The terms “optimisation” and “optimisation” come from the Latin word “Optimus” or “best”, like “the best thing.” Alessio Figalli, a mathematician at the University of ETH Zurich, studies optimal transport. The most efficient allocation of starting points to endpoints. The scope of our research is wider, including clouds, crystals, bubbles, and chatbots.
Dr. Figalli, awarded Fields Medal in 2018, loves mathematics, which is motivated by concrete problems found in nature. He also likes the “eternal sense” of discipline, he said in a recent interview. “It's something that comes here forever.” (Nothing forever, but he admits, mathematics is for “long enough.”) “There's no ambiguity. It's true or false . In 100 years, you can rely on it no matter what.”
The study of optimal transport was introduced almost 250 years ago by the French mathematician and politician Gaspad Monge, motivated by military engineering problems. His idea discovered a wider application to solve logistic problems in the Napoleon era. For example, we have identified the most efficient ways to build fortresses to minimize the cost of transporting materials across Europe.
In 1975, Russian mathematician Leonid Kantrovich shared the Nobel of Economic Sciences to improve strict mathematical theory for optimal allocation of resources. “He had examples of bakeries and coffee shops,” Dr. Figari said. The goal of optimization in this case was to ensure that every baker had delivered all the croissants each day, and that every coffee shop had got all the croissants they wanted.
“This is called a global wellness optimization problem in the sense that there is no competition between bakers and no competition between coffee shops,” he said. “It's not like optimizing one player's utility. It's optimizing the global utility of the population. And that's why it's so complicated. One bakery or one coffee shop Because if you do something different, this will affect everyone else.”
The following conversations with Dr. Figari, held at an event in New York City and interviews organized by the Simmons Laufer Institute of Mathematical Sciences, were condensed and edited for clarity.
How do you finish the sentence “Mathematics is…”? What is mathematics?
For me, mathematics is a creative process and a language that describes nature. The reason mathematics is that they realize that it is the right way for humans to model the Earth, and that they were observing. What's appealing is that it works very well.
Is nature constantly trying to optimize?
Naturally, nature is an optimizer. It has a minimum energy principle – by itself. Of course, it becomes more complicated when other variables enter the equation. It depends on what you are studying.
When I was applying optimal transport for meteorology, I was trying to understand the movement of clouds. It was a simplified model in which several physical variables that could affect cloud movement were ignored. For example, you may ignore friction and wind.
The movement of water particles in the cloud follows the optimum transport path. And here we are transporting billions of points, billions of water particles to billions of points, which is a much bigger problem than 50 coffee shops and 10 bakeries. The numbers are very large. That's why mathematics is needed to study it.
How did the optimal transport capture your interest?
I'm most excited about the application and it turned out to be the fact that math came from a very beautiful and very specific problem.
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 dimension spaces. But that's what allows us to now use mobile phones and Google, and all the latest technology we have. If mathematicians weren't crazy enough to break out of the standard boundaries of the mind, if we only live in three dimensions, then everything doesn't exist. The reality is more than that.
In society, the risk is always that people think mathematics is important when they see their connection to applications. But beyond that, it is important – thinking, the development of new theories brought about through mathematics over time, has led to major changes in society. It's all mathematics.
And in many cases, mathematics came first. It's not about awake to the applied questions and finding the answer. Usually, the answer was already there, but it was there because people had the freedom to think about time and big. The opposite method may work, but there are problems due to problems in a more limited way. Typically, major changes occur due to free thinking.
Optimization has limitations. You can't really optimize your creativity.
Yes, creativity is the opposite. Let's say you're doing very good research in your area. Your optimization scheme will keep you there. But it's better to take risks. Failure and frustration are important. Big breakthroughs, big changes will always come as this will not become an optimization process as one day you're taking yourself out of your comfort zone. Optimizing everything means sometimes the opportunity is missing. I think it's important to be careful about what you really value and optimize.
What have you been working on these days?
One challenge is to use optimal transport in machine learning.
From a theoretical perspective, machine learning is an optimization problem with a system, and several parameters or functions must be optimized so that the machine performs a certain number of tasks.
To classify images, the best transport compares features such as color and textures, and compares these features to alignment between two images (transporting them) and the two images to which the two images are Measure whether it resembles like this. This technique helps to improve accuracy and makes the model more robust to changes and distortions.
These are very high-level phenomena. I'm trying to understand an object with many features, many parameters, and all the features. All functions correspond to one dimension. So if you have 50 features, you're in the 50 dimension space.
The higher the dimensions the object is present, the more complicated the optimal transport problem. It takes too long to solve the problem and you can't do it because there is too much data. This is called the Curse of Dimensions. Recently, people have been trying to consider ways to avoid the curse of dimensions. One idea is to develop a new type of optimal transport.
What is the point of that?
By disrupting some functions, optimum transport is reduced to a lower dimension space. For me, three dimensions are too big and I want to make it a one-dimensional problem. I get some points in 3D space and project them onto the line. I solve the optimal transport on the line, calculate what I should do, repeat this, repeat on many lines. We then use these results in dimension 1 to try to reconstruct the original 3D space by doing a kind of adhesion. That's not an obvious process.
Sounds like the shadow of an object. A two-dimensional square shadow provides information about the three-dimensional cube that casts the shadow.
It's like a shadow. Another example is X-rays, a 2D image of a 3D body. However, if you do the x-rays in a sufficient direction, you can essentially stitch together images to reconstruct your body.
Does conquering the curse of dimensions help with the drawbacks and limitations of AI?
Using some optimal transport techniques, perhaps this will make some of these optimization problems in machine learning more robust, more stable, more reliable, less biased, and safer. It may be. That is the principle of meta.
And in the interaction of pure and applied mathematics, here, the practical and real-world needs are the motivations of new mathematics.
that's right. Machine learning engineering is far ahead. But I don't know why it works. There are very few theorems. There is a huge gap when compared to what we can prove to be achieved. It's impressive, but mathematically, it's still very difficult to explain why. So we can't trust it enough. We want to improve that in many directions and hope that mathematics will help.
