Chapter 759 Entering the world of theoretical mathematics?

Next, for almost half of September, apart from responding to some necessary documents and phone calls every day, Chang Haonan locked himself in the office and solved the mathematical problem he had concluded.

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

But the smooth direction was a little different from what he expected.

Because after spending a few days to come up with a complete problem-solving idea, the project directly given by the system turned out to be a proof question.

To put it simply, "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 definition of differential topological geometry of manifolds, a series of basic conceptual issues need to be solved first before we can really start to get to the point.

This is one of the reasons why theoretical mathematics papers are generally extremely 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 is upgraded to LV3, the ability is indeed a level stronger than before——

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

So much so that after he first wrote down the extremely jumping analysis process on paper according to the logical order given by the system, he had no doubt that no other person could understand the content above.

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

It's a bit like a university-graduated teacher who needs to adjust his way of thinking to roughly the same level as the latter before teaching elementary school students to solve problems.

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

After all, the system is in charge of killing and not burying...

Only on your own.

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

When Chang Haonan handed Yao Mengna a pile of proof procedures that had been compiled into photocopies for her to look at, the latter's expression was probably one of surprise and surprise, mixed with a little bit of dissatisfaction.

"Have you solved the problem?"

"So fast?"

"But what if we agreed to participate together?"

It's just a fatal three-part company.

"That's not true."

Chang Haonan quickly waved his hand:

"There is still a long way to go before we can solve the actual problem. I have only achieved some initial results."

"Phase results..."

Yao Mengna lowered her head suspiciously.

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

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

But in fact, it has directly raised the original turbojet 14 with serious patchwork traces to the performance level of a quasi-third-generation mid-range model, and some indicators are even close to those of the F404.

However, as she watched, her expression gradually froze.

It seems... a bit difficult to understand.

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

Fortunately, apart from some novel definitions, Chang Haonan did not use any particularly outrageous theories.

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

This is very important.

Yao Mengna does not specialize in mathematics and has little ability to judge the rationality of new mathematical theories, but her mathematics scores and application level are both good.

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

It's not exactly reading from heaven.

Just when Chang Haonan was about to ask Yao Mengna to go back and watch slowly, the latter suddenly raised her head:

"Can you give me some pieces of paper?"

Clearly, she's making progress.

Chang Haonan quickly pulled out some 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 spent three weeks sorting out, even if it was 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 brought joy to the latter's expression.

"Great!"

Chang Haonan clapped his hands:

“In this way, we can continue to study specific manifold learning algorithms!”

"Uh...Professor Chang."

Yao Mengna frowned, showing some confusion:

"If I understand correctly, this should be a proof of a conjecture that has been put into application. It seems...not directly related to the specific manifold learning algorithm?"

"Indeed not."

Chang Haonan nodded:

“But since even you can understand my proof process, it means that the topology knowledge applied in the entire theory has not been out of date, and the new algorithms developed next should be understandable and applicable by most people. "

"?"

Yao Mengna felt that this sentence didn't sound right, but she couldn't find any angle to refute it, so she opened her mouth several times in a row and suppressed her blush.

Finally, I had to change the topic:

"How about... let's talk about manifold learning algorithms?"

Chang Haonan was still excited at this time:

"As for 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 use the flow with points outside its neighborhood Replace it with the shortest path between them, so that if the low-dimensional manifold where the high-dimensional data is located and a subset of the Euclidean space are overall equidistant, you can get a very ideal embedding result. "

"The second is to construct a reconstructed weight vector between the sample point and its neighborhood points and keep the weight in each neighborhood unchanged in the low-dimensional space. For each sample point and its neighborhood set, respectively Calculate the reconstruction weight, so that only the computational complexity of O((m+k)k^2*N) is needed to complete the embedding process..."

"..."

"We are each responsible for one of these two ideas, so even if one of them fails, there is still a second way for you to choose first."

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

But as Chen Shengshen said——

Really good work, top-notch work, is done by one person.

This sentence still holds great status in the world of mathematics, including applied mathematics.

Two people working together may 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 part, she was already a little dizzy, so she didn't understand much.

"Well."

Chang Haonan nodded:

"That's settled for now."

……

After sending Yao Mengna away, he did not immediately invest in researching the second idea.

On the one hand, the National Day is coming soon. During this period, he will definitely not be able to devote himself to it as before. It is better to wait until he is finished.

On the other hand, the certification project that was previously hung on the system was not judged as completed.

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

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

Therefore, Chang Haonan turned his attention to the computer screen again and sorted out the proof process by himself.

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

As early as a few years ago, in the fields of image recognition and speech recognition, someone 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 prevent researchers from temporarily assuming that it is true and then starting to apply it.

It's just that the performance of computers these days is really poor, so until now, I haven't applied one, two, or three.

Chang Haonan's proof itself can only have a stabilizing effect on the engineering community.

The mentality is probably equivalent to the empirical formula that has been used for so many years, and it turned out to be right in the end.

It may not even be a surprise.

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

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

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

In addition, this proof may also provide some inspiration for human research on multi-dimensional space, such as Poincaré conjecture and other topics.

But the system’s criterion for judging project completion is the impact the project has on the real world.

How can a purely theoretical proof process be influential?

It seems the best thing to do is to publish it...

And find a theoretical mathematics journal to publish it.