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On Deterministic Sampling Patterns for Robust Low-Rank Matrix Completion
In this letter, we study the deterministic sampling patterns for the com...
12/05/2017 ∙ by Morteza Ashraphijuo, et al. ∙
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A deterministic theory of low rank matrix completion
The problem of completing a large low rank matrix using a subset of reve...
10/02/2019 ∙ by Sourav Chatterjee, et al. ∙
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Matrix completion with deterministic pattern - a geometric perspective
We consider the matrix completion problem with a deterministic pattern o...
01/31/2018 ∙ by Alexander Shapiro, et al. ∙
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Fundamental Conditions for Low-CP-Rank Tensor Completion
We consider the problem of low canonical polyadic (CP) rank tensor compl...
03/31/2017 ∙ by Morteza Ashraphijuo, et al. ∙
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Characterization of Deterministic and Probabilistic Sampling Patterns for Finite Completability of Low Tensor-Train Rank Tensor
In this paper, we analyze the fundamental conditions for low-rank tensor...
03/22/2017 ∙ by Morteza Ashraphijuo, et al. ∙
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Low-rank Matrix Completion in a General Non-orthogonal Basis
This paper considers theoretical analysis of recovering a low rank matri...
12/14/2018 ∙ by Abiy Tasissa, et al. ∙
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Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion
The Euclidean distance geometry problem arises in a wide variety of appl...
04/12/2018 ∙ by Abiy Tasissa, et al. ∙
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A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion
03/09/2015 ∙ by Daniel L. Pimentel-Alarcón, et al. ∙
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Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete d × N matrix is finitely rank-r completable if there are at most finitely many rank-r matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a deterministic sampling condition for finite completability. We use this to also derive deterministic sampling conditions for unique completability that can be efficiently verified. We also show that under uniform random sampling schemes, these conditions are satisfied with high probability if O({r, d}) entries per column are observed. These findings have several implications on LRMC regarding lower bounds, sample and computational complexity, the role of coherence, adaptive settings and the validation of any completion algorithm. We complement our theoretical results with experiments that support our findings and motivate future analysis of uncharted sampling regimes.
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