The words “optimal” and “optimize” come from the Latin “Optimus” or “Best”, such as in “Make the Best of Things”. Alessio Figalli, a mathematician at the University Eth Zurich, studies optimal transport: the most efficient allocation of starting points at end points. The scope of the research is wide, including clouds, crystals, bubbles and chatbots.
Dr. Figalli, who received the Fields medal in 2018, loves mathematics that is motivated by concrete problems found in nature. He also likes the 'feeling of eternity' of the discipline, he said in a recent interview. “It's something that will be here forever.” (Nothing is forever, he admitted, but math will be “long enough”.) “I like that if you prove a statement, you prove it,” he said. “There is no ambiguity, it is true or false. In a hundred years you can trust whatever happens. '
The study of optimal transport was introduced almost 250 years ago by Gaspard Monge, a French mathematician and politician who was motivated by problems in military engineering. His ideas found broader application that solve logistical problems during the Napoleonic era – for example, identifying the most efficient way to build fortifications, to minimize the costs of transporting materials throughout Europe.
In 1975, the Russian mathematician Leonid Kantorovich shared the Nobel in economic science for refining a rigorous mathematical theory for the optimal allocation of resources. “He had an example with bakeries and coffee shops,” said Dr. Figalli. The optimization goal in this case was to ensure that every bakery supplied all its croissants on a daily basis, and every coffee shop received all the desired croissants.
“It is called a worldwide wellness optimization problem in the sense that there is no competition between bakeries, no competition between coffee shops,” he said. “It is not like optimizing the usefulness of one player. It optimizes the global usefulness of the population. And that is why it is so complex: because if a bakery or one coffee shop does something else, this will affect everyone others. “
The next conversation with Dr. Figalli – performed at an event in New York City organized by the Simons Laufer Mathematical Sciences Institute and in interviews before and after – is condensed and edited for clarity.
How would you finish the sentence “Math is …”? What is math?
For me, math is a creative process and a language to describe nature. The reason that math is as it is is because people realized that it was the right way to model the earth and what they observed. What is fascinating is that it works so well.
Does nature always want to optimize?
Nature is of course an optimizer. It has a minimal energy principle of nature automatically. Then it becomes of course more complex when other variables enter the comparison. It depends on what you study.
When I applied optimum transport to meteorology, I tried to understand the movement of clouds. It was a simplified model in which some physical variables that can influence the movement of clouds were neglected. For example, you can ignore friction or wind.
The movement of water particles in clouds follows an optimal transport path. And here you transport billions of points, billions of water particles, to billions of points, so it is a much bigger problem than 10 bakeries to 50 coffee shops. The numbers grow enormously. That is why you need mathematics to study it.
What about optimum transport that has recorded your interest?
I was most enthusiastic about the applications, and because of the fact that math was very beautiful and came out of very concrete problems.
There is a constant exchange between what mathematics can do and what people need in the real world. As mathematicians we can fantasize. We like to increase dimensions – we work in infinite dimensional space, of which people always think it's a bit crazy. But it is what uses to use mobile phones and Google and all modern technology we have. Everything would not exist if mathematicians had not been crazy enough to go from the standard boundaries of the Spirit, where we only live in three dimensions. The reality is much more than that.
In society, the risk is always that people simply regard mathematics as important when they see the connection with applications. But it is important above that – thinking, the developments of a new theory that came through mathematics over time that led to major changes in society. Everything is math.
And often mathematics came first. It is not that you wake up with an applied question and you will find the answer. Usually the answer was already there, but it was because people had the time and freedom to think big. Conversely, it can work, but in a more limited way, problem with problem. Major changes usually happen because of free thinking.
Optimization has its limits. Creativity cannot really be optimized.
Yes, creativity is the opposite. Suppose you do very good research in an area; Your optimization schedule wants you to stay there. But it is better to take risks. Failure and frustration are the key. Large breakthroughs, major changes, always come because you take yourself out of your comfort zone at some point, and this will never be an optimization process. Optimizing everything sometimes results in missing opportunities. I think it is important to really appreciate and be careful with what you optimize.
What do you work on these days?
A challenge is the use of optimum transport in machine learning.
From a theoretical point of view, machine learning is only an optimization problem where you have a system, and you want to optimize some parameters or functions, so that the machine will perform a certain number of tasks.
To classify images, measure optimum transport how similar two images are by comparing functions such as colors or textures and bringing these functions into coordination – transporting them – between the two images. This technique helps improve accuracy, making models more robust for changes or distortions.
These are very high-dimensional phenomena. You try to understand objects with many functions, many parameters, and every function corresponds to a dimension. So if you have 50 functions, you are in 50-dimensional space.
The higher the dimension where the object lives, the more complex the optimum transport problem is – it requires too much time, too much data to solve the problem, and you will never be able to do it. This is called the curse of dimensionality. People recently tried to look at ways to prevent the curse of dimensionality. An idea is to develop a new type of optimum transport.
What is the core of it?
By collapsing some functions, I reduce my optimum transport to a lower-dimensional space. Let's say that three dimensions are too big for me and I want to make it a one -dimensional problem. I take some points in my three -dimensional space and project them on a line. I solve the optimum transport on the line, I calculate what I have to do and I repeat this for many, many lines. Then I try to use these results in Dimension One, to reconstruct the original 3D room by gluing a kind of adhesives together. It is not an obvious process.
It sounds like the shadow of an object-a two-dimensional, square shadow gives some information about the three-dimensional cube that casts the shadow.
It's like shadows. Another example is X-rays, these are 2D images of your 3D body. But if you do X -rays in sufficient directions, you can essentially compile the images and reconstruct your body.
The conquering of the curse of dimensionality would help with the shortcomings and limitations of AI?
If we use a number of optimum transport techniques, this might be a number of these optimization problems in machine learning, more stable, more reliable, less biased, safer. That is the meta principle.
And in the interplay of pure and applied mathematics, here is the practical, real-world needing a motivating new math?
Precisely. The engineering of machine learning is far ahead. But we don't know why it works. There are few statements; Compare what it can achieve with what we can prove, there is a huge gap. It is impressive, but mathematically it is still very difficult to explain why. So we can't trust it enough. We want to make it better in many directions and we want math to help.
