A Weizmann Institute of Science scientist is developing a method for analyzing phenomena that occur in high-dimensional systems, using Brownian motion. This method may lead to insights into the properties of these systems
When mathematicians talk about the dimension of a space, they mean the number of coordinates necessary to describe a point or sample from this space. The universe (in the classical sense) is three-dimensional. The state space of a system with 100 particles, when the state of each particle is described using position and velocity, is 100*2*3=600 dimension. The space of possible images with 1,000 x 1,000 pixels is a dimension of three million (assuming that each pixel is described using three primary colors); And the dimension of a person's possible DNA sequences is measured in the hundreds of millions.
In the age of information and "big data", analysis of information in high-dimensional spaces is gaining momentum and importance. In three-dimensional space it is easy for us to imagine, for example, what the group of points that is at a fixed distance from a certain point looks like (this group is called a count). But is there a way to imagine what the set of points looks like in the image space where a dog appears? What about the group of blood count results (let's say a blood count has about 100 different indicators) that reflect an increased risk of getting diabetes? Is there a way to understand "what this group looks like" and characterize its geometry?
The main challenge in analyzing high-dimensional information is the multiplicity of the number of possibilities. Let's take the image space for example, and assume for a moment that each pixel can have only two values - black or white. The number of options for the image in this case will be 2 to the power of the number of pixels. Even in an image with a relatively poor resolution, we will reach astronomical numbers this way. In general, we can say that in order to "scan" all possible points in space, the number of samples we will need will be exponential in dimension. This phenomenon is called the "curse of dimensionality".
The big question is, is it possible to find principles that will help us overcome the curse of dimensionality. According to a relatively new mathematical theory, in many cases, in high-dimensional systems, phenomena can be found that indicate the existence of an orderly and simple structure within this enormous multitude of possibilities. It turns out that when you look at the system from the right perspective, order surprisingly emerges from disorder. This is exactly what Dr. Ronan Eldan from the Department of Mathematics at the Weizmann Institute of Science is doing. For his research in this field (high-dimensional systems research), he was recently awarded the Ardash Prize, the most important prize awarded by the Israel Mathematics Association.
High-dimensional phenomena are manifested in statistics, computer science and physics, and their understanding is also important in fields such as machine learning. We look for mathematical motifs in these systems that repeat themselves and may represent deep features"
Dr. Ronan Eldan
One of the principles behind these phenomena is the principle of averaging, or the "law of large numbers": as the number of variables in the system increases, their average will be less random. For example, if we average the daily change percentage in the stock market over a long period, we will get an almost constant expression (or one with very low variability). This simple phenomenon is the basis of a theory called "dimension concentration" (one of its pioneers is Prof. Vitali Millman from Tel Aviv University), and it helps mathematicians overcome the curse of high dimensions.
To what extent, and how, can these phenomena be used to solve statistical or algorithmic problems? For example, in image analysis, instead of measuring each pixel individually, it turns out that sometimes a better way to deal with the information is to randomly select areas of the image, and layer them in a certain way. "High-dimensional phenomena are reflected in statistics, computer science and physics, and their understanding is also important in fields such as machine learning," says Dr. Eldan. "We look for repeating mathematical motifs in these systems. Such motifs may represent deep properties, common to high-dimensional systems." In his research, Dr. Eldan discovered unexpected connections between the behavior of high-dimensional systems and Brownian motion. This movement (sometimes also known as "drunk walking"), is the diffusion of tiny particles immersed in a liquid. The mathematical model that describes this movement ("Wiener process") is used to describe many phenomena in physics, biology and economics.
"Drunken walk": Brownian motion is the diffusion of tiny particles immersed in a liquid
"Drunken walk": Brownian motion is the diffusion of tiny particles immersed in a liquid
Dr. Eldan is developing a method that makes it possible to analyze phenomena occurring in high-dimensional systems using Brownian motion. Thanks to what is known about the relationship between Brownian motion and diffusion and a phenomenon that takes place in a high-dimensional system, insights may emerge regarding the properties of that system. In another group of studies, Dr. Eldan focuses on trying to understand how the theory of high dimensions can help in the development of optimization and learning algorithms. In this sense it can perhaps be said that Dr. Alden has achieved significant achievements in the way of turning the "curse of dimensionality" into a blessing.
Paul Ardash published 1,525 scientific articles during his lifetime, most of them in collaboration with other mathematicians. He saw the practice of mathematics as a social activity, a concept that led him to collaborate with 511 scientists.
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