Georgia Institute of Technology College of Sciences

Efficient simulation of SDEs is essential in many applications, particularly for ergodic
systems that demand efficient simulation of both short-time dynamics and large-time
statistics. To achieve the efficiency, dimension reduction is often required in both space
and time. In this talk, I will talk about our recent work on both spatial and temporal
reductions.
For spatial dimension reduction, the Mori-Zwanzig formalism is applied to derive
equations for the evolution of linear observables of the Langevin dynamics for both
overdamped and general cases.
For temporal dimension reduction, we introduce a framework to construct inference-
based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in
time by several orders of magnitudes.
This is a joint work with Dr. Thomas Hudson from the University of Warwick, UK; Dr. Fei
Lu from the Johns Hopkins University and Dr Xiaofeng Felix Ye from SUNY at Albany.

A central goal in modern data science is to design algorithms for statistical inference tasks such as community detection, high-dimensional clustering, sparse PCA, and many others. Ideally these algorithms would be both statistically optimal and computationally efficient. However, it often seems impossible to achieve both these goals simultaneously: for many problems, the optimal statistical procedure involves a brute force search while all known polynomial-time algorithms are statistically sub-optimal (requiring more data or higher signal strength than is information-theoretically necessary). In the quest for optimal algorithms, it is therefore important to understand the fundamental statistical limitations of computationally efficient algorithms.

I will discuss an emerging theoretical framework for understanding these questions, based on studying the class of "low-degree polynomial algorithms." This is a powerful class of algorithms that captures the best known poly-time algorithms for a wide variety of statistical tasks. This perspective has led to the discovery of many new and improved algorithms, and also many matching lower bounds: we now have tools to prove failure of all low-degree algorithms, which provides concrete evidence for inherent computational hardness of statistical problems. This line of work illustrates that low-degree polynomials provide a unifying framework for understanding the computational complexity of a wide variety of statistical tasks, encompassing hypothesis testing, estimation, and optimization.

Let f be a real-valued Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution.

What is the probability that f remains above a certain fixed line for a long period of time?

We give simple spectral(and almost tight) conditions under which this probability is asymptotically exponential, that is, that the limit of log P(f>a on [0,T])/ T, as T approaches infinity, exists.

This limit defines "the persistence exponent", and we further show it is continuous in the level a, in the spectral measure corresponding to f (in an appropriate sense), and is unaffected by the singular part of the spectral measure.

Proofs rely on tools from harmonic analysis.

Joint work with Ohad Feldheim and Sumit Mukherjee, arXiv:2112.04820.

The talk will be on Zoom via the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09