# Probability and Mathematical Physics Seminar

#### Optimization via Langevin dynamics in complex landscapes

Speaker: Reza Gheissari, NYU

Location: Warren Weaver Hall 512

Date: Friday, September 21, 2018, 11 a.m.

Synopsis:

Consider the problem of recovering $$v \in \mathbb S^{N-1}$$ from an i.i.d. Gaussian $$k$$-tensor spiked by the rank-one tensor $$v^{\otimes k}$$. The likelihood function for this is an example of a complex high-dimensional landscape. It is information-theoretically possible to detect the spike at order 1 signal-to-noise ratios, but if $$k>2$$ it is expected that the signal-to-noise ratio needs to diverge in $$N$$ to efficiently recover $$v$$. We seek to understand the mechanisms for, and obstructions to, such planted signal recovery in high dimensions, via locally optimizing algorithms. We study recovery thresholds for a family of "vanilla" algorithms known as Langevin dynamics, on general landscapes consisting of a planted non-linear signal function perturbed by isotropic Gaussian noise. We propose a mechanism for success/failure of recovery via such algorithms in terms of the curvature of the planted signal.

Joint work with G. Ben Arous and A. Jagannath.