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Low-rank matrix completion and denoising under Poisson noise

Andrew D. McRae, Mark A. Davenport

TL;DR

This work addresses recovering a nonnegative, low-rank matrix from count-based Poisson observations, with optional partial observation. It develops three estimators—constrained nuclear-norm (Dantzig-type), nuclear-norm regularized LS, and exact low-rank LS—and proves high-probability Frobenius-norm error bounds that scale with the rank $r$, sampling probability $p$, and row/column-sum structure via $\widetilde{\sigma}(M)$. The results are minimax-optimal (up to constants and logarithmic terms) for matrices with bounded sums and extend to multinomial denoising, including corollaries for matrix-P multinomial models and independent-row multinomial denoising. Computationally, these bounds hold even if one ignores nonnegativity constraints, implying simple SVD-based methods are effectively optimal in many regimes, and point to broader applicability to noisy matrix completion beyond Poisson noise. The work also discusses limitations, incoherence considerations, and future directions toward approximately low-rank models and distribution-specific metrics.

Abstract

This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyze several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed, and maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.

Low-rank matrix completion and denoising under Poisson noise

TL;DR

This work addresses recovering a nonnegative, low-rank matrix from count-based Poisson observations, with optional partial observation. It develops three estimators—constrained nuclear-norm (Dantzig-type), nuclear-norm regularized LS, and exact low-rank LS—and proves high-probability Frobenius-norm error bounds that scale with the rank , sampling probability , and row/column-sum structure via . The results are minimax-optimal (up to constants and logarithmic terms) for matrices with bounded sums and extend to multinomial denoising, including corollaries for matrix-P multinomial models and independent-row multinomial denoising. Computationally, these bounds hold even if one ignores nonnegativity constraints, implying simple SVD-based methods are effectively optimal in many regimes, and point to broader applicability to noisy matrix completion beyond Poisson noise. The work also discusses limitations, incoherence considerations, and future directions toward approximately low-rank models and distribution-specific metrics.

Abstract

This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyze several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed, and maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.

Paper Structure

This paper contains 16 sections, 11 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Low-rank models for count data
  3. Summary of main results for Poisson noise
  4. Summary of main results for multinomial denoising
  5. Computation and implications for general noisy matrix completion
  6. Comparison to prior work
  7. Paper outline
  8. Upper bounds
  9. Poisson noise
  10. Corollary on multinomial estimation
  11. Multinomial denoising with independent rows
  12. Minimax lower bounds
  13. First lower bound
  14. Second lower bound
  15. When do the upper and lower bounds match?
  16. ...and 1 more sections

Key Result

Theorem 1

Let $M$ be a non-negative $m \times n$ matrix with rank $r$. Let $\lambda_{\text{max}} = \max_{ij} M_{ij}$, and let Suppose $\Omega \subset \{1, \dots, m\} \times \{1, \dots, n\}$ is chosen according to a Bernoulli sampling model with sampling probability $p$, and suppose, conditionally on $\Omega$, $X \sim \operatorname{Poisson}(\operatorname{\mathcal{A}}_\Omega(M))$. Set $\epsilon \in (0, 1/2)$

Theorems & Definitions (18)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof : Proof of \ref{['lem:basic_1']}
  • proof : Proof of \ref{['lem:basic_2']}
  • Lemma 3: Theorem 4.9 and Remark 4.11 in Latala2018
  • Lemma 4
  • proof
  • Lemma 5
  • Corollary 1
  • ...and 8 more