High Rank Matrix Completion via Grassmannian Proxy Fusion

TL;DR

This paper introduces a novel Grassmannian-based clustering method for high-rank matrix completion, effectively handling incomplete data by identifying underlying subspaces, especially under low sampling conditions.

Contribution

It proposes a new clustering approach using Grassmannian geometry that improves high-rank matrix completion performance with theoretical support and better low-sampling rate results.

Findings

Performs comparably at high sampling rates

Outperforms existing methods at low sampling rates

Narrowing the gap to theoretical sampling limits

Abstract

This paper approaches high-rank matrix completion (HRMC) by filling missing entries in a data matrix where columns lie near a union of subspaces, clustering these columns, and identifying the underlying subspaces. Current methods often lack theoretical support, produce uninterpretable results, and require more samples than theoretically necessary. We propose clustering incomplete vectors by grouping proxy subspaces and minimizing two criteria over the Grassmannian: (a) the chordal distance between each point and its corresponding subspace and (b) the geodesic distances between subspaces of all data points. Experiments on synthetic and real datasets demonstrate that our method performs comparably to leading methods in high sampling rates and significantly better in low sampling rates, thus narrowing the gap to the theoretical sampling limit of HRMC.