Many inverse problems encountered in sensing and imaging can be formulated as estimating a low-rank matrix from incomplete linear measurements; examples include phase retrieval, matrix completion, blind deconvolution, and so on. Through the lens of matrix factorization, one of the most popular approaches is to employ simple iterative algorithms such as gradient descent to recover the low-rank factors directly, which allow a small memory footprint. Despite wide empirical success, the theoretical underpinnings have remained elusive. In this talk, I will discuss our recent line of efforts in understanding the geometry of the nonconvex loss landscape with the aid of statistical reasoning, and how gradient descent harnesses such geometry in an implicit manner to achieve both computational and statistical efficiency all at once. Furthermore, I will discuss how to adjust vanilla gradient descent to make it provably robust to outliers and ill-conditioning without losing computational and statistical efficiency for low-rank matrix sensing.
Dr. Yuejie Chi received the Ph.D. degree in Electrical Engineering from Princeton University in 2012, and the B.E. (Hon.) degree in Electrical Engineering from Tsinghua University, Beijing, China, in 2007. Since 2018, she is Robert E. Doherty Career Development Professor and Associate Professor with the Department of Electrical and Computer Engineering at Carnegie Mellon University, after spending 5 years at The Ohio State University. She is interested in the mathematics of data science that take advantage of structures and geometry to minimize complexity and improve performance in decision making. Specific topics include mathematical and statistical signal processing, machine learning, large-scale optimization, sampling theory, with applications in sensing, imaging and data science. She is a recipient of the PECASE Award, NSF CAREER Award, AFOSR YIP Award, ONR YIP Award, IEEE SPS Early Career Technical Achievement Award, and IEEE SPS Young Author Paper Award.