Friday, February 1, 2019

Friday, February 1, 2019 — 4:00 PM EST

From Random Landscapes to Statistical inference

Consider the problem of recovering a rank 1 tensor of order k that has been subject to additive Gaussian Noise. It is information theoretically possible to recover the tensor with a finite number of samples via maximum likelihood estimation, however, it is expected that one needs a polynomially diverging number of samples to efficiently recover it. What is the cause if this large statistical-to-algorithmic gap? To understand this interesting question of high dimensional statistics, we begin by studying an intimately related question: optimization of random homogenous polynomials on the sphere in high dimensions. We show that the estimation threshold is related to a geometric analogue of the BBP transition for matrices. We then study the threshold for efficient recovery for a simple class of algorithms, Langevin dynamics and gradient descent. We view this problem in terms of a broader class of polynomial optimization problems and propose a mechanism or success/failure of recovery in terms of the strength of the signal on the high entropy region of the initialization. We will review several results including joint works with Ben Arous-Gheissari and Lopatto-Miolane. 

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