Online Identification and Tracking of Subspaces from Highly Incomplete Information

TL;DR

This paper introduces GROUSE, an efficient online algorithm for tracking subspaces and completing low-rank matrices from highly incomplete data, using simple linear algebra and incremental gradient descent on the Grassmannian.

Contribution

The paper presents GROUSE, a novel online subspace tracking algorithm that is computationally efficient and applicable to matrix completion with highly incomplete observations.

Findings

GROUSE performs well in subspace tracking tasks.

GROUSE effectively completes low-rank matrices from incomplete data.

The algorithm operates in linear time per iteration.

Abstract

This work presents GROUSE (Grassmanian Rank-One Update Subspace Estimation), an efficient online algorithm for tracking subspaces from highly incomplete observations. GROUSE requires only basic linear algebraic manipulations at each iteration, and each subspace update can be performed in linear time in the dimension of the subspace. The algorithm is derived by analyzing incremental gradient descent on the Grassmannian manifold of subspaces. With a slight modification, GROUSE can also be used as an online incremental algorithm for the matrix completion problem of imputing missing entries of a low-rank matrix. GROUSE performs exceptionally well in practice both in tracking subspaces and as an online algorithm for matrix completion.