possibilities, we show in Fig.\ref{fig:GDR_p} $G_f(P), D_f(P)$ and $R_f(P)$ for MNIST, Fashion MNIST and CIFAR10 for two SOTA architectures. The central results are that (i) stability toward noise is always reduced for larger training sets. This observation is natural: when more data needs to be fitted, the function becomes rougher. (ii) Stability toward diffeomorphisms does not behave universally: it can increase with $P$ or decrease depending on the architecture and the training set.
Additionally,  $G_f$ and $D_f$ alone show a much smaller correlation with performance than $R_f$-- see Figs.\ref{fig:te_Df},\ref{fig:te_Gf},\ref{fig:te_Rf_mean} in Appendix \ref{app:more_experiments}.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=\textwidth]{figures/GDR.pdf}
    \caption{\textbf{Stability toward Gaussian noise ($G_f$) and diffeomorphisms ($D_f$) alone, and the relative stability $R_f$.}
    Columns correspond to different data-sets (MNIST, FashionMNIST and CIFAR10) and rows to architectures (ResNet18 and EfficientNetB0). Each panel reports $G_f$ (blue), $D_f$ (orange) and $R_f$ (green) as a function of $P$ and for different cut-off values $c$, as indicated in the legend. \textit{Statistics:} cf. caption in Fig.\ref{fig:R_f}. Error bars -- omitted here -- are shown in Fig.\ref{fig:GDR_error}, Appendix \ref{app:more_experiments}.
    }
    \label{fig:GDR_p}

\end{figure}

\section{Maximum entropy calculation}
\label{app:max_entropy}

Under the constraint on the borders, $\tau_u$ and $\tau_v$ can be expressed in a real Fourier basis as in Eq.\ref{fou}.
By injecting this form into $\|\nabla\tau\|^2$ we obtain:
\begin{equation}
    \|\nabla\tau\|^2 = \frac{\pi^2}{4} \sum_{i,j\in \mathbb{N}^+}
    (C_{ij}^2 + D_{ij}^2) (i^2 + j^2)
\end{equation}
where $D_{ij}$ are the Fourier coefficients of $\tau_v$.
We aim at computing the probability distributions that maximize their entropy while keeping the expectation value of $\|\nabla\tau\|^2$ fixed.
Since we have a sum of quadratic random variables, the equipartition theorem \cite{Statistical_Mechanics} applies: the distributions are normal and every quadratic term contributes in average equally to $\|\nabla\tau\|^2$.  Thus, the variance of the coefficients follows $\frac{T}{i^2 + j^2}$ where the parameter $T$ determines the magnitude of the diffeomorphism.

\section{Boundaries of studied diffeomorphisms }
\label{app:typical_pix_dis_deform_magn}

\paragraph{Average pixel displacement magnitude $\delta$}
\leo{We derive here the large-$c$ asymptotic behavior of $\delta$ (Eq.\ref{de}). This is defined as the average square norm of the displacement field, in pixel units:
\begin{align*}
    \delta^2 &= n^2 \int_{[0, 1]^2} \|\tau(u, v)\|^2 dudv \\
    &= 2Tn^2 \sum_{i^2 + j^2 \leq c^2} \frac{1}{i^2 + j^2} \int_{[0, 1]^2} \sin^2(i \pi u) \sin^2(j \pi v) dudv \\
    &= \frac{Tn^2}{2} \sum_{i^2 + j^2 \leq c^2} \frac{1}{i^2 + j^2} \\
    &\approx \frac{Tn^2}{2} \int_{1 \leq x^2 + y^2 \leq c^2} \frac{1}{x^2 + y^2} dxdy \\
    &= \frac{\pi Tn^2}{4} \int_1^c \frac{1}{r} dr \\
    &= \frac{\pi}{4} n^2 T \log c,
\end{align*}
where we approximated the sum with an integral, in the third step. The asymptotic relations for $\|\nabla\tau\|$ that are reported in the main text are computed in a similar fashion.
}
\leo{In Fig.\ref{fig:emp_c_laws}, we check the agreement between asymptotic prediction and empirical measurements}. If $\delta\ll 1$, our results strongly depend on the choice of interpolation method. To avoid it,
we only consider conditions for which $\delta \geq 1/2$, leading to
\begin{equation}
    T > \frac{1}{\pi n^2 \log c}.
    \label{eq:green_lower_bound}
\end{equation}

\begin{figure}[ht]
    \centering
    \includegraphics[width=\textwidth]{figures/empirical_c_laws.pdf}
    \caption{Left: The characteristic displacement $\delta (c,T)$ is observed to follow $\delta^2 \simeq \frac{\pi}{4} n^2 T \log c$. Right: measurement of $\max_s \Xi$ supporting Eq.\ref{eq::max_Xi}.

    }
    \label{fig:emp_c_laws}
\end{figure}

\label{app:allowed_diffeo}
\paragraph{Condition for diffeomorphism  in the $(T, c)$ plane}
\begin{figure}[ht]
    \centering
    \includegraphics[width=\textwidth]{figures/overhangs_example.pdf}
    \caption{(a) Idealized image at $T=0$. (b) Diffeomorphism of the image. (c) Deformation of the image at large $T$: colors get mixed-up together, shapes are not preserved anymore. (d) Allowed region for vector transformations under $\tau$. For any point in the image $s$ and any direction $u$, only displacement fields for which all the deformed direction $u'$ is non-zero generate  diffeomorphisms. The bound in Eq.\ref{eq:max_xi_bound} ($u' \cdot u > 0$) correspond to the green region. The gray disc corresponds to the bound $\| \nabla \tau \|_{\infty} < 1$.
    }
    \label{fig:overhangs_example}
\end{figure}

For a given value of $c$, there exists a  temperature scale beyond which the transformation is not injective anymore, affecting the topology of the image and creating spurious boundaries, see Fig.\ref{fig:overhangs_example}a-c for an illustration.
Specifically, consider a curve passing by the point $s$ in the deformed image. Its tangent direction is $u$ at the point $s$. When going back to the original image ($s' = s - \tau(s)$) the curve gets deformed and its tangent becomes
\begin{equation}
    u' = u - (u\cdot\nabla) \tau(s).
\end{equation}
A smooth deformation is bijective iff all deformed curves remain curves which is equivalent to have non-zero tangents everywhere
\begin{equation}
    \forall\, s, u \neq 0 \quad \|u'\| \neq 0.
    \label{eq:bij}
\end{equation}
Imposing $\|u'\| \neq 0$ does not give us any constraint on $\tau$. Therefore, we constraint $\tau$ a bit more and allow only displacement fields such that $u\cdot u' > 0$, which is a sufficient condition for Eq.\ref{eq:bij} to be satisfied -- cf. Fig. \ref{fig:overhangs_example}d. By extremizing over $u$, this condition translates into
\begin{equation}
    \tfrac{1}{2}\left(\sqrt{(\partial_x \tau_x - \partial_y \tau_y)^2 + (\partial_x \tau_y + \partial_y \tau_x)^2} - \partial_x \tau_x - \partial_y \tau_y\right) < 1
\end{equation}
or, equivalently,
\begin{equation}
    \label{eq:max_xi_bound}
    \Xi = \tfrac{1}{2}\left(\sqrt{||\nabla \tau||^2 - 2 \det(\nabla\tau)} - \Tr (\nabla\tau)\right) < 1,
\end{equation}
were we identified by $\Xi$ the l.h.s. of the inequality.
We find that the median of the maximum of $\Xi$ over all the image {($\|\Xi(s)\|_\infty$)} can be approximated by (see Fig.\ref{fig:emp_c_laws}b):
\begin{equation}
    \max_s \Xi \simeq \frac{c}{2} \sqrt{\pi^3 T\log c}.
    \label{eq::max_Xi}
\end{equation}
The resulting constraint on $T$ reads
\begin{equation}
T < \frac{4}{\pi^3 c^{2} \log c}.
\label{eq:green_upper_bound}
\end{equation}

\section{Interpolation methods}
\label{app:interp}

When a deformation is applied to an image $x$, each of its pixels gets mapped, from the original pixels grid, to new positions generally outside of the grid itself -- cf. Fig. \ref{fig:overhangs_example}a-b. A procedure (interpolation method) needs to be defined to project the deformed image back into the original grid.

\begin{figure}
    \centering
    \includegraphics[width=.85\linewidth]{figures/interp.pdf}
    \caption{(a) We consider the region between four pixels as the square $[0, 1]^2$ where, after the application of a deformation $\tau$, the pixel $(0,0)$ is mapped into $(u,v)$. (b) \textbf{Bi-linear interpolation}: the value of $x$ in $(u,v)$ is computed by two steps of linear interpolation. First, we compute $x$ in the red crosses, by averaging values on the vertical axis. Then, a line interpolates horizontally the values in the red crosses to give the result. (c) \textbf{Gaussian interpolation}: we denote by $s_i$ the pixel positions in the original grid. The interpolated value of $s$ in any point of the image is given by a weighted sum of $n\times n$ Gaussian centered in each $s_i$ -- in red.}
    \label{fig:interp}
\end{figure}

For simplicity of notation, we describe interpolation methods considering the square $[0,1]^2$ as the region in between four pixels -- see an illustration in Fig. \ref{fig:interp}a. We propose here two different ways to interpolate between pixels and then check that our measurements do not depend on the specific method considered.

\paragraph{Bi-linear Interpolation} The bi-linear interpolation consists, as the name suggests, of two steps of linear interpolation, one on the horizontal, and one on the vertical direction -- Fig. \ref{fig:interp}b. If we look at the square $[0, 1]^2$ and we apply a deformation $\tau$ such that $(0,0) \mapsto (u,v)$, we have
\begin{equation}
    x(u,v) = x(0,0)(1-u)(1-v) + x(1,0)u(1-v) + x(0,1)(1-u)v + x(1,1)uv.
\end{equation}

\paragraph{Gaussian Interpolation} In this case, a Gaussian function\footnote{$G(s) = (2\pi \sigma^2)^{-1/2} e^{-s^2/2\sigma^2}$.} is placed on top of each point in the grid -- cf. Fig.\ref{fig:interp}. The pixel intensity $x$ can be evaluated at any point outside the grid by computing
\begin{equation}
    x(s) = \frac{\sum_i x(s_i)G(s - s_i)}{\sum_i G(s - s_i)}.
\end{equation}
In order to fix the standard deviation $\sigma$ of $G$, we introduce the \textit{participation ratio} $n$. Given {$\Psi_i = G(s, s_i)\vert_{s=(0.5, 0.5)}$}, we define
\begin{equation}
    n = \frac{\left(\sum_i \Psi_i^2\right)^2}{\sum_i \Psi_i^4}.
\end{equation}
The participation ratio is a measure of how many pixels contribute to the value of a new pixel, which results from interpolation. We fix $\sigma$ in such a way that the participation ratio for the Gaussian interpolation matches the one for the bi-linear ($n=4$), when the new pixel is equidistant from the four pixels around. This gives $\sigma = 0.4715$.

Notice that this interpolation method is such that it applies a Gaussian smoothing of the image even if $\tau$ is the identity. Consequently, when computing observables for $f$ with the Gaussian interpolation, we always compare $f(\tau x)$ to $f(\tilde x)$, where $\tilde x$ is the smoothed version of $x$, in such a way that $f(\tau^{[T=0]}x) = f(\tilde x)$.

\paragraph{Empirical results dependence on interpolation} Finally, we checked to which extent our results are affected by the specific choice of interpolation method. In particular, {blue} and {red} colors in Figs\ref{fig:R_f}, \ref{fig:R_f_mnist} correspond to bi-linear and Gaussian interpolation, respectively. The interpolation method only affects the results in the small displacement limit ($\delta\to0$).

\underline{Note}: throughout the paper, if not specified otherwise, bi-linear interpolation is employed.

\newpage
\section{Stability to additive noise \textit{vs.} noise magnitude}
\label{app:G_f}
\begin{figure}[htbp]
    \centering
    \hspace*{-0.7cm}
    \includegraphics[trim={0 0 7.73cm 0}, clip, width=.9\textwidth]{figures/gaussian_noise.pdf}
    \caption{\textbf{Stability to isotropic noise $G_f$} as a function of the noise magnitude $\|\eta\|$ for CIFAR10 ({left}) and ImageNet ({right}).
    The color corresponds to two different classes of SOTA architecture: ResNet and EfficientNet.
    The slope 2 at small $\|\eta\|$ identifies the linear regime. For larger noise magnitudes, non-linearities appear.}
    \label{fig:G_f_eta}
\end{figure}
We introduced in Section \ref{sec::typical_grad} the stability toward additive noise:
\begin{equation}
        G_f =  \frac{\langle \|f(x + \eta) - f(x)\|^2\rangle_{x, \eta}}{\langle \|f(x) - f(z)\|^2\rangle_{x, z}}.
\end{equation}
We study here the dependence of $G_f$ on the noise magnitude $\|\eta\|$.
In the $\eta \to 0$ limit, we expect the network function to behave as its first-order Taylor expansion, leading to $G_f \propto \|\eta\|^2$. Hence, for small noise, $G_f$ gives an estimate of the average magnitude of the gradient of $f$ in a random direction $\eta$.

\paragraph{Empirical results} Measurements of $G_f$ on SOTA nets trained on benchmark data-sets are shown in Figure \ref{fig:G_f_eta}. We observe that the effect of non-linearities start to be significant around $\|\eta\| = 1$. For large values of the noise -- i.e. far away from data-points -- the average gradient of $f$ does not change with training.

\section{Numerical experiments}

In this Appendix, we provide details on the training procedure, on the different architectures employed and some additional experimental results.

\subsection{Image classification training set-up:}
\label{app::trainings}
\begin{itemize}
    \item Trainings are performed in \texttt{PyTorch}, the code can be found here \href{https://github.com/leonardopetrini/diffeo-sota}{github.com/leonardopetrini/diffeo-sota}.
    \item Loss function: cross-entropy.
    \item Batch size: 128.
    \item Dynamics:
    \begin{itemize}
        \item Fully connected nets: ADAM with \texttt{learning rate }$=0.1$ and no scheduling.
        \item Transfer learning: SGD with \texttt{learning rate }$=10^{-2}$ for the last layer and $10^{-3}$ for the rest of the network, {\texttt{momentum }$= 0.9$} and {\texttt{weight decay }$= 10^{-3}$}. Both learning rates decay exponentially during training with a factor $\gamma = 0.975$.
        \item All the other networks are trained with SGD with {\texttt{learning rate }$= 0.1$}, {\texttt{momentum }$= 0.9$} and {\texttt{weight decay }$= 5\times 10^{-4}$}. The learning rate follows a cosine annealing scheduling \cite{loshchilov_sgdr_2016}.
    \end{itemize}
    \item Early-stopping is performed -- i.e. results shown are computed with the network obtaining the best validation accuracy out of 250 training epochs.

    \item For the experiments involving a training on a subset of the training date of size $P<P_{\max}$, the total number of epochs is accordingly re-scaled in order to keep constant
    the total number of optimizer steps.
    \item Standard data augmentation is employed: different random translations and horizontal flips of the input images are generated at each epoch. As a safety check, we verify that the invariance learnt by the nets is not purely due to such augmentation (Fig.\ref{fig:aug_R}).
    \item Experiments are run on 16 GPUs NVIDIA V100. Individual trainings run in $\sim1$ hour of wall time. We estimate a total of a few thousands hours of computing time for running the preliminary and actual experiments present in this work.
\end{itemize}
The stripe model is trained with an approximation of gradient flow introduced in \cite{geiger2019disentangling}, see \cite{paccolat2020compressing} for details.

\begin{figure}[hbt]
    \centering
    \includegraphics[width=\textwidth]{figures/aug.pdf}
    \caption{\textbf{Effect of data augmentation on $R_f$.} Relative stability to diffeomorphisms $R_f$ after training with different data augmentations: "none" (1st group of bars in each plot) for no data augmentation, "translation" (2nd bars) corresponds to training on randomly translated (by 4 pixels) and cropped inputs, and "diffeo" (3rd bars) to training on randomly deformed images with max-entropy diffeomorphisms ($T = 10^{-2}$, $c = 1$). Results are averaged over 5 trainings of ResNet18 on MNIST ({left}), FashionMNIST ({center}), CIFAR10 ({right}). Colors indicate different cut-off values when probing the trained networks.
    Different augmentations have a small quantitative, and no qualitative effect on the results. As expected, augmenting the input images with smooth deformations makes the net more invariant to such transformations.}
    \label{fig:aug_R}
\end{figure}

\paragraph{A note on computing stabilities at init. in presence of batch-norm} We recall that batch-norm (BN) can work in either of two modes: \textit{training} and \textit{evaluation}. During training, BN computes the mean and variance on the current batch and uses them to normalize the output of a given layer. At the same time, it keeps memory of the running statistics on such batches, and this is used for the normalization steps at inference time (evaluation mode).
When probing a network at initialization for computing stabilities, we put the network in evaluation mode, except for batch-norm (BN), which operates in train mode. This is because BN running mean and variance are initialized to 0 and 1, in such a way that its evaluation mode at initialization would correspond to not having BN at all, compromising the input signal propagation in deep architectures.

\newpage
\subsection{Networks architectures}
\label{app::netsarch}

All networks implementations can be found at \href{https://github.com/leonardopetrini/diffeo-sota/tree/main/models}{github.com/leonardopetrini/diffeo-sota/tree/main/models}. In Table \ref{tab:nets}, we report salient features of the network architectures considered.

\vspace{.5em}
\begin{table}[h!]
  \begin{center}
  \caption{\textbf{Network architectures, main characteristics.} We list here (columns) the classes of net architectures used throughout the paper specifying some salient features (depth, number of parameters, etc...) for each of them.\\}
  \label{tab:nets}
  \small{

    \begin{tabular}{c c c c}
        \toprule
        &\\
      \textit{features} & \textbf{FullConn} & \textbf{LeNet} & \textbf{AlexNet}\\
      & & \tiny{\cite{lecun_backpropagation_1989}} & \tiny{\cite{alexnet}} \\
      \midrule
      depth & 2, 4, 6 & 5 & 8 \\
      num. parameters & 200k & 62k & 23 M\\
      FC layers & 2, 4, 6 & 3 & 3 \\
      activation & ReLU & ReLU & ReLU  \\
      pooling & / & max & max  \\
      dropout & / & / & yes  \\
      batch norm & / & / & /  \\
      skip connections & / & / & / \\
      \bottomrule
        \toprule
        &\\
      \textit{features} & \textbf{VGG} & \textbf{ResNet} & \textbf{EfficientNetB0-2}\\
      & \tiny{\cite{simonyan_very_2015}} & \tiny{\cite{he_deep_2016}} & \tiny{\cite{tan_efficientnet_2019}} \\
      \midrule
      depth  & 11, 16, 19 & 18, 34, 50 & 18, 25 \\
      num. parameters  & 9-20 M & 11-24 M & 5, 9 M \\
      FC layers &  1 & 1 & 1 \\
      activation &  ReLU & ReLU & swish \\
      pooling & max & avg. (last layer only) & avg. (last layer only) \\
      dropout & / & / & yes + dropconnect \\
      batch norm & if 'bn' in name & yes & yes \\
      skip connections & / & yes & yes (inv. residuals) \\
      \bottomrule
    \end{tabular}
    }
  \end{center}
\end{table}

\newpage
\subsection{Additional figures}
\label{app:more_experiments}
We present here:
\begin{itemize}
    \item Fig.\ref{fig:R_f_mnist}: $R_f$ as a function of $P$ for MNIST and FashionMNIST with the corresponding predicted slope, omitted in the main text.
    \item Fig.\ref{fig:Rf_depth}: Relative diffeomorphisms stability $R_f$ as a function of depth for simple and deep nets.
    \item Figs\ref{fig:te_Df},\ref{fig:te_Gf}: diffeomorphisms and inverse of the Gaussian stability $D_f$ and $1/G_f$ \textit{vs.} test error for CIFAR10 and the set of architectures considered in Section \ref{sec::performance}.
    \item Fig.\ref{fig:te_Rf_mean}: $D_f$, $1/G_f$ and $R_f$ when using the mean in place of the median for computing averages $\langle \cdot \rangle$.
    \item Fig.\ref{fig:te_Rf_P}: curves in the $(\epsilon_t, R_f)$ plane when varying the training set size $P$ for FullyConnL4, LeNet, ResNet18 and EfficientNetB0.
    \item Figs\ref{fig:te_Rf_error}, \ref{fig:GDR_error}: error estimates for the main quantities of interest -- often omitted in the main text for the sake of figures' clarity.
\end{itemize}
\begin{figure}[hbt]
    \centering
    \includegraphics[width=\linewidth]{figures/R_f_mnist.pdf}
    \caption{\textbf{Relative stability to diffeomorphisms $R_f(P)$ at $\delta=1$.} Analogous to Figure \ref{fig:R_f}-right but here we have MNIST (a-b) and FashionMNIST (c-d) in place of CIFAR10.
    Stability monotonically decreases with $P$.
    The triangles give a reference for the predicted slope in the stripe model -- i.e. $R_f \sim P^{-1}$ -- see Section \ref{sec:stripe}.
    The slopes in case of ResNets are compatible with the prediction. For EfficientNets, the second panel of Fig.\ref{fig:R_f} suggests that stability to diffeomorphisms is less important. Here, we also see that it builds up more slowly when increasing the training set size.
    Finally, blue and red colors indicate different interpolation methods used for generating image deformations, as discussed in Appendix \ref{app:interp}. Results are not affected by this choice.

    }
    \vspace*{-.5cm}
    \label{fig:R_f_mnist}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/R_f_depth.pdf}
    \caption{\textbf{Relative stability to diffeomorphisms as a function of depth.} $R_f$ as a function of the layers relative depth (i.e. $\frac{\text{current layer depth}}{\text{total depth}}$) where "0" identifies the output of the 1st layer and "1" the last. The relative stability is measured for the output of layers (or blocks of layers) inside the nets for simple architectures (1st column) and deep ones (2nd column) at initialization (dashed) and after training (full lines). All nets are trained on the full CIFAR10 dataset. $R_{f_0} \approx 1$ independently of depth at initialization while it decreases monotonically as a function of depth after training. \textit{Statistics:}
     Each point is obtained by training 5 differently initialized networks; each network is then probed with 500 test samples in order to measure $R_f$. The results are obtained by log-averaging over single realizations.}
    \label{fig:Rf_depth}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/te_Df.pdf}
    \caption{\textbf{Test error $\epsilon_t$ \textit{vs.} stability to diffeomorphisms $D_f$} for common architectures when trained on the full 10-classes CIFAR10 dataset ($P = 50k$) with SGD and the cross-entropy loss; the EfficientNets achieving the best performance are trained by transfer learning from ImageNet ($\star$) -- more details on the training procedures can be found in Appendix \ref{app::trainings}. The color scale indicates depth, and the symbols the presence of batch-norm ($\diamond$) and skip connections ($\dagger$).
    $D_f$ correlation with $\epsilon_t$ (corr. coeff.: 0.62) is much smaller than the one measured for $R_f$ -- see Fig.\ref{fig:R_f}.
    \textit{Statistics:}
     Each point is obtained by training 5 differently initialized networks; each network is then probed with 500 test samples in order to measure $D_f$. The results are obtained by log-averaging over single realizations.}
    \label{fig:te_Df}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/te_Gf.pdf}
    \caption{\textbf{Test error $\epsilon_t$ \textit{vs.} inverse of stability to noise $1 / G_f$} for common architectures when trained on the full 10-classes CIFAR10 dataset ($P = 50k$) with SGD and the cross-entropy loss; the EfficientNets achieving the best performance are trained by transfer learning from ImageNet ($\star$) -- more details on the training procedures can be found in Appendix \ref{app::trainings}. The color scale indicates depth, and the symbols the presence of batch-norm ($\diamond$) and skip connections ($\dagger$).
    $G_f$ correlation with $\epsilon_t$ (corr. coeff.: 0.85) is less important than the one measured for $R_f$ -- see Fig.\ref{fig:R_f}.
    \textit{Statistics:}
     Each point is obtained by training 5 differently initialized networks; each network is then probed with 500 test samples in order to measure $G_f$. The results are obtained by log-averaging over single realizations.}
    \label{fig:te_Gf}
\end{figure}
\begin{figure}
    \centering
    \hspace*{-1.1cm}
    \includegraphics[width=1.1\linewidth]{figures/te_Rf_mean.pdf}
    \caption{\textbf{Test error $\epsilon_t$ \textit{vs.} $D_f$, $1/G_f$ and $R_f$ where $\langle \cdot \rangle$ is the mean.} Analogous to Figs\ref{fig:te_Df}-\ref{fig:te_Rf_error}, we use here the mean instead of the median to compute averages over samples and transformations.
    }
    \label{fig:te_Rf_mean}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/te_Rf_P.pdf}
    \caption{\textbf{Test error $\epsilon_t$ \textit{vs.} relative stability to diffeomorphisms $R_f$ for different training set sizes $P$.} Same data as Fig.\ref{fig:te_Rf}, we report here curves corresponding to training on different set sizes for 4 architectures. The other architectures considered together with the power-law fit are left in background.
    For a small training set, CNNs behave similarly.
    \textit{Statistics:}
     Each point is obtained by training 5 differently initialized networks; each network is then probed with 500 test samples in order to measure $R_f$. The results are obtained by log-averaging over single realizations.}
    \label{fig:te_Rf_P}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/te_Rf2_error.pdf}
    \caption{\textbf{Test error $\epsilon_t$ \textit{vs.} relative stability to diffeomorphisms $R_f$ with error estimates.} Same data as Fig.\ref{fig:te_Rf}, we report error bars here.
    \textit{Statistics:}
     Each point is obtained by training 5 differently initialized networks; each network is then probed with 500 test samples in order to measure $R_f$. The results are obtained by log-averaging over single realizations.}
    \label{fig:te_Rf_error}
\end{figure}
\begin{figure}
    \centering
    \hspace*{-1cm}
    \includegraphics[width=1.15\linewidth]{figures/other_datasets.pdf}
    \caption{\textbf{Test error $\epsilon_t$ \textit{vs.} $D_f$, $G_f$ and $R_f$ (on the columns) for different data sets (on the rows).} The corresponding correlation coefficients are shown in Table \ref{tab:corr_coeff}. Lines 1-2: MNIST and SVHN both contain images of digits and show a similar $\epsilon_t(R_f)$. Line 3: FashionMNIST results are comparable to the CIFAR10 ones shown in the main text. Line 4: Tiny ImageNet32 is a re-scaled (32x32 pixels) version of ImageNet with 200 classes and 100'000 training points. The task is harder than the other data sets and is such that we could not train simple networks (FC, LeNet) on it -- i.e. the loss stays $\mathcal{O}(1)$ throughout training -- so these are not reported here.}
    \label{fig:te_Rf_otherdata}
\end{figure}
\newpage
\begin{figure}
    \centering
    \hspace*{-1cm}
    \includegraphics[width=1.15\linewidth]{figures/cifar_deltacut.pdf}
    \caption{\textbf{Test error $\epsilon_t$ \textit{vs.} $D_f$, $G_f$ and $R_f$ for CIFAR10 and varying $\delta$ and cut-off $c$.} Titles report the values of the varying parameters together with corr. coeffs. Parameters corresponding to allowed diffeo are indicated by the green background. Red and blue colors correspond to different interpolation methods. Overall, results are robust when varying these parameters.}
    \label{fig:te_Rf_deltacut}
\end{figure}
\begin{table}[p]
  \begin{center}
  \caption{\textbf{Test error vs. stability: correlation coefficients for different data sets.}}
  \label{tab:corr_coeff}
  \small{
    \begin{tabular}{c c c c}
        &\\
    \toprule
      \textit{data-set} & $D_f$ & $G_f$ & $R_f$ \\
      \midrule
      MNIST & 0.71 & -0.43 & 0.75 \\
      SVHN & 0.87 & -0.28 & 0.81 \\
      FashionMNIST & 0.72 & -0.68 & 0.94\\
      Tiny ImageNet & 0.69 & -0.66 & 0.74 \\
      \bottomrule
    \end{tabular}
    }
  \end{center}
\end{table}
\begin{figure}[htbp]
    \centering
    \includegraphics[width=\textwidth]{figures/GDR_error.pdf}
    \caption{\textbf{Stability toward Gaussian noise ($G_f$) and diffeomorphisms ($D_f$) alone, and the relative stability $R_f$ with the relative errors.} Analogous to Fig.\ref{fig:GDR_p} in which error estimates are omitted to favour clarity. Here we fix the cut-off to $c=3$ and show error estimates instead.
    Columns correspond to different data-sets (MNIST, FashionMNIST and CIFAR10) and rows to architectures (ResNet18 and EfficientNetB0). Each panel reports $G_f$ (blue), $D_f$ (orange) and $R_f$ (green) as a function of $P$ and for different cut-off values $c$, as indicated in the legend. \textit{Statistics}: Each point in the graphs is obtained by training 16 differently initialized networks on 16 different subsets of the data-sets; each network is then probed with 500 test samples in order to measure stability to diffeomorphisms and Gaussian noise. The resulting $R_f$ is obtained by log-averaging the results from single realizations. As we are plotting quantities in log scale, we report the relative error (shaded).
    }
    \label{fig:GDR_error}
\end{figure}

\section*{Checklist}
\begin{enumerate}

\item For all authors...
\begin{enumerate}
  \item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
    \answerYes{}
  \item Did you describe the limitations of your work?
    \answerYes{} Our work seeks to discover empirical laws and, as such, raises more questions than it answers. We list some of them in the conclusion.
  \item Did you discuss any potential negative societal impacts of your work?
    \answerNo{} We don't foresee possible negative societal consequences of our work. On the contrary, we think our work can positively contribute to the effort that the machine learning Community is putting into developing interpretable AI.
  \item Have you read the ethics review guidelines and ensured that your paper conforms to them?
    \answerYes{}
\end{enumerate}

\item If you are including theoretical results...
\begin{enumerate}
  \item Did you state the full set of assumptions of all theoretical results?
    \answerNA{}
	\item Did you include complete proofs of all theoretical results?
    \answerNA{}
\end{enumerate}

\item If you ran experiments...
\begin{enumerate}
  \item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
    \answerYes{}, we provide links to github repositories, code and instructions on how to run it are found there.
  \item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
    \answerYes{} we provide major details in the main text and specific details in Appendix E.1.
	\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
    \answerYes{} We specify that -- to the benefit of figures clarity -- error estimates are not always reported in the main text. In these cases, we provide error estimates in Appendix E.3.
	\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
    \answerYes{} We provide GPUs type and the wall time needed to run each individual training performed. We provide an estimate for the total number of compute time employed for this work.
\end{enumerate}

\item If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
\begin{enumerate}
  \item If your work uses existing assets, did you cite the creators?
    \answerYes{} We use existing data-sets and architectures, all properly cited, to the best of our knowledge.
  \item Did you mention the license of the assets?
    \answerYes{} For the models we did not implement ourselves, the license is present in the \texttt{models} folder of our github repository.
  \item Did you include any new assets either in the supplemental material or as a URL?
    \answerYes{} We provide a new library for generating max-entropy diffeomorfisms on images together with the link to the corresponding github repository.
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    \answerNA{}
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\item If you used crowdsourcing or conducted research with human subjects...
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    \answerNA{}
\end{enumerate}

\end{enumerate}
\section{Introduction}
\input{1Intro}

\section{Maximum-entropy model of diffeomorphisms}\label{sec::maxentropy}
\input{2MaxEntropyModel}

\newpage
\section{Measuring the relative stability to diffeomorphisms}\label{sec::compression-p}
\input{3DiffeoSensitivity}

\section{Relative stability to diffeomorphisms indicates performance}
\label{sec::performance}
\input{4Performance}

\section{Stability toward diffeomorphisms {\it vs.} noise}
\input{5DiffeoVSGauss}

\newpage
\section{A minimal model for learning invariants}\label{sec:stripe}
\input{6StripeModel}

\section{Conclusion}

We have introduced a novel empirical framework to characterize how deep nets become invariant to diffeomorphisms. It is jointly based on a maximum-entropy distribution for diffeomorphisms, and on the realization that  stability of these transformations relative to generic ones $R_f$ strongly correlates to performance, instead of just the diffeomorphisms stability considered in the past.

The ensemble of smooth deformations we introduced may have interesting applications. On the one hand, it could be used to complement traditional data-augmentation techniques (whose effect on relative stability is discussed in  Fig.\ref{fig:aug_R} of the Appendix). On the other hand, this ensemble could be used to crudely estimate $R_f$ for data entering the batch during training. It would allow one to build a regularizing term entering the loss, so as to penalize  predictors that have poor relative stability.

redpajama_set_name: RedPajamaArXiv


MPACT has several tools to help guide you through the special education process. Below you will find helpful sample forms, sample letters, and fact sheets that will provide you with information on your eligibility, evaluation, effective communication tips, individualized education plans, and much more.
Worksheet for school districts to use for reporting the number of students who have an IEP in the district.
This fact sheet explains Missouri disability categories, definitions, and criteria.
This form is often used by schools when requesting that a child be evaluated for special education.
This form documents the decision made by the parent and LEA that a triennial reevaluation is unnecessary.
Process for re-evaluation of a student.
This is an example of how to write a letter to request an Independent Educational Evaluation paid for by the school district.
This fact sheet offers an in-depth look at evaluations and IEP timelines.
Review of Existing Data Documentation Form is a component of the IEP.
Student invitation to an IEP meeting regarding transition.
This form is used to make amendments to a child's IEP.
This is an example of how to write a letter to follow up with the school about your child's IEP.
Answering the following questions prior to your child's next IEP. They should help you see the big picture and identify key issues.
Download this parent participation checklist before your child's next IEP meeting.
This fact sheet is a brief overview of the 8 principles of IDEA.
Helpful questions when reviewing your child's present level of academic achievement and functional performance (PLAAFP).
This chart walks you through the 10 steps of the IEP Process.
Special Education: What I Need to Know This free training helps you to understand the many aspects of obtaining special education services for your child with a disability. This training is also available as an MPACT online training.
School districts must give parents notification prior to taking any action impacting a child's IEP.
Notification of IEP Meeting form.
Use this form to keep track of the documents in your child's file so they are easier to find quickly.
Form that is used to log a student's time in special education on a weekly basis.
This is an example of how to write a letter to request records from your school.
This fact sheet provides an overview about extended school year.
Information on the Missouri Migrant and English Language Learning (MELL) program.
Missouri's Office of Special Education administers state and federal funds to support services for students and adults with disabilities. The Office of Special Education has valuable information about special education in Missouri.


Figure 20.2 shows a flow chart for a process line for the manufacture of acid casein. As can be seen, the plant downstream of acidification is almost identical to the one used for production of rennet casein.
We can provide the client with the complete production line on turn-key basis. Customized Design of Calcium Chloride Plant We will design the CaCl2 plant basing on the client's practical conditions, like fuel, electricity, and its special requirements on the final product.
Calcium is the most abundant and one of the most important minerals in the body. It is essential for cell signaling and the proper functioning of muscles, nerves, and the heart.
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The Bottom Line: Recommendations for Calcium Intake and Bone Health; ... Throughout the lifespan, bones are constantly being broken down and built up in a process known as remodeling. Bone cells called osteoblasts build bone, while other bone cells called osteoclasts break down bone.
How to Remove Calcium Buildup in Pipes How to Remove Calcium Buildup in Pipes. What You'll Need. Water softener system . Pots ... If you repeat the process at least once a year (if you don't get a water softener system) you'll keep your wallet and your pipes happy. Comments.
Di Calcium Phosphate Process DCP (Dicalcium Phosphate) animal feed grade, dicalcium phosphate product . dicalcium phosphate production line in dubai united arab . 18% feed grade DCP dicalcium Phosphate plant is the stone processing production .
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[S946] British GRO births index, Warrington Mar 1930 8c 222.
[S1030] BMD Lancs deaths, Wigan & Leigh Mar 2007 1A001 1.
KAY (nee Scotson) Peacefully in hospital on 29th March 2007 HELEN aged 77 years. Beloved wife of the late SID. Much loved Mum to Susan, Stephen and Paul. Dear mother-in-law to Denis, Vivienne and Janet. A loving and devoted Nan to Pamela, Sarah, Rachael, Christ, Sophie, Ian, Greg, Jonathan and Great Nan to Jard. Much loved sister, sister-in-law and auntie. Good friend and neighbour. Mum you were the heart of our family.
[S1028] British GRO marriage index, Leigh Mar 1955 10d 75.


For sale is ONE kinetic copper wind sculpture with a theme of Dancing Octopus. The tentacles are positioned so the dual spinners will always spin in opposite directions with winds. The entire wind sculpture including the pole and its fixture is made of pure copper with brass hardware. The only metal is the stainless steel ball bearings that are completely covered by brass hardware. Two ball bearings make the two tentacle-mounted wheels move independently and opposite from each other. The pole is made of heavy gauge copper and secured into the ground by an all copper 4-point anchoring system. With its heavy duty construction, this kinetic sculpture can withstand strong winds up to 70 mph. The copper tentacles are not coated and will age naturally. If you want to preserve its original copper shine, you may spray a layer of clear coat on it. We have had great success with clear coats made by Rust-Oleum and Permalac (Peacock Labs). The craftsmanship on this kinetic sculpture is impeccable. The wheels are finely balanced and maintenance free. The sculpture is entirely weather proof and made to last for decades. We proudly back this beautiful wind sculpture with lifetime warranty. Size: The sculpture is 6 ft tall. Each spinning unit is 10 inches tall and 15 inches in diameter.