Rishabh Dudeja

Bio: I am a postdoctoral researcher in the Department of Statistics at Harvard University. My research interests lie at the intersection of high-dimensional statistics and applied probability. Currently, I am excited about understanding phase transitions, universality, and computational-statistical gaps in high-dimensional inference problems. Before joining Harvard, I obtained a Ph.D. in Statistics from Columbia University and a B.Tech. in Electrical Engineering from the Indian Institute of Technology, Delhi.

Talk Title: High-Dimensional Asymptotics for Phase Retrieval with Structured Sensing Matrices.

Talk Abstract: Phase Retrieval is the problem of recovering an unknown complex-valued signal vector from the magnitudes of several linear measurements. This problem arises in applications like X-ray crystallography, where it is infeasible to acquire the phase of the measurements. In this talk, I will describe some results regarding the analysis of this problem in the high-dimensional asymptotic regime where the number of measurements and the signal dimension diverge proportionally so that their ratio remains fixed. The measurement mechanism in phase retrieval is specified by a sensing matrix. A limitation of existing high-dimensional analysis of this problem is that they model this matrix as a random matrix with independent and identically distributed (i.i.d.) Gaussian entries. In practice, this matrix is highly structured with limited randomness. I will describe a correction to the i.i.d. sensing model, known as the sub-sampled Haar sensing model, which faithfully captures a crucial orthogonality property of realistic sensing matrices. For the Haar sensing model, I will present a precise asymptotic characterization of the performance of commonly used spectral estimators for solving the phase retrieval problem. This characterization can be leveraged to tune certain parameters involved in the spectral estimator optimally. The resulting estimator is information-theoretically optimal. Next, I will describe an empirical universality phenomenon: the performance curves derived for the Haar model accurately describe the observed performance curves for realistic sensing matrices. Finally, I will present recent progress towards obtaining a theoretical understanding of this universality phenomenon that causes practical sensing matrices to behave like Haar sensing matrices.