The “optimal” and “optimize” words derive from the Latin “optimus” or “better”, as in “make the best of things”. Alessio Figalli, mathematical at the Eth Zurich University, studies optimal transport: the most efficient allocation of the starting points for endpoints. The flow of the investigation is wide, including clouds, crystals, bubbles and chatbots.
Dr. Figalli, to which the field of fields in 2018 was assigned, loves mativated mathematics by concrete problems found in nature. He also likes the “sense 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 mathematics will be around “long enough”) “I like the fact that if you show a theorem, it shows it,” he said. “There is no ambiguity, it’s true or false. In one hundred years, you can rely on it, it doesn’t matter what. “
The optimal transport study was introduced almost 250 years ago by Gaspard Monge, a French and political mathematician who was motivated by problems in military engineering. His ideas found a wider application that solved logistical problems during the Napoleonic era, for example by identifying the most efficient way to build fortifications, in order to minimize the costs of transporting materials throughout Europe.
In 1975, the Russian mathematician Leonid Kantorovich shared the Nobel In Economic Sciences for having perfected a rigorous mathematical theory for the optimal allocation of resources. “He had an example with bakeries and cafés,” said dr. Figalli. The objective of optimization in this case was to guarantee that every day each oven would deliver all its croissants and that each cafeteria had desired all the croissants.
“It is called a problem of optimizing global well -being, in the sense that there is no competition between bakeries, no competition between the cafés,” he said. “It’s not like optimizing the usefulness of a player. It is optimizing the global usefulness of the population. And that’s why it is so complex: because if a bakery or a cafeteria does something different, this will influence all the others. “
The following conversation with Dr. Figalli – conducted in an event in New York City organized by the Simons Laufer Mathematical Sciences Institute and in the interviews before and after – was condensed and treated for clarity.
How would you end the phrase “Mathematics is …”? What is mathematics?
For me, mathematics is a creative process and a language to describe nature. The reason why mathematics is so it is because humans have understood that it was the right way to model the earth and what they were observing. What is fascinating is that it works so well.
Nature always try to optimize?
Nature is of course an optimizer. It has a minimum energy principle: nature alone. So obviously it 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 movement of the clouds. It was a simplified model in which some physical variables that can influence the movement of the clouds have been neglected. For example, you could ignore the friction or the wind.
The movement of water particles in the clouds follows an optimal transport path. And here you are transporting billions of points, billions of water particles, to billions of points, so it is a much larger problem than 10 from baker to 50 cafés. The numbers grow enormously. That’s why you need mathematics to study it.
What about optimal transport has captured your interest?
I was very excited by applications and by the fact that mathematics was very beautiful and came from very concrete problems.
There is a constant exchange between what mathematics can do and what people require in the real world. As mathematics, we can fantasize. We like to increase the dimensions: we work in infinite dimensional space, that people always think is a little crazy. But that’s what allows us now to use Cell Phones and Google and all the modern technology we have. Everything did not exist if the mathematicians had not been quite mad to come out of the standard boundaries of the mind, where we live only in three dimensions. The reality is much more than this.
In society, the risk is always that people see mathematics as important when they see the connection with applications. But it is important beyond: the thought, the developments of a new theory that has come in mathematical time that led to great changes in society. Everything is mathematical.
And often mathematics 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 time and freedom to think big. The opposite can work, but in a more limited way, a problem for problem. Great changes usually occur because of free thought.
Optimization has its limits. Creativity cannot be really optimized.
Yes, creativity is the opposite. Suppose you are doing excellent research in an area; Your optimization scheme would make you stay there. But it is better to take risks. Bankruptcy and frustration are fundamental. Great discoveries, big changes, they always come because in a moment you are pulling out of your comfort zone and this will never be a process of optimization. The optimization of everything translates into missing opportunities sometimes. I think it is important to really evaluate and pay attention to what optimizes.
What are you working on these days?
A challenge is the use of optimal transport in automatic learning.
From a theoretical point of view, automatic learning is only a problem of optimization in which you have a system and you want to optimize some parameters or functionality, so that the machine carries out a certain number of activities.
To classify the images, the optimal transport measures how similar two images are by comparing characteristics such as colors or plots and alignment these features – transporting them – between the two images. This technique helps to improve accuracy, making models more robust for changes or distortions.
These are very highly dimensional phenomena. You are trying to understand objects that have many features, many parameters and each function corresponds to a size. So if you have 50 features, you are in the 50-dimensional space.
The greater the size in which the object lives, the more complex the optimal transport problem is: it requires too much time, too many data to solve the problem and you will never be able to do it. This is called curse of dimensionality. Recently people have tried to examine ways to avoid the curse of dimensionality. An idea is to develop a new type of optimal transport.
What is the essence?
By collapsing some features, I reduce my optimal transport in a low -size space. Let’s say that three dimensions are too big for me and I want to make it a unidimensional problem. I take some points in my three -dimensional space and project them on a line. I solve optimal transport on the line, calculate what I should do and I repeat it for many, many lines. So, using these results in the one size, I try to reconstruct the original 3-D space through a sort of gluing together. It is not an obvious process.
It seems a bit the shadow of an object: a two -dimensional and square shadow provides some information on the three -dimensional cube that launches the shadow.
It’s like shadows. Another example are the X-rays, which are 2-D images of your 3D body. But if you make X -rays in sufficient directions you can essentially put the images together and reconstruct your body.
Would conquer the curse of dimensionality would help with the urges and limitations of the AI?
If we use some optimal transport techniques, perhaps this could make some of these problems of optimization in the most robust, more stable, more reliable, less distorted, safer automatic learning. This is the goal principle.
And, in the interaction of pure and applied mathematics, here is the practical and real world need to motivate the new mathematics?
Exactly. Automatic learning engineering is much forward. But we don’t know why it works. There are few theorems; By comparing what can achieve with what we can demonstrate, there is a huge gap. It is impressive, but mathematically it is still very difficult to explain why. So we can’t trust enough. We want to improve in many directions and we want mathematics to help.