Fast Algorithms for Robust PCA via Gradient Descent

05/25/2016
by   Xinyang Yi, et al.
0

We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observations do we need, how many corruptions can we tolerate) via polynomial-time algorithms is by now understood. This paper presents and analyzes a non-convex optimization approach that greatly reduces the computational complexity of the above problems, compared to the best available algorithms. In particular, in the fully observed case, with r denoting rank and d dimension, we reduce the complexity from O(r^2d^2(1/ε)) to O(rd^2(1/ε)) -- a big savings when the rank is big. For the partially observed case, we show the complexity of our algorithm is no more than O(r^4d d (1/ε)). Not only is this the best-known run-time for a provable algorithm under partial observation, but in the setting where r is small compared to d, it also allows for near-linear-in-d run-time that can be exploited in the fully-observed case as well, by simply running our algorithm on a subset of the observations.

READ FULL TEXT

page 11

page 12

research
11/04/2014

Fast Exact Matrix Completion with Finite Samples

Matrix completion is the problem of recovering a low rank matrix by obse...
research
05/26/2016

Provable Efficient Online Matrix Completion via Non-convex Stochastic Gradient Descent

Matrix completion, where we wish to recover a low rank matrix by observi...
research
10/15/2019

A greedy anytime algorithm for sparse PCA

The taxing computational effort that is involved in solving some high-di...
research
05/25/2018

Randomized Robust Matrix Completion for the Community Detection Problem

This paper focuses on the unsupervised clustering of large partially obs...
research
05/24/2016

Matrix Completion has No Spurious Local Minimum

Matrix completion is a basic machine learning problem that has wide appl...
research
03/03/2018

Fast and Sample Efficient Inductive Matrix Completion via Multi-Phase Procrustes Flow

We revisit the inductive matrix completion problem that aims to recover ...
research
02/19/2023

iPCA and stability of star quivers

Integrated principal components analysis, or iPCA, is an unsupervised le...

Please sign up or login with your details

Forgot password? Click here to reset