Current Projects

High-dimensional Probability and its Applications to Data Analysis and Asymptotic Convex Geometry

The study of data with a large number of features is made challenging by the high-dimensional structure of the data. A common theme is to look instead at lower-dimensional projections to see what insight they provide into the high-dimensional data. Analogous questions also arise in high-dimensional probability and statistics, where one looks at lower-dimensional projections of high-dimensional measures, as well as in asymptotic convex geometry, where the measures of interest is often the normalized volume measures on a convex body. While fluctuations of lower-dimensional projections are universal, my research focuses on understanding non-universal features such as tail behavior that can be used to identify the original measures. This uses a variety of tools from large deviation theory, Fourier analysis and asymptotic approximation.

Comparison of the importance sampling algorithm (IS), the LDP estimates, the sharp large deviation estimates (SLD), and the exact values


Preprints:

  • Yin-Ting Liao and Kavita Ramanan. Geometric sharp large deviations for random projections of \ell^n_p spheres.

arXiv:2001.04053, 2020. Submitted to Electronic Journal of Probability.

  • Steven Soojin Kim, Yin-Ting Liao, and Kavita Ramanan. An asymptotic thin shell condition and large deviations for random multidimensional projections. arXiv:1912.13447, 2020. To appear in Advances in Applied Mathematics.

  • Yin-Ting Liao and Kavita Ramanan. A refined asymptotic thin-shell condition and applications to asymptotic convex geometry. Preprint.

  • Yin-Ting Liao and Kavita Ramanan. Quenched sharp large deviations for multi-dimensional projections of $\ell^n_p$ balls. Preprint.

  • Yin-Ting Liao and Kavita Ramanan. Estimating probabilities in asymptotic convex geometry: a case study. Preprint.

Dynamics on large-scale networks

Interacting particle systems on networks have wide-ranging applications including in epidimeology, neural networks, social networks, etc. Recent complementary work has initiated the study of approximating rare events. We are interested in further developing large deviaion principles for empirical measures of interacting diffusions and Markov chains on sparse (possibly random) graphs.

Past Projects

Estimating the smallest eigenvalue of a kernel matrix

We investigate asymptotic properties of the smallest eigenvalue of the random kernel matrix and show that the smallest eigenvalue converges to an exponential random variable when scaled properly.

Accelerating Markov Chain Monte Carlo

We provide a lower bound of the worst-case analysis of the asymptotic variance over general Markov chains with a given invariant probability, reversible as well as non-reversible ones, and construct an optimal transition matrix that achieves this lower bound.