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Fundamental conditions on the sampling pattern for union of low-rank subspaces retrieval

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Abstract

This paper is concerned with investigating the fundamental conditions on the locations of the sampled entries, i.e., sampling pattern, for finite completability of a matrix that represents the union of several subspaces with given ranks. In contrast with the existing analysis on Grassmannian manifold for the conventional matrix completion, we propose a geometric analysis on the manifold structure for the union of several subspaces to incorporate all given rank constraints simultaneously. In order to obtain the deterministic conditions on the sampling pattern, we characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern, which is closely related to finite completion. We also give a probabilistic condition in terms of the number of samples per column, i.e., the sampling probability, which leads to finite completability with high probability. Furthermore, using the proposed geometric analysis for finite completability, we characterize sufficient conditions on the sampling pattern that ensure there exists only one completion for the sampled data.

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Acknowledgments

This work was supported in part by the U.S. National Science Foundation under Grant CCF-1814803 and in part by the U.S. Office of Naval Research under Grant N000141410667.

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  1. Columbia University, New York, NY, USA

    Morteza Ashraphijuo & Xiaodong Wang

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  1. Morteza Ashraphijuo
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  2. Xiaodong Wang
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Correspondence to Xiaodong Wang.

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Ashraphijuo, M., Wang, X. Fundamental conditions on the sampling pattern for union of low-rank subspaces retrieval. Ann Math Artif Intell 87, 373–393 (2019). https://doi.org/10.1007/s10472-019-09662-6

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  • DOI: https://doi.org/10.1007/s10472-019-09662-6

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