"Within this group of objects," says Prof. Eldan, "one can speak of a subset consisting of the results of tests that indicate a particular medical condition, such as diabetes. Understanding the geometry of such a subset has direct implications for our ability to identify diabetes from all the tested insureds in the future, because this understanding gives insights into the question of whether I can study this subset using machine learning." He says. "Understanding its geometry allows me to give the algorithm some features that actually tell what kind of objects to look for, and in what class.", says Prof. Eldan, but adds that this understanding is not only practical, but has pure mathematical importance.
"Basically, ‘big data’ is statistics that operate in high-dimensional spaces," explains Prof. Eldan, "but instead of coordinates, we enter data. In fact,” he adds, "almost everything we do in data science and machine learning—developing algorithms that allow a computer to learn from examples—can be phrased as a mathematical problem regarding high-dimensional objects.
Using Prof. Eldan's methodology, it is possible to apply the theory of Brownian motion to the analysis of objects of high dimensions, thus enabling a new point of view that leads to insights into their behavior. "My methodology utilizes a seemingly-unrelated theory, which concerns the diffusion of particles," explains Prof. Eldan. "We use the theory of diffusion to help describe objects of high dimensions." Prof. Eldan says that through application of the method he developed, he can understand “several basic aspects related to the question of what a set that lies in a high-dimensional space looks like."
Despite the abstract thinking required to understand this scientific achievement, some of the discoveries themselves have very practical implications. For example, Prof. Eldan’s new methodology has led to the solution of one of the major problems in the field of high-dimensional geometry: Bourgain’s slicing problem (named after the Belgian mathematician Jean Bourgain). In addition, the method has direct implications for machine learning: "It helps to show that machine learning algorithms that have already been designed can solve a larger set of problems and can also be used to develop new algorithms," he explains.
Prof. Eldan is currently on sabbatical at Princeton University with his family, and says he received the call informing him that he had won the Blavatnik Award early in the morning. "It was six-o’-clock in the morning, and the main thing that went through my mind was that it would wake my daughter," he says with a smile, and adds he was very excited to hear that he had received the prestigious award.
"I didn't really want to be a mathematician from childhood," he recalls. "I really liked physics and engineering, and as a kid I liked building things." During his military service he began taking mathematics and physics courses at the Open University and eventually completed a degree in mathematics. "I finally focused on mathematics, in part because there were no physics labs at the Open University,” he said. He went on to pursue a master's degree and a doctorate at Tel Aviv University on "Distribution of Mass in Convex Bodies," and then a postdoctoral fellowship at Microsoft's research lab near Seattle. In 2015, he returned to Israel, to serve as a faculty member at the Weizmann Institute of Science.
As a pure mathematician, Prof. Eldan is doing his research without a laboratory or complicated experimental software. "Most of the time it's me with myself, and I just think. It happens while I'm on the train or waiting in line. I work with students, which is perhaps the most fulfilling part of academic life, and I'm sure I would not have gotten anywhere without the mentors I had, but in the end, most ideas arise when you’re sitting by yourself and thinking." When asked if he has ways to force those moments to happen, he jokes, "If I was stuck in a room with exposed concrete walls, no windows, and most importantly no internet, it would be the most effective way for me to make progress in math."
Interviewed by Assaf Uni