My house collapsed since I became the top scorer in the college entrance examination

Chapter 78 Krylov Space Matrix
ps: (The title of the previous question has been slightly changed to a Krylov space matrix problem, so that the protagonist can use random matrices to solve the problem, but if it is a problem of solving sparse linear equations, the current protagonist If I want to impress the two big guys with my knowledge reserve, it may feel a little inconsistent.

So in order to make this pretense more rounded, I changed the title, sorry for that)
The following is the body part:
“Suppose G is an n × 8 real matrix, each element of which independently satisfies the standard normal distribution with probability ō(m)/n and is zero with probability 1ō(m)/n. We want to show that Krylov The condition number of the space matrix K:=[G∣AG∣AG∣.∣A^(m-1)G] has an upper bound of exp(ō(m)) ​​with high probability.”

Looking at this question, Xiao Ran frowned unconsciously. The Krylov space matrix is ​​an atypical random matrix, and the condition number is the ratio of the largest singular value to the smallest singular value.

The maximum singular value is a norm of the matrix, which can be understood as the data scale of the problem, while the minimum singular value can be understood as the non-degenerate degree of the matrix, so this can be understood as the relative degree of matrix degeneration.

In this question, the maximum singular value is not difficult to estimate. The difficulty is how to estimate the minimum singular value of this random matrix.

After scratching his head, Xiao Ran was gradually attracted by this question.

"Lao Lu, didn't you say you kept the truth at home? Why did you act like a mouse meeting a cat when your sister-in-law came over?" When his wife walked away, Lao Liu glanced at Lao Lu, his tone full of contempt.

Hearing this, Lao Lu exhaled slowly and looked solemn: "I really only dare to say one sentence at home, and I don't dare to say the second sentence. I say the same thing. Is there any problem?"

Old Liu: .
"It's not me. You are the head of the family after all. Sometimes you have to be tougher!" Lao Liu patted Lao Lu on the shoulder and taught him his experience.

How are you better than me?
Lao Lu glanced at him sideways and said slowly: "I will go to your house to talk to Sister Su Mei some other time and ask her how you are so tough."

Lao Liu's hand on Lao Lu's shoulder suddenly paused, and then he took it back as if nothing had happened: "Eh? What is Xiao Ran looking at? He is so fascinated that he has not spoken for a long time?"

After saying that, he buried his head and walked towards Xiao Ran, as if there was something attracting him there.

At this time, Xiao Ran was completely engrossed. His scrawled messy formulas and ideas were all on the draft paper. For a while, he didn't even notice when Lao Liu came to him.

"what!"

Walking next to Xiao Ran, Old Liu raised his eyebrows when he saw clearly what he was writing, and exclaimed, "Are you studying the Krylov space matrix problem that Old Lu and I were arguing about?"

After rubbing his chin, he once again looked at Xiao Ran in surprise, who was immersed in writing, and then looked down at the various determinants he had written, "Old Lu, come here!"

Lao Liu waved to Lao Lu who was not far away without raising his head.

"What's wrong?" Lao Lu came over, confused.

"Keep your voice down, your student is studying the question we just discussed."

"Let me take a look." Hearing this, Lao Lu quickly turned his head and took a look, "Well, it's true, this child really has a pure love for mathematics. He never forgets to study mathematics when he comes to my house."

His tone expressed satisfaction with Xiao Ran.

"You really found a treasure." Old Liu said sourly, with an expression of envy.Lao Lu waved his hand proudly and pretended to be reserved: "With my level, I estimate that I can only teach him for another two or three years. By then, if he wants to make a breakthrough in mathematics, he will have to rely on his own luck." ”

"Okay, okay, who are you pretending to show!" Old Liu scolded with a smile, then lowered his head and looked at Xiao Ran's draft, thoughtfully: "Do you think Xiao Ran can solve this problem?"

Hearing this, Lao Lu also carefully looked at the various determinants listed by Xiao Ran and frowned: "This question is a bit strange. Its elements satisfy the sparse Gaussian distribution, but the result to be proved is the Gaussian distribution. This means we need a tool to establish the connection between the two."

"But what exactly should this tool be used for? To be honest, I only have some rough ideas. What I want is to use Markov's inequality to estimate the probability. This mainly uses the property of the joint Gaussian distribution to be two independent vectors that obey the joint Gaussian distribution. The sum still obeys the joint Gaussian distribution, but after this, I am not sure whether the polynomial bound can be obtained after replacing the Gaussian distribution with a uniform distribution or a Bernoulli distribution."

“In addition, the difficulty of this question mainly lies in how to estimate the minimum singular value of this random matrix. If you want to estimate the minimum singular value of a random matrix, the main difficulty is how to break through the independence between elements in the random matrix theory. If If this step cannot be solved, the proof of this problem will be impossible.”

Random matrix theory originated from the study of physical models. In early experiments, people found that the distribution of eigenvalues ​​and singular values ​​of some large random matrices often approaches certain specific distributions, and thus proposed such concepts as the semicircle law and the circular law. Laws about limit distributions such as the Marchenko-Pastur law.

The assumptions and conclusions of these laws are similar to the central limit theorem in classical probability theory (that is, the distribution of the sum of a large number of independent random numbers often approaches a normal distribution), which requires the assumption that the matrix elements are independent of each other except for specific structures, and then Let the dimensions go to infinity.

Despite this, the limit is a limit after all, and from the perspective of inequality estimation, it is still not very convenient to use.

Starting around the late 80s, people began to study the estimation of singular values ​​in a non-asymptotic sense, the core part of which is the estimation of the minimum singular value.

The development of random matrices also dealt with the situation where independent and identically distributed matrix elements obeyed Gaussian distribution from the beginning, and gradually relaxed the requirements. It began to no longer require Gaussian distribution and identical distribution, and more and more accurate estimates were obtained.

But the most difficult condition to relax is still independence. This requirement is first changed to require that each row of the matrix be independent of each other.

The second is that the matrix is ​​required to have additional structures, such as symmetry, but otherwise be independent of each other.

The third is to require that the correlation between matrix elements decay exponentially with the distance between the positions in the matrix.
"Judging from Xiao Ran's draft, he seems to be using the VC-dimensional entropy method to apply the representative function, but this has stricter requirements for the minimum singular value. Under such conditions, he may not be able to obtain effective results using the entropy method. result"

"Unless he can find a tool to estimate the VC-dimension and bypass the entropy method"

The more Lao Lu looked, the more he frowned.

Raising his head, he asked: "Old Liu, where did you find such a difficult problem?"

Lao Liu smiled sheepishly: "This is a question that this year's Fields Medal winner accidentally raised when he gave a report at the International Congress of Mathematicians last month. I was somewhat interested in this question at the time, so I asked him Come here and plan to publish an SCI paper on this issue."

As he spoke, he sighed and said helplessly: "But after studying for a long time, I still can't solve the problem of independence between the elements. Only then did I think of you. Your research on random matrices is a little deeper than mine. I was wondering if there was any way you could provide me with some inspiration."

"It turns out that it was all in vain!" Old Liu rolled his eyes at Old Lu and said leisurely: "Forget it, I'd better go back and study it myself."

(End of this chapter)