Hammer of Technology

This is the most concerned issue of the KLS conjecture.

As mathematicians further abstract, the KLS conjecture can be understood as the shape of the watermelon in high-dimensional space is a container encapsulating gas, and finding the best cut surface is to find the bottleneck of this container.Imagine one, if the watermelon was turned into a dumbbell-shaped container with a gas molecule moving randomly through it, the thinner the connection in the middle of the dumbbell, the harder it would be for the molecule to get to the other side.

So now the real problem that Professor Han wants to solve is to find out how thin the convex container can be at the thinnest point in the high-dimensional space.

To put it more simply and rudely, it is to prove whether there is such a constant c, which is a fixed value in any dimension. If there is, then it means that this watermelon cannot be like a dumbbell in high-dimensional space, with large two sides and a connection in the middle Portions can be very fine.Because this constant c determines that its shape cannot have such a thin connection part.

And if this cannot be proved, then everything is possible, gas molecules may move on one side of the container for a long time in high-dimensional space, and it is difficult to go to the other side

So by solving this problem, the existing computer random walk time can be optimized accordingly.

If it is put into mathematics, if this proposition is solved, it can speed up the volume research in the high-dimensional space of approximate convex body.

But in fact, although this is a geometric problem, the breakthroughs in the previous research on this problem were the contributions of scientists in the computer field.

As early as nine years ago, a computer scientist used random positioning technology to reduce the upper bound of the dimension of this problem when studying this problem, but the effect was not obvious.

Six years ago, two Ph.Ds from the University of Washington improved the previous random positioning technology and further reduced the KLS factor, which is the factor used to describe the existence of bottlenecks, to the fourth root of the dimension.

If they can reduce the only power exponent to almost 0, then the 0th power of this number is always equal to 1, which proves that the KLS factor is a constant that has nothing to do with the dimension, thus completely ending this problem, these two I did try, but failed in the end, and the proof process was proved to be wrong, so I just left some ideas for future generations to learn from.

The topic that Professor Han is applying for now is to solve this problem.

For others, this is just an ordinary opening report, but after reading it, Ning Wei suddenly had a flash of inspiration in his mind, because he found that this problem can be solved with the statistical knowledge he just sorted out recently. solve.

Yes, you don't need to use algebraic geometry, and you don't need too advanced computer technology. You only need to use the content of statistics to solve this problem.And if this problem is solved, his statistics graduation thesis can be completed, and at the same time, Lao Han probably has nothing to do in the near future, so he can rejoin the EDA project team as he wishes.

Yes, at this moment Ning Wei only felt that some things in this world were too coincidental.

After searching for a graduation thesis proposition for a long time, it turned out that because of a casual agreement with Yu Xingwei before, and then an opening report came, it was solved.

Not to mention anything else, if the idea in his brain can withstand verification at this time, and the mathematics community has not yet solved this conjecture, then this paper, let alone the first section of SCI, does not seem to be a big problem if it is published as a top journal .

More importantly, the laboratory seems to be able to achieve multiple results again.

Ning Wei subconsciously looked at the funding that Professor Han applied for, 30.

"Um, Professor Lu, I suddenly thought of something urgent, so I'm going back to the dormitory first. Other things can be done as you think, I'm fine." Ning Wei said, and put Professor Han's opening report back on Lu Changbin's desk superior.

"Ah? Urgent matter?" Lu Changbin was a little puzzled. This little guy was just reading Lao Han's report in a leisurely manner. Why did he suddenly have an urgent matter?

"Yes, it's urgent." Ning Wei said.

"Okay, then go quickly." Lu Changbin said hastily.

"Goodbye, Professor Lu."

……

Ning Wei hurried back to the dormitory, and there were three people in the dormitory who were reading books at the desk, and writing papers.

No way, the last hurdle of graduating from 985 university is really not so easy, especially for the scumbags who are not very interested in mathematics, sometimes when writing a thesis, you really need to look up formula by formula, one page book Turning a page of the book is even more exasperating because you often can't find the knowledge points you need to use.

Fortunately, several people have some ideas in their hearts. They didn't ask Ning Wei to guide them. They didn't think for themselves, and they argued that they couldn't pass the test, which was even more disgusting.

After greeting the roommates, Ning Wei woke up the computer.

In the dormitory these days, Sanyue, the virtual little milk cat, has successfully won the love of several people in the dormitory, especially when this milk cat is anxious, the way it stares and bluffs people is very popular.

But like Yu Xingwei, none of the other three regarded Sanyue as a smart display program, they just regarded it as a pet applet developed by Ning Wei when he was bored living alone in the laboratory.

Xu Ruixuan also mentioned copying the program to his computer as a surprise for Xu Mo, but after being directly rejected by Ning Wei, he gave up on it.

Ning Wei occasionally spoke to Sanyue, and was also regarded by the three as a kind of dispatch from the god of learning.

After all, it is somewhat understandable for a guy with outstanding abilities to be a little eccentric.

For example, Ning Wei not only has the eccentricity of teasing virtual cats, but also the eccentricity of hiding, and he has been hiding for three years.

This is the conclusion reached by the three of them when Ning Wei lived in the laboratory and had night talks together every night.

After all, they couldn't accept the fact that Ning Wei suddenly became enlightened. The most reasonable explanation is that in the past three years, Ning Wei has been using his high IQ mind, insightful and caring eyes to silently watch the three mentally handicapped, and Try hard to suppress its IQ so that it can fit into them perfectly.

It wasn't until the senior year that everyone was going to be separated, and the unfair stimulation of the avoidance student finally caused the child to explode...

In this way, everything can have a perfect explanation.

……

Ning Weike didn't care at all what the three people in the dormitory thought of him.

Anyway, everyone is familiar with it.

At this time, he has temporarily withdrawn from the March program, and then began to search for content about the KLS conjecture in the library's retrieval system.

Soon he found the paper mentioned in Professor Han's opening report in the NeurIPS2016 conference report. Through this paper, he roughly understood the previous thinking on solving this problem, and then began to compare it with the solution in his mind.

After careful analysis of the brain, Ning Wei first started programming, and he needed to verify whether his ideas were correct.

Computer deep learning, from a statistical point of view, can basically be regarded as a recursive generalized linear model.

To put it simply, the idea derived from Ningwei's brain is to reduce the upper bound of the KLS factor by using a recursive method. After repeated iterations, the KLS conjecture problem is pulled back to the upper bound of d0 to solve this problem.

Now Ningwei is no longer satisfied with writing the paper directly when he has an idea in his brain, but has to try to solve the problem by himself first, and then write the paper.

"Huh? What are you doing for Ningwei?" Xu Ruixuan got up to pour water, just in time to see what Ningwei was beating, and asked casually.

"I've just decided on the topic of my final thesis, let's do a small experiment first." Ning Wei replied without turning his head.

Everyone is used to this state. When Ning Wei is concentrating on something, he is often defiant.

"Huh...it turns out that a good person has to program and verify before writing a thesis. I love it..." Xu Ruixuan sighed, and then went straight back to his seat.

"I'm going, are you so arrogant? Let me take a look..."

One sentence attracted two other people to watch. After reading it, they said "I love it" twice, and then returned to their seats.

I can't help it, I don't understand it, but if I don't sigh a few words, I will look not very smart.

Of course, this small disturbance will not affect Ning Why at all.

It took him an afternoon to roughly verify that his thinking was indeed correct. This method can indeed prove that the KLS factor is a constant that has nothing to do with dimension.That is to say, no matter which dimension is upgraded to, there is indeed a constant, and the area of ​​at least one section of the convex body is larger than this constant.

Then there is the thesis.

After the hands-on verification, the paper is written more smoothly.

Please ask Liu Cong to bring dinner, it took about five hours, and Ning Wei finished this graduation thesis, which he spent a week studying and got inspiration from Professor Han’s project——"Using Recursive Method to Prove that KLS Factor Is A dimension-independent constant".

The topic is simple and rude, but Ning Wei thinks it is very beautiful.

After all, this result is enough to prove that in high-dimensional space, it is impossible for a convex object to have a structure like a dumbbell.So walking randomly in an n-dimensional convex body, the speed of walking through the entire graph is faster than everyone expected before.

Ning Wei felt that when Professor Han read his paper, he would be very pleased, so Ning Wei was quite satisfied with his paper.

But unfortunately, this is his graduation thesis, and it has to be sent to his graduation thesis supervisor first.

With this in mind, Ning Wei checked the paper again, corrected a few typos, saved the paper, and sent it directly to Professor Yuan via email.

March was then re-run.

On weekends and at night without electricity, you can leave March here and let it continue to think about the things you are interested in, although the little guy's thinking mode must be very simple now.

Ning Wei has already made up his mind that when he goes to the capital after graduation, he will arrange a special room for Sanyue and equip it with a workstation, so that his learning ability will be stronger and he will be better trained.If the dormitory arranged by the school is inconvenient, he will rent a suite in the school.

Anyway, he is already rich now...

I don't care about this expense.

The three people who worked on the thesis for a day in the dormitory were already asleep. Ning Wei couldn’t make a sound, so he teased the little guy with the mouse. Jumping around the screen is also quite interesting.

It wasn't until a few minutes later that Sanyue was too lazy to pay attention to the mouse dangling in front of her eyes, and lay down in front of the screen in a daze, Ning Wei then dropped the mouse with satisfaction, then took off her coat, and climbed directly onto the bed.

Hmm, yes, is this a biological instinct that has evolved to sense the fatigue that comes with repetitive movements?

……

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131 Go your own way and let others have no way

Yuan Chengming got up early in the morning, came to the office, and opened the mailbox.

These days, students have been submitting graduation thesis one after another, and he has to read each of them, and then propose revisions.

If there are only some minor problems, just reply to the email and point them out one by one, and then the students can print them and submit them officially.

If the problem is very big, it is difficult to explain it clearly by email, and you have to make an appointment with the student on WeChat or phone to give face-to-face guidance.

There are two papers submitted for preliminary review today.

Seeing Ning Wei's name, Yuan Chengming was happy.

This young man is indeed a fast shooter for thesis. He came to him last week to decide on the topic. In just a few days, the thesis was handed in.

I don't know if this kid met the SCI District [-] requirements he proposed.

However, Yuan Chengming didn't read the paper impatiently, but made himself hot tea first, and prepared to drink it slowly.

After downloading the thesis from the attachment and opening it, Professor Yuan was stunned when he saw the title.

What the hell is this? "Using recursion to prove that the KLS factor is a constant that has nothing to do with dimension"?

For the first time, Yuan Chengming saw his lack of knowledge in the current stage of mathematics research from a final thesis topic.

Because at the first moment, he found that the word KLS factor was a little strange in his mind.

Well, I have been teaching statistics all these years, and the branch of mathematics development has become more and more detailed. It is normal to not know much about these.So comforting himself in his heart, Professor Yuan began to log into the library's search catalog to find things about the KLS factor.

A proposition about this conjecture was soon found.

Oh, it turned out to be a conjecture in space geometry, but I have been tirelessly trying to solve this problem. Besides mathematicians, why are there so many elites in the computer field?

After carefully studying the predecessors' papers, Yuan Chengming finally sorted out the context of this issue.

Then it got more complicated.

So, if Ningwei's paper is correct, then he used statistical methods to prove a world-class space geometry problem that the computer science community has always been interested in?

You must know that the mathematician who proposed this conjecture was originally the winner of the Fields Medal. Before his death in 2018, he was still thinking about whether this conjecture could be proved.

After figuring out this proposition, Yuan Chengming opened this graduation thesis with apprehension. After all, this thesis could not solve a high-dimensional space geometry problem that has plagued mathematicians all over the world for more than 20 years, and he was just a humble college statistician. As a professor of science, I feel a little unworthy to ask him to guide this thesis.

Even if this paper is really going to be submitted to a journal, it is impossible for other editors to hand it over to him for review.

Since Riemannian geometry has theoretically proved the existence of high-dimensional spaces, these mathematical propositions have become more and more metaphysical.

Unsurprisingly, after reading the whole paper, Yuan Chengming once again felt how important it is to expand the breadth of knowledge. His knowledge reserve of linear algebra and spatial analytic geometry is still insufficient, and he can understand the recursion used in this paper. Method, through step-by-step iterations, the method used to finally deduce the KLS factor as a constant, and the idea of ​​​​each derivation.

But whether it is right or not is beyond the scope of his knowledge.

Of course, this is also normal.

If the derivation process of a world-class mathematical problem can be read by any mathematics professor in any research direction, and he can know whether it is right or wrong, then there will probably not be so many quarrels in the entire academic world.

Yuan Chengming wanted to slap himself in the face and make him talk badly. He was definitely qualified to publish a paper in the first area of ​​SCI. No, this paper was qualified to be published in the top journal. As a result, as a graduation thesis, his supervisor, let alone advising, directly reviewed it. They dare not try.

It was rumored that he might become Jiang University's first tutor for an undergraduate graduation thesis, but he was not sure whether his own student's thesis was right or wrong. It was a shame.

But there was no way, he still picked up the phone and called Kong Mingde directly.

"Dean Kong, I have a graduation thesis here. You may need to organize people to discuss it together. This thesis uses statistical methods to prove that the KLS factor is a constant that has nothing to do with dimension."

"Huh? What did you say? Old Yuan, you mean that you received a graduation thesis to solve the structural problems of convex objects in high-dimensional space?"

Well, it really is the dean who studies linear algebra. Knowing this proposition, it is not like him. He has to check the information first to study this stuff.

"Yes! Solved, I think there is nothing wrong with the argumentation process, but this thing is beyond the scope of my research, so..."

"Which student's thesis is this again? This year's seniors are really hidden dragons and crouching tigers?" Before Yuan Chengming was given time to finish speaking, he said pleasantly on the other side of the phone.

"Dean Kong, you are overthinking, the thesis is for Ning!" Yuan Chengming simply patted the Dean's dream.

"What? What did Ning do? Why did Ning become interested in statistics and spatial geometry again? What was he thinking? The subject of PDE is no longer studied? Didn't he just published another article in "Nature"? Is it a paper on the general solution of the NS equation with a limited range of values?"

"How do you say this? Maybe the child wants to change his mind and rest his head?"

"Stop talking nonsense. I'll organize people to review it right now. You can come up too. Go to the conference room on the third floor. Wait a minute. Send me the paper first, and I'll help him take a look first."

"Okay, let's post it now. By the way, do I need to call Ning Wei?"

"No need, let's finish the trial first. By the way, you are not in a hurry. Someone from the Academy of Computer Science and Technology should also be interested in this issue. I will say hello to the Academy of Computer Science and the laboratory first. Let's come together Interrogation. I'll call you when everyone arrives."

"Okay, then I'll send you the paper first!"

"it is good!"