ps: (The title of the previous question has been changed slightly, and it has been changed to a Krylov space matrix problem, so that the protagonist can solve it in the way of a random matrix, but if it is a solution problem of sparse linear equations, with the current knowledge reserve of the protagonist, I am afraid that there is a sense of disobedience.)

So in order to make this forced outfit more mellow, I still changed the title, forgive me)

Here is the body part:

"Let G be a real matrix n×8, each of which independently satisfies the standard normal distribution with a probability of ō(m)/n, and zero with a probability of 1−ō(m)/n, we want to prove that Krylov's space matrix K:=[G∣AG∣A²G∣...... ∣A^(m-1)G] has an upper bound of exp(ō(m)) at high probability. ”

Looking at this question, Xiao Ran's brows furrowed unconsciously, the Krylov space matrix is an atypical random matrix, and the conditional number is the ratio of the maximum singular value to the minimum singular value.

The maximum singular value is a norm of the matrix, which can be understood as the data size of the problem, and the minimum singular value can be understood as the degree to which the matrix is not degenerate, so this can be understood as the relative degree of matrix degradation.

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

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

......

"Lao Lu, didn't you say that you say everything at home? Why is it that when my sister-in-law comes over, you are like a mouse seeing a cat? When his wife walked away, Lao Liu glanced at Lao Lu, and his tone was full of contempt.

Lao Lu slowly exhaled when he heard this, his face was solemn: "I really only dare to say one sentence at home, and I don't dare to say a second sentence. ”

Lao Liu: ......

"It's not me saying it, you're the head of the family, sometimes you have to be tough when you should be tough!" Lao Liu patted Lao Lu's shoulder and taught him experience.

How are you better than me?

Lao Lu glanced at him obliquely, and said slowly, "Oh, yes? Then I'll go to your house another day and talk to Sister Su Mei and ask her how tough she is. ”

Lao Liu's hand on Lao Lu's shoulder suddenly paused, and then took it back casually: "....... Hey? What is Xiao Ran looking at, so fascinated that he hasn't spoken for a long time? ”

As he spoke, he buried his head and walked towards Xiao Ran's side, as if there was something that attracted him there.

At this time, Xiao Ran was already completely engrossed, and the scratch paper was full of his scribbled and chaotic formulas and ideas, and he didn't even find out when Lao Liu came to his side for a while.

"Huh!"

Walking to Xiao Ran's side, Lao Liu, who saw clearly what he was writing, suddenly raised his eyebrows and exclaimed, "This is studying the Krylov space matrix issue that Lao Lu and I are arguing about?" ”

Rubbing his chin, he looked at Xiao Ran who was immersed in writing again in surprise, and then looked down at the various determinants he wrote, "Lao Lu, come here!" ”

Lao Liu beckoned to Lao Lu not far away without raising his head.

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

"Be quiet, your student is working on the question we just discussed."

"I'll see." When Lao Lu heard this, he hurriedly poked his head over and took a look, "Well, it's true, this kid really has a pure love for mathematics, and he doesn't forget to study mathematics when he comes to my house." ”

The tone was ten thousand satisfied with Xiao Ran.

"You've really picked up a treasure." Lao Liu said sourly, with envy in his expression.

Lao Lu waved his hand proudly, and pretended to be reserved: "With my level, I can only teach him for another two or three years, and at that time, if he wants to make a breakthrough in mathematics, he will have to rely on his own creation." ”

"Okay, okay, who will you show it!" Lao Liu scolded with a smile, then lowered his head and looked at Xiao Ran's draft, thoughtful: "Do you think Xiao Ran can solve this problem?" ”

Lao Lu Wenyan also carefully glanced at the various determinants listed by Xiao Ran, and frowned: "This question is a bit strange, its elements satisfy the sparse Gaussian distribution, but to prove the result, it is the Gaussian distribution that needs to be satisfied, which means that we need a tool to establish the connection between the two......"

"But what should be used for this tool, to be honest, I only have some superficial ideas, I think of using Markov's inequality to estimate probability, which is mainly based on the fact that the property of the joint Gaussian distribution is to obey the sum of two independent vectors of the joint Gaussian distribution, and still obey the joint Gaussian distribution, but after that, I am not sure whether the polynomial bounds can be obtained after the Gaussian distribution is replaced by a uniform distribution or Bernoulli distribution......"

"In addition, the difficulty of this problem mainly lies in how to estimate the minimum singular value of this random matrix, and 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.

Random matrix theory originated from the study of physical models, and it was found in early experiments that the distribution of eigenvalues and singular values of some large random matrices often converged to some specific distributions, and thus proposed laws on limit distributions such as semicircular law, circular law and Marchenko-Pastur law.

The assumptions and conclusions of these laws are similar to the central limit theorem in classical probability theory (i.e., the distribution of the sum of a large number of random numbers that are independent of each other tends to approach a normal distribution), which requires the assumption that the matrix elements are independent of each other except for a specific structure, and then that the dimensions tend to be infinite.

Still, a limit is a limit, and from the point of view of inequality estimation, it is not very easy to use.

Since the end of the 80s of the last century, people have begun to study the estimation of singular values in a non-progressive sense, and the core part of this is the estimation of the minimum singular value.

The development of random matrices also first dealt with the situation that independent and identically distributed matrix elements obeyed the Gaussian distribution from the beginning, and gradually relaxed the requirements, and began to not require the Gaussian distribution, did not require the same distribution, and obtained more and more accurate estimates.

But the most difficult condition to relax is still independence, which requires, first, to require that the rows of the matrix be independent of each other.

The second is that the matrices are required to have additional structures, such as symmetry, and are otherwise independent of each other.

The third is to require that the correlation between matrix elements decays exponentially with the distance of their positions in the matrix......

"Judging from Xiao Ran's draft, it seems that he is using the VC-dimensionality method to apply entropy to the schematic function, but this requires more stringent minimum singular values, and he may not be able to get effective results by using the entropy method under such conditions......"

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

The more Lao Lu looked, the tighter his brows wrinkled.

Raising his head, he asked, "Lao Liu, where did you find such a problem?" ”

Lao Liu smiled a little embarrassedly: "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. ”

Saying that, he sighed and said helplessly: "But after studying for a long time, I still can't solve the problem of the independence between the elements in it, and then I thought of you, your research on random matrices is a little deeper than mine, and I want to see if you have any way to provide me with some inspiration." ”

"It turned out that it was in vain!" Lao Liu glanced at Lao Lu as he spoke, and said leisurely: "Forget it, I'll go back and study it myself." ”