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

"How to solve the independence between elements? I always feel that my intuition is right. The idea of ​​applying the entropy method to the representative function using VC-dimension should be correct, but how to establish the connection between them afterwards? .”

Xiao Ran's brows wrinkled tighter and tighter. Just at that moment, he seemed to have a flash of inspiration, but when he thought about it carefully, he couldn't catch the inspiration.

It seems that there is only one layer of window paper missing, but it can't be pierced.

This feeling made him very irritable, but the more irritable he was, the less he could remember it.

Xiao Ran pinched his eyebrows and felt that his brain was a little swollen.

This step is already the limit, and the next argument process has no idea at all.

Who asked this question and why is it so perverted!

Xiao Ran had the urge to complain. No wonder with the level of Lao Lu and Lao Liu, they discussed this question for a long time.

As expected, this was something that a high school student like him who had not even gone to college could figure out.

"If you can't solve it, don't force yourself." He had been thinking hard for more than ten minutes as he watched Xiao Ran get stuck on how to estimate the new minimum singular value of the "sum" of independent semi-definite random matrices.

Seeing Xiao Ran's frown getting tighter, Lao Lu, who had been watching quietly for a while, couldn't help but sigh, patted Xiao Ran's shoulder, and said gently: "This question is from Fields this year." It’s normal that you can’t solve the problems raised by prize winners when giving reports at mathematician conferences.”

"Uh, teacher, Professor Liu." Only then did Xiao Ran notice that the two of them had arrived beside him at some point. He looked up at the time and saw that more than an hour had passed.

Judging from the looks like Lao Lu and Lao Liu, they must have been standing next to him for a long time, but they never made a sound to disturb him.

Suddenly he felt a little embarrassed and said: "Sorry, I was a little lost in doing this question and didn't notice you."

"It's okay. Seeing that you are so obsessed with mathematics, as a teacher, it's not too late for me to be happy." Lao Lu laughed, his tone becoming more and more pleasant.

"The food is already cooked, let's go eat first."

"I'll study it for a while." Xiao Ran shook his head, still feeling a little unwilling.

"Don't put too much pressure on yourself." Lao Lu advised.

"When doing mathematics research, it is useless to just think hard. Sometimes you can relax yourself and don't let your nerves be too tense. On the contrary, you will have unexpected effects."

"Haha, Lao Lu is right. The most taboo thing in mathematics is to be impatient and become fat in one breath." Lao Liu also said cheerfully.

"I have been thinking hard about this problem for more than ten days, but I still can't figure out a few steps. It would be damned if you could solve it in more than an hour."

"Let's eat first. In fact, I'm very surprised that you can do this. If you can't put down this question, you can copy it down and study it slowly when you go back. The same thing."

Both Lao Lu and Lao Liu said this, and Xiao Ran could only smile bitterly, feeling that he would not be able to figure it out for a while.

So he nodded, "Then just listen to the teacher."

Just as he stood up, his movement suddenly stalled.

etc!

I think I still have a big killer weapon that I haven’t used yet, right?

Sudden inspiration: can greatly improve wisdom in a short period of time and enter a state of overwhelming inspiration, which lasts for one hour.

My buddy is a cheating player, why did he forget this thing!

He sat down again subconsciously.

"What's wrong?" Seeing Xiao Ran sitting down, Lao Lu asked in surprise.

"Uh." Xiao Ran scratched his head, considered his tone, and said, "Teacher, I suddenly have another inspiration. Can you give me half an hour or ten minutes?"

"I'll give you ten minutes." Lao Lu looked at Xiao Ran seriously, "Are you sure?"

"I'll give it a try." Xiao Ran couldn't guarantee it, so he could only say vaguely.

Give it a try. Old Liu gave Old Lu a strange look.

Lao Lu, you student seems a little too crazy!

"Okay, it's about ten minutes." Lao Lu didn't think it was anything. When he was young, it was common for him to not even bother to eat at the end of the day in order to study a math problem.

Seeing that the two of them agreed, Xiao Ran calmed down and thought silently in his heart. His talent suddenly appeared and he activated it!

Um.

It doesn't seem like there's anything wrong with it, huh? ? ! !

The moment this idea came to his mind, his brain seemed to be hit by an electric current. With a bang, Xiao Ran felt his brain heat up.

Then countless inspirations burst out from his mind.

With only one hour left, Xiao Ran did not dare to relax for a moment, and quickly turned his attention to the Krylov space matrix.

"The idea of ​​applying the entropy method to illustrative functions using VC-dimension is correct. Then how to establish a pair of matrix inverse concentration inequalities between them! This is it! Use matrix inverse concentration inequalities to estimate the VC-dimension and wrap around The over-entropy method directly uses the ε-net theorem!

Then to establish the non-negative condition for the increment of the partial determinant, Markov's inequality can be used here."

"It's done! It's done!" Xiao Ran murmured to himself with a hint of excitement, and he kept writing the pen in his hand.

Got a brain!

I really have a brain!

From the process to various formulas, theorems, and determinants, everything was so clear and clear in Xiao Ran's eyes at this moment, like a beauty slowly undressing in front of him.

Lao Lu and Lao Liu were dumbfounded at this moment!

Looking at Xiao Ran who was writing crazily with a crazy smile on his face, Lao Liu couldn't help but swallowed, tugged on Lao Lu's sleeves, and asked in a low voice: "Are you okay, my student? Why do I look a little weird?" Something's wrong?"

"Go, go, you don't see something right!" Lao Lu glared at Lao Liu dissatisfied and said so, but a trace of worry and concern flashed in his eyes.

Unknowingly, ten minutes had already passed, but Xiao Ran was still writing furiously, with a focus on his face and a hint of fanaticism.

Seeing this situation, Lao Lu and the other two people did not dare to disturb him, mainly because Xiao Ran's expression was really permeable, and he had a sense of déjà vu as a crazy scientist doing human experiments in science fiction novels.

It doesn’t look like acting!

Time seemed to pass very slowly at this moment, and every minute and every second was torture for the two of them.

Another ten minutes passed.

"I figured it out!" At this time, Xiao Ran suddenly murmured excitedly. Looking at the proof process he had written, a hint of excitement appeared in the shock.

"What? What has been solved?" Lao Liu has not yet reacted.

Lao Lu couldn't wait to grab the draft of the argumentation process on Xiao Ran's desk and read it carefully from beginning to end.

Xiao Ran's handwriting was very illegible, and it was difficult for Lao Lu to read it, but he could still see Xiao Ran's thoughts from various formulas and determinants.

First, Xiao Ran classified the problem of the minimum singular value of the original matrix as the lower bound of the absolute value of the inner product of a row in the matrix and the unit vector.

It is then discussed that when the elements in a row of the matrix are Bernoulli random variables, the lower bound on the absolute value of the inner product with the unit vector depends on whether there is an arithmetic structure defined in the field of additive number theory in the unit vector. , here the inverse Littlewood-Offord theory is used to control structured vectors and unstructured vectors respectively to obtain the minimum singular value of an m×m matrix, which loses the independence between any elements.

Then in the second step, the matrix inverse concentration inequality is used to fill in the inverse Littlewood-Offord theory, and finally it is demonstrated that the vector structure corresponds to the global small ball probability bound.

At this point, Xiao Ran's argumentation process has been completed.

The difficulty and creativity of this proof can be broken down into three points: one is the need to define the correct structure when decomposing the unit sphere, and the other is the need to use the matrix inverse concentration inequality to estimate the VC-dimensionality and bypass the entropy method to directly use the ε-net theorem. .

The third is to establish non-negative conditions for the increment of partial determinants to utilize Markov's inequality.

These three points respectively correspond to the steps to reduce the time complexity from O(n^2.34) to O(n^2.32) and then to O(n^2.28) and then simplify the original algorithm.

After reading Xiao Ran's entire argument, Lao Lu was speechless for a long time, and then sighed with emotion: "The future generations are terrible!"

As for Lao Liu, he was completely dumbfounded at this moment!

There is only one thought in my heart, where did this bastard Lao Lu dig out this little monster?