Chapter 759 Entering the World of Theoretical Mathematics?

Next, for almost half of September, Chang Haonan, in addition to replying to some necessary documents and phone calls every day, locked himself in the office to solve the mathematical problem he had summarized.

It should be said that with the help of the system, this matter went very smoothly.

But the direction of success was somewhat different from his expectations.

Because after spending a few days to conceive a complete solution, the project directly given by the system turned out to be a proof question.

Simply put, "For any set of high-dimensional data X, there must be a mapping relationship that maps X to a set of data Y in a locally simple Euclidean space."

Of course, the actual content is more complicated.

Because it also involves the differential topological geometric definition of manifolds, it is necessary to solve a series of basic conceptual problems before we can really get to the point.

This is also one of the reasons why theoretical mathematics papers are generally very long and difficult to understand.

In short, it took Chang Haonan about three weeks to completely organize the entire proof process on paper.

After the system level was upgraded to LV3, the ability was indeed a level stronger than before-

When giving the "problem-solving steps", Chang Haonan's understanding ability in related fields would also be strengthened.

So much so that at the beginning, after writing down the highly jumpy analysis process on paper according to the logical order given by the system, he had no doubt that no one else could understand the content on it.

For most of the time afterwards, Chang Haonan tried his best to solve this problem.

It was a bit like a college-graduated teacher who needed to adjust his way of thinking to a level roughly equivalent to that of elementary school students before teaching them to solve problems.

And the difficulty of this matter was often no less than solving the problem itself.

After all, the system was in charge of killing but not burying...

You can only rely on yourself.

In a blink of an eye, it was the end of September.

When Chang Haonan handed Yao Mengna a stack of proof processes that had been sorted into photocopies to let her see, the latter's expression was probably surprised with surprise, and mixed with a little dissatisfaction.

"Have you solved the problem?"

"So fast?"

"But we agreed to participate together?"

It was a deadly three-in-one.

"That's not the case."

Chang Haonan waved his hand quickly:

"It's still a long way to go before solving the actual problem. I have only achieved some interim results."

"Interim results..."

Yao Mengna lowered her head with a suspicious look.

According to her experience, the "interim results" mentioned by Chang Haonan are often not interim results.

For example, when working on the turbojet 14, Chang Haonan also said that he had only conducted preliminary research on the compressor principle.

But in fact, the turbojet 14, which was originally heavily pieced together, was directly raised to the performance level of the quasi-third-generation medium thrust, and some indicators were even close to the F404.

However, as she watched, her expression gradually solidified.

It seems... a bit difficult to understand.

From the content, this seems to be a purely mathematical process.

Fortunately, in addition to some novel definitions, Chang Haonan did not use any particularly extraordinary theories.

The part about topology did use some cutting-edge results, but the overall framework still did not break away from tradition, but only made some limited expansions based on the predecessors.

This is very important.

Yao Mengna is not specialized in mathematics, and basically has no ability to judge the rationality of new mathematical theories, but her mathematics grades and application level are good.

This means that although she will face some obstacles, she will still be able to understand what Chang Haonan wrote on the paper in the end.

It's not completely reading a book of heaven.

Just when Chang Haonan was about to let Yao Mengna go back and read it slowly, the latter suddenly looked up:

"Can you give me a few sheets of paper?"

Obviously, she has made progress.

Chang Haonan quickly pulled out a few sheets of printing paper and a ballpoint pen and handed them over.

Yao Mengna immediately lowered her head and began to write and draw on it.

...

In the end, Yao Mengna took the things back.

What Chang Haonan sorted out in three weeks, even if it was based on the answer, was not something she could fully understand in a few hours.

However, just a few days later, Yao Mengna found Chang Haonan again.

"I understand."

These four short words instantly made the latter's expression happy.

"Great!"

Chang Haonan clapped his hands:

"In this way, we can continue to study the specific manifold learning algorithm!"

"Uh... Professor Chang."

Yao Mengna frowned, revealing a puzzled expression:

"If I understand correctly, this should be a proof of a conjecture that has been put into application. It seems... It has no direct relationship with the specific manifold learning algorithm?"

"It really doesn't."

Chang Haonan nodded:

"But since even you can understand my proof process, it means that the topological knowledge used in the entire theory is not out of date, and the new algorithm developed next should also be understood and applied by most people."

"?"

Yao Mengna felt that this sentence sounded wrong, but she couldn't find any angle to refute it, so she opened her mouth several times in a row, and her face turned red.

Finally, I had to change the topic:

"How about... let's talk about the manifold learning algorithm?"

Chang Haonan was still excited at this time:

"Regarding the specific algorithm, I currently have two ideas. One is to replace the geodesic distance between the sample point xi and its neighborhood points with the Euclidean distance between them; and replace the points outside its neighborhood with the shortest path between them on the manifold. In this way, if the low-dimensional manifold where the high-dimensional data is located is equidistant from a subset of the Euclidean space, then a very ideal embedding result can be obtained."

"The second is to construct a reconstruction weight vector between the sample point and its neighborhood points and keep the weights in each neighborhood unchanged in the low-dimensional space. For each sample point and its neighborhood set, the reconstruction weight is calculated separately. In this way, only O((m+k)k^2*N) computational complexity is required to complete the embedding process..."

"..."

"We are responsible for each of these two ideas. In this way, even if one of them doesn't work, there is still a second way. You can choose first."

The reason why Chang Haonan made such an arrangement was not entirely for training.

Rather, as Chen Xingshen said,

Really good work, first-class work, is done by one person.

This sentence is still very important in mathematics, including applied mathematics.

When two people work together, it is possible to produce an effect of 1+1 less than 1.

"I choose the first one."

Yao Mengna made her choice almost without hesitation.

Because when Chang Haonan talked about the second one, she was already a little dizzy, so she didn't understand it very well.

"Okay."

Chang Haonan nodded:

"That's it for the time being."

...

After sending Yao Mengna away, he did not immediately devote himself to the research of the second idea.

On the one hand, the National Day is coming soon, and he certainly can't devote himself to it as much as before. It's better to wait until he is done.

On the other hand, the proof project that was previously hung on the system has not been judged to be completed.

Although the level 3 system can carry out 3 projects at the same time, it means that he still has a vacancy at this time.

But for an obsessive-compulsive disorder, this is like playing a game with unclear question marks, which is unbearable.

Therefore, Chang Haonan turned his attention back to the computer screen and the proof process he had sorted out.

In fact, manifold learning was not a new term even in 1999.

As early as a few years ago, people in the field of image recognition and speech recognition had already proposed this concept and put forward a hypothesis that "high-dimensional data can be processed through its inherent low-dimensional manifold structure."

Although this hypothesis has never been proven, it does not affect the researchers to temporarily assume that it is valid and then start to apply it.

It's just that the performance of computers these days is really sorry, so so far, no one, two, three applications have been made.

And Chang Haonan's proof itself, for the engineering community, can probably only play a stabilizing effect.

The mentality is probably equivalent to using the empirical formula for so many years, and it is indeed correct in the end.

It may not even be a surprise.

But for the theoretical mathematics community, it should still have some value.

Although he did not directly prove the above hypothesis, at least he has proved the universal existence of low-dimensional manifold structure.

It is equivalent to taking a big step in this direction.

In addition, this proof may also provide some inspiration for human research on multidimensional space, such as the Poincare conjecture.

But the system's criterion for judging the completion of a project is the impact of the project on the real world.

How can a purely theoretical proof process have an impact?

It seems that the best way is to publish it...

And find a theoretical mathematics journal to publish it.