Title: Introduction to High-dimensional Probability and Stochastic Localization
Abstract: In high-dimension, the computational complexity grows exponentially with the dimension and geometric objects are not as intuitive as we might think. To deal with this problem, mathematicians imposed probabilistic methods, making the problems more tractable. In this talk, I will give some examples of how things get worse in higher dimensions and how to solve this using the probabilistic method.
Then, I will talk about concentration inequalities, which are useful tools for analyzing high-dimensional problems, especially Poincaré Inequality(PI). PI is related to the rate of convergence in the Markov process, and Poincaré constant(PC) plays an important role in this. However, the order of the Poincaré constant was known as linear order of dimension.
In 2013, Ronen Eldan invented the stochastic localization(SL) scheme, a stochastic process of probability measures attained by random Gaussian tilt at every step. Eldan and other mathematicians improved the order of PC and SL became the cornerstone of this rapid development. Also, SL has many applications in other fields. I will briefly introduce the SL and some results and applications.Title: Introduction to High-dimensional Probability and Stochastic Localization
Abstract: In high-dimension, the computational complexity grows exponentially with the dimension and geometric objects are not as intuitive as we might think. To deal with this problem, mathematicians imposed probabilistic methods, making the problems more tractable. In this talk, I will give some examples of how things get worse in higher dimensions and how to solve this using the probabilistic method.
Then, I will talk about concentration inequalities, which are useful tools for analyzing high-dimensional problems, especially Poincaré Inequality(PI). PI is related to the rate of convergence in the Markov process, and Poincaré constant(PC) plays an important role in this. However, the order of the Poincaré constant was known as linear order of dimension.
In 2013, Ronen Eldan invented the stochastic localization(SL) scheme, a stochastic process of probability measures attained by random Gaussian tilt at every step. Eldan and other mathematicians improved the order of PC and SL became the cornerstone of this rapid development. Also, SL has many applications in other fields. I will briefly introduce the SL and some results and applications.