Chapter 758 Manifold Learning

This question raised by Yao Mengna is not difficult for Chang Haonan to understand.

It's just difficult to solve.

To be honest, this involves a series of issues such as text mining, data visualization, information retrieval, data mining, machine learning and even artificial intelligence.

If we can achieve fully automated production as envisioned by Yao Mengna, it will be Industry 4.0.

At this point in time in 1999, it was obviously unrealistic.

But the impossibility of fully realizing this whole set of things does not mean that there are no parts that can serve as breakthroughs.

For example, data mining and information retrieval are very hot research directions around the millennium.

Its core purpose is to extract valuable knowledge from massive databases and large amounts of complex information, and further improve the utilization of information.

In fact, before Chang Haonan was reborn, the field of aircraft design and manufacturing had already begun to apply this technology, and he himself had been exposed to a lot of it.

But at that time, as an ordinary technician with an engineering background, he did not have much theoretical foundation.

As for the system, we first need to construct a complete and feasible idea.

This resulted in a lot of nouns in his mind, but he didn't know which one was the key to breaking the situation——

In fact, he was faced with the dilemma of being unable to extract valuable information from a large amount of complicated information.

"information……"

Chang Haonan pulled a piece of paper from the side and wrote two words in the middle of the paper.

In an idealized model, it would be best if one piece of data could accurately and uniquely describe a meaning.

That is one-dimensional data.

The word problems I did in elementary school and middle school were generally like this.

In fact, most of the problems faced in real life are also this kind of problem.

For slightly more complex situations, a set of data is often needed to fully describe a meaning.

But at the same time, this set of data often cannot only describe this one meaning.

In order to mathematically describe this phenomenon of a set of (multiple) data corresponding to multiple meanings, it is necessary to expand a set of data in different dimensions.

This is a situation pushed into reality by mathematical theory.

On the other hand, in most cases, the information collected in reality is already expanded high-dimensional data.

And if you want computers to process these high-dimensional data...

Chang Haonan thought for a long time and wrote down three basic conditions on the paper:

1. Compress the original high-dimensional data to reduce the dimensionality of the original high-dimensional data, thereby saving storage space and reducing the computational complexity of high-dimensional data.

2. Eliminate, or at least reduce, the noise hidden in the original high-dimensional data.

3. Extract high-quality data features to improve the performance of subsequent data representation and classification tasks.

He went through these three items in his mind, and then tried to get the system to give a result.

no response.

Obviously, this cannot be counted as a "complete and feasible" idea.

……

Unknowingly, Chang Haonan was sitting at his desk until it was almost time for lunch.

Still haven't come up with a good idea.

Until a cry from the abdomen woke it up from deep thought.

I'm actually a little hungry.

Yao Mengna looked at a noun and three sentences on the paper, and knew that Chang Haonan probably had no idea, so she simply stood up and said:

"How about we go have a meal first?"

"Alright."

Chang Haonan is not the kind of person who is obsessed with getting into trouble.

What's more, when it comes to things like mathematics, I can't figure it out just by thinking about it.

Without inspiration, nothing can be said.

It's better to relax first and change your mind.

Fifteen minutes later, the three of them (along with Zhu Yadan) were sitting around a round table on the second floor of the cafeteria.

This is a small restaurant with an a la carte menu. The price is a bit more expensive than the large canteen below. In addition, it is on an extra floor, so there are not many people eating here.

On the contrary, the small supermarket next to it has a lot of people coming and going.

Chang Haonan had a steaming plate of mutton soup noodles in front of him, but he was not in a hurry to move his chopsticks. Instead, he was looking at the people going up and down the stairs not far away.

In the 90s, instant noodles were still a very popular ready-to-eat food.

When Chang Haonan was studying as an undergraduate, people's conditions were generally poor, and not many people had spare money to afford food.

But by this time in 99, it was no longer uncommon for college students to keep a few bags or even a box in their dormitories.

"You said..."

Chang Haonan suddenly said:

“How do companies that produce instant noodles ensure that they don’t miss out or overfill seasoning packets?”

Yao Mengna, who was eating with her head down, was stunned for a moment, and immediately realized that Chang Haonan was still thinking about the question she just raised.

Stuffing seasoning packets into instant noodles and driving rivets into airplanes are actually similar in terms of mathematical models.

Companies that produce instant noodles are obviously unlikely to have sophisticated equipment and technology.

"Probably...weighing?"

Yao Mengna guessed:

"The seasoning packet accounts for about 10% of the weight of the entire package of instant noodles. If you add less or more, it should be easy to detect."

"Hmm...but there is an error in the weight of the dough itself, and there are several kinds of seasoning packets. Weighing can only prove that the total amount is ok, but it cannot guarantee that it is not wrong..."

Chang Haonan shook his head and denied.

Zhu Yadan next to her looked at Chang Haonan on the left and Yao Mengna on the right. She really didn't know why these two people suddenly discussed this issue.

"that……"

Although she felt that it was a bit of a trick in front of the two doctors, she couldn't help it in the end:

"Before the packaging step, wouldn't it be enough to find someone to watch next to the assembly line?"

Yao Mengna held her forehead with one hand:

"We are just thinking about how we can achieve the same effect without using this person."

"Is this..."

Zhu Yadan shrank his head instantly:

"I just said it casually...but sometimes the role of the human brain may not be replaced..."

Calm returned to the table, save for the occasional faint sound of chewing.

But Chang Haonan still didn't move his chopsticks.

"you're right."

A few minutes later, when Zhu Yadan was about to finish the fried noodles on the plate in front of him, Chang Haonan suddenly said:

"The human brain can parse high-dimensional data in a certain way to gain a perception of the external world."

"?"

Zhu Yadan raised his head with questions in his head, but looking at Chang Haonan's thinking, he was very self-aware and did not interrupt.

"In other words, external information with high dimensions must be underlying a nonlinear manifold structure in a low-dimensional space..."

Nearly 70 years ago, American statistician Harold Hotelling proposed the principal component analysis method to reduce the dimensionality of high-dimensional data.

He believed that the greater the variance, the more information was provided, and conversely, the less information was provided, so he constructed several principal components with large variance and high information content through the linear combination of the original components, and then performed matrix singular value decomposition to reduce the data dimension. .

However, the principal component analysis method is only equivalent to finding the best linear mapping in the sense of minimizing the projection distance, but in reality there are not so many simple linear problems.

However, this idea can be used for reference.

Chang Haonan put down the mutton soup noodles that he had only eaten one bite, stood up and left the canteen quickly.

Zhu Yadan, who was responsible for security, quickly followed.

Yao Mengna's reaction was a little slow. Just as she was about to get up, she realized that she hadn't paid yet, so she had to take out her wallet and walked to the cashier helplessly.

Chang Haonan, who returned to the office, found the piece of paper again.

A few more lines were written below the three basic conditions.

Given a set of high-dimensional data X={x1, x2,...,xn}RD, n is the number of data samples, and D is the dimension of the high-dimensional data.

Assume that the data samples in x come from or approximately come from the data Y={y1, y2,…, yn}Rd in the low-dimensional embedding space.

Find a mapping relationship from high-dimensional observation space to low-dimensional embedding space such that yi=(xi), and a one-to-one reconstruction mapping relationship ^-1 such that xi=^-1(yi).

As he wrote this, Chang Haonan showed a satisfied smile on his face.

Although he still has not given a complete idea, he has at least analyzed the three abstract basic conditions into a concrete mathematical problem.

For theoretical research, clearly raising questions is almost half of the way to success.

Thinking of this, he returned to the top of the paper and wrote six words again.

Manifold learning methods.