TITLE: Analysis and recovery of high-dimensional data with low-dimensional structures

ABSTRACT:

High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning and data recovery. Many data sets in image analysis and signal processing are in a high-dimensional space but exhibit a low-dimensional structure. We are interested in building efficient representations of these data for the purpose of compression and inference, and giving performance guarantees depending on the intrinsic dimension of data. I will present two sets of problems: one is related with manifold learning; the other arises from imaging and signal processing where we want to recover a high-dimensional, sparse vector from few linear measurements. In the first problem, we model a data set in $R^D$ as samples from a probability measure concentrated on or near an unknown $d$-dimensional manifold with $d$ much smaller than $D$. We develop a multiscale adaptive scheme to build low-dimensional geometric approximations of the manifold, as well as approximating functions on the manifold. The second problem arises from source localization in signal processing where a uniform array of sensors is set to collect propagating waves from a small number of sources. I will present some theory and algorithms for the recovery of the point sources with high precision.

BIO: Dr. Wenjing Liao is an assistant professor in the School of Mathematics at Georgia Tech. She obtained her Ph.D in mathematics at University of California, Davis in 2013, and B.S. at Fudan University in 2008. She was a visiting assistant professor at Duke University from 2013 to 2016, as well as a postdoctoral fellow at Statistical and Applied Mathematical Sciences Institute from 2013 to 2015. She worked at Johns Hopkins University as an assistant research scientist from 2016 to 2017. She works on theory and algorithms in the intersection of applied math, machine learning and signal processing. Her current research interests include multiscale methods for dimension reduction, regression on data, convex and non-convex optimization, source localization and sensor calibration in signal processing.