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Matrix ALPS: Accelerated Low Rank and Sparse Matrix Reconstruction

Anastasios Kyrillidis, Volkan Cevher

arXiv:1203.3864v4cs.ITmath.IT

TL;DR

The paper addresses recovery of sparse-plus-low-rank matrices from corrupted and incomplete linear measurements. It proposes Matrix ALPS, a projected-gradient method with memory-based acceleration, and derives RIP-based convergence guarantees while reporting improved computational efficiency without sacrificing stability.

  • Problem

    Recovering a sparse-plus-low-rank matrix from incomplete and noisy linear measurements requires identifying both rank-k and s-sparse components from undersampled observations.

    Source: §I · p. 1

  • Method

    Matrix ALPS combines first-order projected updates over low-rank and sparse structures with memory-based momentum and convergence analysis based on sparse- and rank-RIP properties.

    Source: C. Contributions · p. 2, §III · p. 2, §IV · p. 4

  • Results

    Under stated sparse- and rank-RIP conditions, the iterates satisfy convergence recursions, including δ4s(A) ≤ 0.075, δ4k(A) ≤ 0.04, and δ2s+3k(A) ≤ 0.07.

    Source: §III · p. 3

  • Takeaways & Limitations

    The memory-based scheme provides computational advantages over convex and non-convex approaches while retaining stable recovery behavior.

    Source: §IV · p. 4, §VI · p. 5

Abstract

from arXiv · show

We propose Matrix ALPS for recovering a sparse plus low-rank decomposition of a matrix given its corrupted and incomplete linear measurements. Our approach is a first-order projected gradient method over non-convex sets, and it exploits a well-known memory-based acceleration technique. We theoretically characterize the convergence properties of Matrix ALPS using the stable embedding properties of the linear measurement operator. We then numerically illustrate that our algorithm outperforms the existing convex as well as non-convex state-of-the-art algorithms in computational efficiency without sacrificing stability.

I. INTRODUCTION

The paper studies recovery of a sparse-plus-low-rank matrix from potentially incomplete and noisy linear measurements. It situates this problem alongside compressed sensing and affine rank minimization, where observations are far fewer than ambient variables.

I. INTRODUCTION

  • The framework covers applications including MRI signal processing, collaborative filtering, hyperspectral imaging, and large-scale data processing.Source: §I · p. 1
  • Sparse-plus-low-rank recovery seeks rank-k and s-sparse components from incomplete, noisy measurements of their sum.The target is X* = L* + M*, observed through a linear operator with an additive noise term.Source: §I · p. 1

A. Compressed sensing and affine rank minimization

B. Fusing low-dimensional embedding models

The paper combines low-rank and sparse reconstruction through the SpaRCS framework and motivates Matrix ALPS as an accelerated memory-based extension. Its setting assumes a general measurement operator satisfying both sparse- and rank-RIP with substantially undersampled observations.

B. Fusing low-dimensional embedding models

C. Contributions

  • The recovery setting assumes an arbitrary operator satisfying sparse- and rank-RIP while p ≪ mn observations constrain the unknown matrix.The identity operator includes complete-data RPCA as a special case.Source: C. Contributions · p. 2
  • The paper claims improved restricted-isometry guarantees for noisy recovery and empirical support for better recovery performance with reduced complexity.Source: C. Contributions · p. 2
  • Matrix ALPS uses previous low-rank and sparse estimates to form a low-cost gradient surrogate and applies momentum through memory.The momentum parameter τ may be selected constantly or adaptively; adaptive analysis is reserved for extended work.Source: C. Contributions · p. 2
  • The method represents estimates using projections onto rank and support structures, with empirical error measured by ∥y − AX∥2.Source: C. Contributions · p. 2

II. THE SPARCS ALGORITHM

  • SpaRCS iteratively expands candidate subspaces and supports, then fits rank-k and s-sparse estimates by least squares.Source: §II · p. 2

III. IMPROVED CONVERGENCE GUARANTEES

The convergence analysis imposes incoherence-style structure on the true components and derives recovery conditions from sparse- and rank-RIP properties. It separately analyzes active subspace and support expansion as mechanisms for capturing the true low-rank and sparse components.

III. IMPROVED CONVERGENCE GUARANTEES

  • The analysis assumes the true low-rank component is non-sparse and incoherent, while the sparse component is non-low-rank with uniformly random support.Source: §III · p. 2
  • The proof relies on a lemma for operators satisfying both sparse- and rank-RIP over combined sparse and low-rank structures.Source: §III · pp. 2–3
  • The convergence theorem improves the stated conditions for Algorithm 1 relative to the paper’s comparison with prior state-of-the-art constants.Source: §III · p. 3
  • Under δ4s(A) ≤ 0.075, δ4k(A) ≤ 0.04, and δ2s+3k(A) ≤ 0.07, Algorithm 1 produces iterates decomposable into low-rank and sparse components satisfying stated recursions.Source: §III · p. 3
  • The analysis does not guarantee that intermediate estimates remain uniformly bounded or retain random support patterns for arbitrary problem configurations.Source: §III · p. 3

A. Subspace and support exploration

B. Least-squares estimates over low rank subspaces

Matrix ALPS estimates low-rank and sparse components over expanded candidate structures using least-squares reductions and projected updates. The algorithm maintains memory variables and stops when successive matrix estimates satisfy a relative tolerance or the iteration limit.

B. Least-squares estimates over low rank subspaces

C. Least-squares estimates over sparse support sets

The sparse-support least-squares analysis identifies sparse energy over expanded support sets, while Matrix ALPS combines first-order updates with memory-based acceleration and analyzes constant-momentum convergence.

C. Least-squares estimates over sparse support sets

IV. THE MATRIX ALPS FRAMEWORK

  • Matrix ALPS forms a low-cost gradient surrogate from previous low-rank and sparse estimates, then uses memory to add Nesterov-style momentum.The method replaces least-squares optimization steps with first-order gradient descent updates.Source: §IV · p. 4
  • The low-rank projection can require O(kmn) arithmetic operations with Lanczos for constant accuracy, limiting scalability in practical applications.The paper notes alternatives to SVD and approximate projections as possible remedies.Source: §IV · p. 4
  • Under rank-RIP and sparse-RIP conditions with constant momentum τ_i = 1/4, the noiseless algorithm is characterized by a second-order linear system.Source: §IV · p. 4

SVP [16]

  • The resulting first-order system converges because every eigenvalue satisfies |λ_j(Δ̂)| ≤ 1, implying lim_i→∞ w(i) = 0.Source: SVP [16] · p. 5
  • The paper reserves the proof and noisy analogue of Theorem 2 for an extended version.Source: SVP [16] · p. 5

V. EXPERIMENTS

Experiments evaluate Matrix ALPS on robust matrix completion and video background subtraction, reporting stronger phase-transition performance and faster execution than a GoDec comparison.

V. EXPERIMENTS

  • MATRIX ALPS achieves better phase-transition performance on robust matrix completion under shared realizations and tolerance η = 10^-4.The experiments use 30% uniformly sampled entries and evaluate noisy and noiseless configurations.Source: §V · p. 5
  • For video background subtraction, GoDec and MATRIX ALPS were applied to a 144 x 176 x 200 sequence using the same low-rank projection operators.Source: §V · p. 5

VI. CONCLUSIONS

  • The paper concludes that its memory-based scheme offers computational advantages over both convex and non-convex approaches.Source: §VI · p. 5

VII. APPENDIX

The appendix develops projection and subspace relations used in the sparse-support analysis and documents the computational implementation and video benchmark context.

VII. APPENDIX

  • Matrix ALPS code is available for MATLAB at http://lions.epfl.ch/MatrixALPS.Source: A. Proof of Lemma 2 · p. 5
  • For video background subtraction, median execution time over 10 Monte-Carlo iterations was 34.8 seconds for GoDec versus 15.8 seconds for MATRIX ALPS.Source: Figure 2 · p. 6
  • The proof bounds sparse-support projection terms using triangle inequalities, tools from [13], and Lemma 3.2 in.Source: A. Proof of Lemma 2 · p. 6

F . Thus: PSL

The appendix derives an inequality for the sparse estimate by applying analogous projection and bounding arguments.

F . Thus: PSL

  • The sparse-estimate derivation combines corresponding inequalities to obtain the target bound.The proof follows the same ideas used for the right-hand-side calculation and Corollary 1.Source: F . Thus: PSL · p. 7

B. Proof of Lemma 3

The proof decomposes the relevant error expression, applies optimality conditions and cited lemmas, and bounds the resulting quadratic through its largest root.

C. Proof of Inequality (7)

The proof derives the low-rank error recursion by transforming the relevant expressions, completing the squares, and invoking Lemma 3 to obtain inequality (7).

D. Proof of Theorem 2

Theorem 2 is proved by combining low-rank and sparse error recursions into a coupled first-order system, whose stability under RIP conditions implies convergence to zero.

  • The proof constructs orthonormal rank-1 matrices spanning the low-rank range and uses orthogonal projections to analyze the low-rank estimate.Source: D. Proof of Theorem 2 · p. 9
  • Projection bounds rely on RIP-controlled singular-value inequalities and cited lemmas.Source: D. Proof of Theorem 2 · pp. 9–10
  • The low-rank and sparse estimates satisfy separate recursive bounds that are combined into a coupled set of inequalities.Source: D. Proof of Theorem 2 · p. 10
  • The coupled inequalities are converted into a two-dimensional first-order system using a new state variable y(i).Source: D. Proof of Theorem 2 · p. 10
  • The resulting linear system characterizes the evolution of the two state variables and is unfolded to obtain inequality (9).Source: D. Proof of Theorem 2 · pp. 10–11
  • Under rank-RIP, sparse-RIP, and joint RIP conditions, the associated system has distinct real eigenvalues inside the unit circle.Source: D. Proof of Theorem 2 · p. 11
  • Because the system has zero forcing and eigenvalues with magnitude below one, its equilibrium is zero and the state converges to zero.Source: D. Proof of Theorem 2 · p. 11
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