Provable Subspace Tracking from Missing Data and Matrix Completion

TL;DR

This paper introduces a provable method for subspace tracking with missing data, providing the first complete guarantees and extending to robust scenarios, with applications to matrix completion.

Contribution

A simple modification of robust subspace tracking algorithms that provably solves subspace tracking with missing data and robust scenarios, with theoretical guarantees.

Findings

Guarantees subspace estimates are close to true subspaces over time

Handles piecewise constant subspace changes

Backed by extensive numerical experiments

Abstract

We study the problem of subspace tracking in the presence of missing data (ST-miss). In recent work, we studied a related problem called robust ST. In this work, we show that a simple modification of our robust ST solution also provably solves ST-miss and robust ST-miss. To our knowledge, our result is the first `complete' guarantee for ST-miss. This means that we can prove that under assumptions on only the algorithm inputs, the output subspace estimates are close to the true data subspaces at all times. Our guarantees hold under mild and easily interpretable assumptions, and allow the underlying subspace to change with time in a piecewise constant fashion. In contrast, all existing guarantees for ST are partial results and assume a fixed unknown subspace. Extensive numerical experiments are shown to back up our theoretical claims. Finally, our solution can be interpreted as a provably correct mini-batch and memory-efficient solution to low-rank Matrix Completion (MC).