Local Convergence of an Algorithm for Subspace Identification from Partial Data

TL;DR

This paper analyzes the local convergence of GROUSE, an iterative algorithm for subspace identification from partial data, demonstrating linear convergence under specific randomness and incoherence assumptions.

Contribution

It provides the first detailed convergence analysis of GROUSE, including the case of full vector observations, linking it to incremental SVD and gradient methods.

Findings

Convergence is linear under certain randomness and incoherence conditions.

The full observation case simplifies the convergence analysis.

GROUSE relates to incremental SVD and gradient projection algorithms.

Abstract

GROUSE (Grassmannian Rank-One Update Subspace Estimation) is an iterative algorithm for identifying a linear subspace of R^n from data consisting of partial observations of random vectors from that subspace. This paper examines local convergence properties of GROUSE, under assumptions on the randomness of the observed vectors, the randomness of the subset of elements observed at each iteration, and incoherence of the subspace with the coordinate directions. Convergence at an expected linear rate is demonstrated under certain assumptions. The case in which the full random vector is revealed at each iteration allows for much simpler analysis, and is also described. GROUSE is related to incremental SVD methods and to gradient projection algorithms in optimization.