To lie or not to lie in a subspace

TL;DR

This paper provides deterministic conditions to determine when partially observed data from a union of subspaces truly lies in a single subspace and when such a subspace is unique, aiding in subspace identification.

Contribution

It introduces necessary and sufficient deterministic conditions for subspace fitting and uniqueness with incomplete data, advancing subspace analysis methods.

Findings

Conditions for data lying in a subspace are established.

Criteria for subspace uniqueness are provided.

The results enable reliable subspace identification from partial observations.

Abstract

Give deterministic necessary and sufficient conditions to guarantee that if a subspace fits certain partially observed data from a union of subspaces, it is because such data really lies in a subspace. Furthermore, Give deterministic necessary and sufficient conditions to guarantee that if a subspace fits certain partially observed data, such subspace is unique. Do this by characterizing when and only when a set of incomplete vectors behaves as a single but complete one.