"How do you solve the problem of independence between elements...... I've always felt that my instincts were right, and that the idea of applying entropy to demonstrative functions with VC-dimensionality should be correct, but how do I establish a connection between them after that......"

Xiao Ran's brows furrowed more and more, just now, he seemed to have a flash of inspiration, but when he thought about it carefully, he couldn't grasp that inspiration.

Obviously, it seems that there is only such a layer of window paper, but it can't be broken.

This feeling made him very irritable, but the more irritable he was, the more he couldn't remember.

Pinching his eyebrows, Xiao Ran felt his brain swell a little.

At this point, the limit is reached, and the rest of the argument process is completely untrained......

Who came up with 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, this question was discussed for a long time.

This is really not something that he, a high school student who hasn't even gone to college, can solve.

"If you can't solve it, you don't have to force yourself." Watching Xiao Ran get stuck in the step of how to give a new minimum singular value estimation of the "sum" of an independent semi-positive definite random matrix, he had been thinking about it for more than ten minutes.

Seeing that Xiao Ran's brows were getting tighter and 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 a question raised by this year's Fields Medal winners when they gave a report at the Mathematicians Conference, and it is normal that you can't solve it." ”

"Uh, teacher, Professor Liu." Xiao Ran only then noticed that the two of them came to his side at some point, and looked up at the time, more than an hour had passed.

Looking at Lao Lu and Lao Liu, it is estimated that they have been standing next to him for a long time, but they have never made a sound to disturb him.

Suddenly a little embarrassed: "I'm sorry, I'm a little fascinated by doing this question, and I didn't notice you." ”

"It's okay, seeing that you are so obsessed with mathematics, it is too late for me to be relieved as a teacher." Lao Lu laughed, his tone becoming more and more pleasant.

"The meal is ready, let's go eat first."

"I'll look into it a little longer." Xiao Ran shook his head, still feeling a little unwilling.

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

"It's useless to do mathematical research, blindly meditate, sometimes give yourself a proper relaxation, don't let your nerves be too tight, but there will be unexpected effects."

"Hehe, what Lao Lu said is right, the most taboo thing about doing mathematics is to be impetuous and become fat in one go." Lao Liu also said happily.

"I've been thinking about this problem for more than ten days, but I still can't think of a few steps, if you can solve it in more than an hour, then it's hell."

"Let's eat first, in fact, I am very surprised that you can do this, if you can't put this question down, you can copy it down, go back and study it slowly, it's the same."

Lao Lu and Lao Liu both said this, and Xiao Ran had no choice but to smile bitterly, feeling that he really couldn't solve it for a while.

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

As soon as he got up, all of a sudden, he froze when he got up.

Wait a minute!

I don't think I have a big killer yet......

Inspiration: It can greatly improve your wisdom in a short period of time, and enter a state of inspiration for an hour.

Dude is an open player, how did you forget this thing!

He subconsciously sat down again.

"What's wrong?" Looking at Xiao Ran's action of sitting down, Lao Lu asked in surprise.

"Ugh." Xiao Ran scratched his head, considered his tone, and said, "Teacher, I suddenly have a hint of inspiration, which can give me half an hour...... No, ten minutes of time? ”

"Give you ten minutes......" Lao Lu glanced at Xiao Ran seriously, "Are you sure?" ”

"I'll give it a try." Xiao Ran didn't dare to guarantee it, so he had to be ambiguous.

Try...... Lao Liu glanced at Lao Lu with a strange face.

Lao Lu, you student...... It seems a little crazy!

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

Seeing that the two agreed, Xiao Ran stabilized his mind, and said silently in his heart, and his talent flashed and activated!

Well......

It doesn't seem to be a counter...... Well??!!

The moment this thought came to his mind, his brain seemed to be hit by an electric current, and with a bang, Xiao Ran only felt that his brain was hot.

Countless inspirations burst out of his mind.

There was only an hour, Xiao Ran didn't dare to relax for a moment, and hurriedly focused his attention on the Krylov space matrix,

"The idea of applying the entropy method to the demonstrative function with VC-dimensionality is correct, and then how to establish the connection between them...... Matrix inverse inequality...... Right! That's it! Use the matrix inverse set inequality to estimate the VC-dimensionality and bypass the entropy method and use the ε-net theorem directly!

Then establish a non-negative condition for the increment of the partial determinant...... Here you can use Markov's inequality......"

"That's it! It's done! Xiao Ran muttered to himself with a hint of excitement, and the pen in his hand did not stop for a moment.

Grow your brain!

Really brainy!

From the process, to various formulas, theorems, and determinants, they are all so clear and clear in Xiao Ran's eyes at this moment, just like a beauty slowly unbuttoning her clothes in front of him.......

Lao Lu and Lao Liu are already stupid at the moment!

Looking at Xiao Ran, who was writing frantically and had a crazy smile on his face, Lao Liu couldn't help but swallow his saliva, pulled Lao Lu's sleeve, and asked in a low voice, "Are you okay as a student?" Why do I look something wrong? ”

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

Unconsciously, ten minutes had already passed, but Xiao Ran was still writing hard, with a hint of fanaticism on his face.

Seeing this situation, Lao Lu and the others didn't dare to disturb it, mainly because Xiao Ran's expression was really a bit infiltrating, and there was a sense of déjà vu in science fiction novels about mad scientists who did human experiments.

It's not like acting!

Time seems to pass very slowly at this moment, and every minute and second is a torment for Lao Lu and the two.

Another ten minutes passed.

"I've got it!" At this time, Xiao Ran suddenly muttered excitedly, looking at the proof process he had written, and there was a hint of excitement in shock.

"What? What's the solution? Lao Liu hasn't reacted yet.

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 is very sloppy, and Lao Lu has a hard time reading it, but he can still see Xiao Ran's ideas from various formulas and determinants.

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

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

Then, in the second step, the inverse Littlewood-Offord theory is populated with matrix inverse set inequalities....... Finally, it is proved that the vector structurality corresponds to the global spherical probability boundary.

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: first, the need to define the correct structurality when decomposing the unit sphere, and second, the need to use the matrix inverse set inequality to estimate the VC-dimensionality and bypass the entropy method and directly use the ε-net theorem.

The third is to establish a non-negative condition for the increment of the partial determinant to take advantage of Markov's inequality.

These three points correspond to the steps of reducing the time complexity from O(n^2.34) to O(n^2.32) to O(n^2.28) and then simplifying the original algorithm.

After reading Xiao Ran's entire argument process, Lao Lu didn't say anything for a long time, and only after a long time did he sigh: "The future life is terrifying!" ”

As for Lao Liu, he is completely stupid at the moment!

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