Global Optimality Guarantees for Nonconvex Unsupervised Video Segmentation

Abstract

In this paper, we consider the problem of unsupervised video object segmentation via background subtraction. Specifically, we pose the nonsemantic extraction of a video's moving objects as a nonconvex optimization problem via a sum of sparse and low-rank matrices. The resulting formulation, a nonnegative variant of robust principal component analysis, is more computationally tractable than its commonly employed convex relaxation, although not generally solvable to global optimality. In spite of this limitation, we derive intuitive and interpretable conditions on the video data under which the uniqueness and global optimality of the object segmentation are guaranteed using local search methods. We illustrate these novel optimality criteria through example segmentations using real video data.

Olfactory search with finite-state controllers

“…In this sense, we can view finite-state controllers as belonging to the class of model-based, or planning, approaches such as Bayesian POMDPs. Under some conditions, policy gradient methods can attain global maxima, whereas in general, they are only guaranteed to reach local maxima ( 37 ). What can be done when the agent does not know how the environment interacts with itself through the sensorimotor interface?…”

Nonsmooth rank-one matrix factorization landscape

“…While the exact recovery of a low-rank matrix via convex optimization is well understood [7,9], its non-convex counterpart inf (X,Y )∈R m×r ×R n×r XY T − M 1 (1) remains elusive, where M ∈ R m×n and A 1 := m i=1 n j=1 |A ij | for any A ∈ R m×n . Note that minimizing the Frobenius norm squared instead yields approximate recovery and is better understood [2,32,14].…”

“…At best, this would imply convergence guarantees for local search algorithms when initialized in a neighborhood of the global minima. In order to prove convergence to a global minimum from any random initial point, as observed in [21,25,15,1], it is necessary to analyze the landscape. We do so in the rank-one case and obtain the following theorem.…”