The words “optimal” and “optimize” stem from “optimus” or “best”, as in “do the best of things”. Alessio Figalli, a mathematician at Eth Zurich University, studies optimal transport: the most efficient division of starting points for the last points. The extension of the investigation is wide, including clouds, crystals, bubbles and chatbots.
Dr. Figalli, who was awarded the field of fields in 2018, likes the math that is motivated by the concrete problems found in nature. He also likes the “feeling of eternity” of discipline, “he said in a recent interview.” Something a thing that will be here forever. ” during. “)” I like the fact that if you try a theorem, you try it, “he said. . “
The optimal transportation was presented almost 250 years ago by Gaspard Monge, a French mathematician and politician who was motivated by problems in military engineering. His ideas found wider applications for solving logistical problems during the Napoleonic era – for example, identifying the most efficient way to build fortifications in order to minimize the costs of transporting materials throughout Europe.
In 1975, Russian mathematician Leonid Kantorovich divided Nobel in economic science for refining a rigorous mathematical theory for optimal resource allocation. “He had an example with bakeries and cafes,” said Dr. Figalli. The purpose of optimism in this case was to ensure that on a daily basis each bakery gave all its croissants, and each cafe has desired all the croissants.
“It is called a global problem of Wellness optimism in the sense that there is no competition between bakeries, no competition between cafes,” he said. “It is not how to optimize the benefit of a player. Is optimizing the global benefit of the population. And that’s why it’s so complex: because if a bakery or a cafe does something different, it will affect everyone else. “
The following conversation with Dr. Figalli – performed at a New York City event organized by the Simons Laufer Institute of Mathematics Sciences and in pre -and post interviews – has been condensed and edited for clarity.
How would you end the sentence “Mathematics is …”? What is math?
For me, mathematics is a creative process and a language to describe nature. The reason that mathematics is how it is because people realized it was the right way to model the land and what they observed. What is interesting is that it works so well.
Are you always seeking nature to optimize?
Nature is of course an optimizing. It has a principle of minimal energy-Nature itself. Then it certainly becomes more complex when other variables enter the equation. It depends on what you are studying.
When I was applying optimal transport to meteorology, I was trying to understand the cloud movement. It was a simplified model where some physical variables that could affect the cloud movement were neglected. For example, you can ignore friction or the smell.
The movement of water particles in the clouds follows an optimal transport route. And here you are transporting billions of points, billions of water particles, in billions of points, so it is a much larger problem than 10 bakeries in 50 coffee shops. Numbers grow extremely a lot. That is why you need math to study it.
What about optimal transport to capture your interest?
I was more excited by applications and the fact that math was very beautiful and came from very concrete problems.
There is a constant exchange between what mathematics can do and what people look for in the real world. As mathematicians, we can fantasize. We like to enhance the dimensions – we work in the endless dimensional space that people always think is a little crazy. But it is what allows us to use cell phones and Google and all the modern technology we have. Everything would not exist if mathematicians were not crazy about getting out of the standard limits of mind, where we live in only three dimensions. The reality is much more than that.
In society, the danger is always that people simply see mathematics as important when they see the connection with applications. But it is important beyond that – thinking, the developments of a new theory that came through mathematics over time that led to major changes in society. Everything is mathematics.
And often math came first. It’s not that you wake up with an applied question and find the answer. Usually the answer was already there, but it was there because people had the time and freedom to think greatly. The other way around it can work, but in a more limited way, problem by problem. Big changes usually occur due to free thinking.
Optimization has its boundaries. Creativity cannot be really optimized.
Yes, creativity is the opposite. Suppose you are doing very good research in one area; Your optimism scheme will stay there. But it is better to take risks. Failure and frustration are essential. Great advances, big changes, always come because at one point you are getting yourself from your comfort zone, and this will never be an optimization process. Optimizing everything results in the possibilities that are ever missing. I think it’s important to really appreciate and be careful with what you optimize.
What are you working on these days?
One challenge is to use optimal transport in machine learning.
From a theoretical point of view, learning machinery is just a problem of optimism where you have a system, and you want to optimize some parameters, or features, so that the machine can do a certain number of tasks.
To classify images, optimal transport measures how similar two images are comparing features like colors or textures and placing these features in approximation – by transporting them – between the two images. This technique helps improve accuracy, making models more powerful to change or distortions.
These are very high dimensional phenomena. You are trying to understand objects that have many features, many parameters, and each feature corresponds to one dimension. So if you have 50 features, you are in 50-dimensional space.
The higher the dimension where the object lives, the more complex the optimal problem of transport is – it takes a lot of time, a lot of data to solve the problem, and you will never be able to do it. This is called the dimension curse. Recently people have tried to look at ways to avoid dimension curse. One idea is to develop a new type of optimal transport.
What is its essence?
By collapsing some features, I reduce my optimal transport to a lower dimensions space. Let’s say that three dimensions are very big for me and I want to make it a one-dimensional problem. I get a few points in my three-dimensional space and design them in a line. I solve optimal transport in line, calculate what I have to do, and repeat this for many, many lines. Then, using these results in dimension One, I try to rebuild the original 3-D space from a type of climbing together. It is not an obvious process.
This type sounds like the shade of a two-dimensional, square shade, provides some information about the three-dimensional cube that casts the shadow.
Likes like shadows. Another example is X-rays, which are 2-D images of your 3-D body. But if you make X -ray in enough directions, you can essentially join the images and rebuild your body.
The invasion of the dimension curse would help in its shortcomings and limitations?
If we use some optimal transport techniques, maybe this can make some of these optimism problems in learning the most powerful, more stable, reliable, less biased, safer. This is the principle of Meta.
And, in the interaction of pure and applied mathematics, here the practical need of the real world is the motivation of the new mathematics?
Exactly the machine learning engineering is far away. But we don’t know why it works. There are few theorems; By comparing what it can achieve with what we can testify, there is a big gap. It is impressive, but mathematically it is still very difficult to explain why. So we can’t trust them enough. We want to make it better in many ways, and we want math to help.