The words “optimal” and “optimize” derive from the Latin “optimus” or “best”, as in “draw the best party of things”. Alessio Figalli, mathematician at Eth Zurich University, studies optimal transport: the most effective allowance of starting points at final points. The scope of the survey is wide, including clouds, crystals, bubbles and chatbots.
Dr. Figalli, who received the field medal in 2018, loves mathematics motivated by concrete problems found in nature. He also likes the “sense of eternity” of the discipline, he said in a recent interview. “This is something that will be there forever.” (Nothing is forever, he conceded, but the mathematics will be there for “long enough”.) “I like the fact that if you prove a theorem, you prove it,” he declared. “There is no ambiguity, it’s true or false. In a hundred years, you can count, 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 wider applications solving logistical problems in the Napoleonic era – for example, identifying the most effective means of building fortifications, in order to minimize the transport costs of materials across Europe.
In 1975, the Russian mathematician Leonid Kantorovich shared the Nobel in economics To refine a rigorous mathematical theory for the optimal allocation of resources. “He had an example with bakeries and cafes,” said Dr. Figalli. The objective of optimization in this case was to ensure that daily, each bakery delivered all its croissants, and each coffee has obtained all the desired croissants.
“This is called a global problem of optimizing well-being in the sense that there is no competition between bakeries, no competition between cafes,” he said. “It’s not like optimizing the usefulness of a single player. It optimizes the global utility of the population. And that’s why it’s so complex: because a bakery or coffee is doing something different, it will influence everyone. »»
The following conversation with Dr. Figalli – led during an event in New York organized by the Simons Laufer Mathematical Sciences Institute and in the interviews before and after – was condensed and published for more clarity.
How would you end up the phrase “Mathematics are …”? What is mathematics?
For me, mathematics are a creative process and a language to describe nature. The reason why mathematics is the way he is because humans 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 seek to optimize?
Nature is naturally an optimizer. It has a minimum energy principle – nature in itself. Then, of course, it becomes more complex when other variables enter the equation. It depends on what you are studying.
When I applied optimal transport to meteorology, I was trying to understand the movement of the clouds. It was a simplified model where certain physical variables which can influence the movement of the clouds have been neglected. For example, you can ignore friction or wind.
The movement of water particles in the clouds follows an optimal transport path. And here, you transport billions of points, billions of water particles, to billions of points, it is therefore a much larger problem than 10 bakeries at 50 cafes. The numbers are increasing enormously. This is why you need mathematics to study it.
What about optimal transport has captured your interest?
I was very excited by the applications and by the fact that the mathematics were very beautiful and came from 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 the dimensions – we work in an infinite dimensional space, that people always think a little crazy. But that’s what now allows us to use mobile phones and Google and all the modern technology we have. Everything would not exist if the mathematicians had not been crazy enough to get out of the standard limits of the mind, where we only live in three dimensions. The reality is much more than that.
In society, the risk is always that people simply see mathematics as important when they see the link with applications. But it is important beyond that – thought, the developments of a new theory that has gone through mathematics over time that has led to major changes in society. Everything is mathematical.
And often, the calculations 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 great. The inverse in progress can work, but more limited, problem by problem. Great changes generally occur because of free reflection.
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 scheme would make you stay there. But it is better to take risks. Failure and frustration are essential. Large breakthroughs, great changes, always come because at one point, you withdraw from your comfort zone, and it will never be an optimization process. Optimization of everything sometimes results in missing opportunities. I think it is important to really appreciate and pay attention to what you optimize.
What are you working on these days?
A challenge is to use optimal transport in automatic learning.
From a theoretical point of view, automatic learning is only an optimization problem where you have a system, and you want to optimize certain parameters or features, so that the machine makes a certain number of tasks.
To classify images, optimal transport measures how similar two images are by comparing features such as colors or textures and putting these features in alignment – transport them – between the two images. This technique improves precision, which makes the models more robust to changes or distortions.
These are very high dimension phenomena. You try to understand objects that have many features, many parameters and each feature corresponds to a dimension. So if you have 50 features, you are in a space of 50 dimensions.
The higher the dimension in which the object experiences, the more complex the optimal transport problem – it requires too much time, too much data to solve the problem and you can never do it. This is called the curse of dimensionality. Recently, people have tried to seek ways to avoid the curse of dimensionality. An idea is to develop a new type of optimal transport.
What is the essential?
By collapsing certain features, I reduce my optimal transport to a lower dimension space. Let’s say that three dimensions are too big for me and I want to make it a one -dimensional problem. I take a few points in my three -dimensional space and project them on a line. I solve the optimal transport on the line, I calculate what I have to do, and I repeat for many, many lines. Then, using these results in size, I try to reconstruct the original 3D space by a kind of collage together. It is not an obvious process.
It looks likely in the shadow of an object – a two -dimensional and square shadow provides information on the three -dimensional cube that throws the shadow.
It’s like shadows. Another example is the X -rays, which are 2D images of your 3D body. But if you make X -rays in enough directions, you can essentially reconstruct the images and rebuild your body.
The conquest of the curse of dimensionality would help the gaps and limitations of AI?
If we use optimal transport techniques, this could perhaps make some of these optimization problems in automatic learning, more robust, more stable, more reliable, less biased, safer. It is the meta of the principle.
And, in the interaction of pure and applied mathematics, here the practical and real need is motivating mathematics?
Exactly. Automatic learning engineering is far away. But we don’t know why it works. There are few theorems; By comparing what he can achieve as we can prove, there is a huge gap. It’s impressive, but mathematically, it is always very difficult to explain why. So we cannot trust him enough. We want to improve it in many directions, and we want mathematics to help.
