[Paper Review] Binary component decomposition Part II: The asymmetric case

Richard Kueng, Joel A. Tropp|arXiv (Cornell University)|Jul 31, 2019

Sparse and Compressive Sensing Techniques21 references4 citations

TL;DR

This paper proposes a tractable algorithm for decomposing a low-rank matrix into a binary factor (entries in {±1} or {0,1}) and an unconstrained weight matrix, establishing conditions for existence, uniqueness, and robustness. The key contribution is a deterministic condition under which the decomposition is uniquely identifiable and computable via convex optimization, extending prior work on symmetric binary factorizations.

ABSTRACT

This paper studies the problem of decomposing a low-rank matrix into a factor with binary entries, either from $\{\pm 1\}$ or from $\{0,1\}$, and an unconstrained factor. The research answers fundamental questions about the existence and uniqueness of these decompositions. It also leads to tractable factorization algorithms that succeed under a mild deterministic condition. This work builds on a companion paper that addresses the related problem of decomposing a low-rank positive-semidefinite matrix into symmetric binary factors.

Motivation & Objective

• To establish theoretical foundations for binary component decompositions where one factor is constrained to {±1} or {0,1} and the other is unconstrained.
• To resolve fundamental questions about existence and uniqueness of such decompositions in the asymmetric case.
• To develop efficient, tractable algorithms for computing these decompositions under mild deterministic conditions.
• To extend the theory of binary matrix factorization beyond symmetric, positive-semidefinite settings to general rectangular matrices.
• To provide robustness guarantees against gross errors in the data matrix.

Proposed method

• Uses semidefinite programming to relax the non-convex binary constraint and derive convex optimization formulations.
• Introduces the concept of permeance ν(S) to quantify the geometric structure of binary matrices and link it to algorithmic tractability.
• Employs random matrix theory and concentration inequalities to analyze the behavior of random binary matrices and their interaction with Gaussian vectors.
• Applies the Gaussian loadings model (GLM) to model random binary factors and derive probabilistic bounds on decomposition performance.
• Leverages the singular value decomposition (SVD) as a baseline for low-rank approximation and compares it to the binary factorization framework.
• Uses Rademacher complexity and empirical process theory to bound the supremum of linear forms over the range of the binary matrix, enabling generalization bounds.

Experimental results

Research questions

• RQ1 Under what conditions does a low-rank matrix admit a unique decomposition into a binary factor and an unconstrained weight matrix?
• RQ2 Can such decompositions be computed efficiently using convex optimization techniques?
• RQ3 How does the geometric structure of the binary factor, quantified by permeance ν(S), affect the identifiability and stability of the decomposition?
• RQ4 What is the robustness of the decomposition to gross errors in the observed matrix?
• RQ5 How does this asymmetric binary decomposition relate to the symmetric binary factorization studied in the companion paper?

Key findings

• A unique binary component decomposition exists and can be computed efficiently if the binary factor satisfies a mild deterministic condition related to its permeance ν(S).
• The proposed algorithm achieves exact recovery under the permeance condition, with theoretical guarantees on uniqueness and robustness to gross errors.
• The expected supremum of linear forms over the range of a random sign matrix is bounded by √n, which supports the tractability of the optimization problem.
• With high probability, the absolute inner product between a unit vector in the range of the binary matrix and a random column of the factor matrix exceeds (2/3)√(ν(S)n), ensuring sufficient signal strength for recovery.
• The permeance ν(S) serves as a key geometric parameter that controls the stability and identifiability of the decomposition, with higher values enabling more robust recovery.
• Theoretical bounds derived via Gaussian process and Rademacher complexity techniques ensure that the algorithm succeeds with high probability under the stated conditions.