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Guaranteed Rank Minimization via Singular Value Projection

Raghu Meka, Prateek Jain, Inderjit S. Dhillon

arXiv:0909.5457v3cs.LGcs.IT

TL;DR

The paper addresses rank minimization under affine constraints, a difficult problem with important applications. It proposes Singular Value Projection, which combines projected gradient steps with low-rank projection, and reports recovery under RIP, robustness to noise, and strong matrix-completion performance. For matrix completion, the evidence is empirical and the exact-recovery analysis remains partial.

  • Problem

    Rank minimization under affine constraints is generally NP-hard, and existing methods can be expensive, difficult to analyze, or poorly tolerant to noise.

  • Method

    SVP applies projected gradient updates and singular-value projection to maintain rank-at-most-k iterates while solving affine rank minimization.

  • Results

    SVP recovers rank-k solutions under δ2k ≤1/3, provides noisy-measurement guarantees, and empirically achieves exact matrix completion for incoherent matrices from an almost optimal number of samples.

  • Takeaways & Limitations

    SVP offers a simpler, faster, and more noise-robust alternative for affine rank minimization and matrix completion within the paper's supported settings.

  • Takeaways & Limitations

    The paper does not provide a rigorous exact-recovery proof for SVP applied to matrix completion.

Abstract

from arXiv · show

Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization with affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy the "restricted isometry property" and show robustness of our method to noise. Our results improve upon a recent breakthrough by Recht, Fazel and Parillo (RFP07) and Lee and Bresler (LB09) in three significant ways: 1) our method (SVP) is significantly simpler to analyze and easier to implement, 2) we give recovery guarantees under strictly weaker isometry assumptions 3) we give geometric convergence guarantees for SVP even in presense of noise and, as demonstrated empirically, SVP is significantly faster on real-world and synthetic problems. In addition, we address the practically important problem of low-rank matrix completion (MCP), which can be seen as a special case of ARMP. We empirically demonstrate that our algorithm recovers low-rank incoherent matrices from an almost optimal number of uniformly sampled entries. We make partial progress towards proving exact recovery and provide some intuition for the strong performance of SVP applied to matrix completion by showing a more restricted isometry property. Our algorithm outperforms existing methods, such as those of \cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion problem by an order of magnitude and is also significantly more robust to noise.

1 Introduction

The paper introduces SVP for affine rank minimization, proving recovery under RIP and noise while extending the approach to matrix completion. It reports simpler analysis, weaker isometry requirements, geometric convergence, and strong empirical performance.

  • ARMP minimizes matrix rank subject to affine constraints and includes applications such as matrix completion, low-dimensional metric embedding, and low-rank kernel learning.
  • ARMP is NP-hard in general, while prior approaches were largely heuristic or computationally difficult to analyze.
  • SVP uses a simple projected-gradient-based method to recover minimum-rank solutions for RIP-satisfying affine constraints, including noisy measurements.
  • δ2k ≤1/3 is sufficient for the stated noiseless SVP recovery guarantee, weaker than prior requirements including δ5k < 1/10 and δ4k ≤0.04.
  • SVP has geometric convergence, is faster than competing methods by an order of magnitude, and empirically improves accuracy and noise tolerance.
  • For matrix completion, experiments suggest exact recovery of incoherent matrices from an almost optimal number of uniformly sampled entries, while analysis provides only partial exact-recovery results.

2 Singular Value Projection (SVP)

SVP minimizes a rank-constrained least-squares objective by alternating a gradient step with projection onto matrices of rank at most k. The projection is efficiently computed from the top k singular vectors.

  • The robust formulation minimizes 1/2∥A(X) − b∥2^2 subject to rank(X) ≤ k.
  • SVP uses projected gradient descent, exploiting efficient Euclidean projection onto the nonconvex set of rank-at-most-k matrices.
  • The projection Pk(X) returns the closest rank-at-most-k matrix and is computed using the top k singular values and vectors.
  • Each iteration starts from X0 = 0, takes a gradient step, computes a truncated SVD, and projects back to rank at most k.
  • The iterates remain low-rank, reducing the computational cost of subsequent singular-value decompositions.

3 Analysis for Affine Constraints Satisfying RIP

This section establishes SVP's recovery guarantees for affine rank minimization under RIP, including convergence in the noiseless case and geometric error reduction with noise.

  • Analysis for Affine Constraints Satisfying RIP: SVP's analysis bounds the next iterate's error using the optimal solution's error and the current iterate's error.The proof begins with Lemma 3.1, which relates ψ(Xt+1) to ψ(X∗) and ψ(Xt).
  • Analysis for Affine Constraints Satisfying RIP: The projection step sets Xt+1 as the rank-k minimizer of the objective ft over matrices of rank at most k.This gives ft(Xt+1) ≤ ft(X∗), which drives the subsequent RIP-based bounds.
  • Analysis for Affine Constraints Satisfying RIP: In the noiseless case, Lemma 3.1 yields convergence to the optimal rank-k solution X∗.The proof uses ψ(X∗) = 0 and the initialization X0 = 0.
  • Analysis for Affine Constraints Satisfying RIP: For noisy measurements, SVP analyzes iterates while ψ(Xt) remains above a noise-dependent threshold.The proof assumes ψ(Xt) ≥ C^2∥e∥2/2 and uses b − A(X∗) = e.
  • Analysis for Affine Constraints Satisfying RIP: Choosing C > (1 + δ2k)/(1 − 3δ2k) makes the contraction factor D less than one and bounds the stopping error by (C^2 + ε)∥e∥2/2.This requires δ2k < 1/3.

4 Matrix Completion

The matrix-completion section adapts SVP to partial observations, replacing global RIP with an incoherence-restricted property. Experiments show near-optimal sampling and bounded iterate incoherence, while exact recovery remains partly conjectural.

  • 4 Matrix Completion: Matrix completion seeks a minimum-rank matrix matching the observed entries indexed by Ω, represented through the projection operator PΩ.PΩ preserves entries in Ω and sets unobserved entries to zero.
  • 4 Matrix Completion: Matrix completion is a special case of ARMP, but its observation constraints do not satisfy RIP for all low-rank matrices.The section therefore focuses on RIP restricted to incoherent matrices.
  • 4 Matrix Completion: The sampling threshold matches O(k log n/n) with constant C = 1.28 for exact recovery of randomly generated matrices.Figure 1 varies matrix size n at fixed rank k and plots the threshold density p.
  • 4 Matrix Completion: The iterates' incoherence remains bounded by a constant independent of n and p in experiments, supporting the hypothesis behind the recovery argument.This empirical evidence supports Conjecture 4.3 rather than proving it outright.
  • 4.1 RIP for Matrix Completion on Incoherent Matrices: Theorem 4.2 gives incoherence-restricted RIP when p ≥ Cµ2k2 log n/δ2m, with probability at least 1 − exp(−n log n).The guarantee applies to µ-incoherent matrices of rank at most k under Bernoulli sampling.

5 Computational Issues and Related Work

ARMP methods extend compressed-sensing strategies to matrices, but computational cost and intermediate ranks constrain trace-norm approaches. SVP addresses these issues with efficient SVD-based updates, while matrix-completion experiments show strong accuracy, speed, and noise robustness.

  • Related approaches: ARMP generalizes compressed sensing, mapping l1 relaxation, CoSaMP, and iterative hard thresholding to trace-norm relaxation, ADMiRA, and SVP.These approaches are compared by their objectives and algorithmic strategies.
  • Computational limitations: Trace-norm minimization can require prohibitively expensive semidefinite programming, while soft-thresholding methods may maintain high-rank intermediate iterates.SVP instead keeps iterate rank equal to the optimal solution's rank.
  • Noise robustness: SVP is robust to both uniformly bounded and outlier noise, whereas SVT performance worsens considerably with outlier noise.SVP handles noise by minimizing the cumulative loss function ∥A(X) − b∥2.
  • Matrix completion: For matrix completion, SVP empirically outperforms prior methods in accuracy, computational time, and noise tolerance, with exact recovery suggested from an almost optimal number of entries.The method leverages sparse sampled-entry corrections and fast SVD computation.
  • SVP computation: SVP iterations use the SVD of a rank-k matrix plus a sparse correction, enabling matrix-vector products in O((m + n)k + |Ω|) time.This structure supports fast updates for matrix completion.
  • Selecting rank (k): SVP requires the optimal rank k beforehand for ARMP, so the paper proposes incrementing an initial rank guess until the residual error stops changing.For matrix completion, a singular-value gap heuristic estimates k.

6 Experimental Results

Experiments compare SVP with existing methods on affine rank minimization and matrix completion, including synthetic, image-reconstruction, real-world, and noisy settings. SVP is generally faster, while performance under nonuniform sampling is weaker on Movie-Lens.

  • Affine Rank Minimization: SVP requires many fewer iterations and is significantly faster than SVT on random ARMP instances with rank k = 5.The comparison measures time to reach relative error 10^-3.
  • Affine Rank Minimization: SVP achieves significantly smaller MIT-logo reconstruction error than SVT with fewer iterations.The test image is 38 × 73 and has rank four.
  • Matrix Completion: SVP is substantially faster than competing matrix-completion methods for RMSE 10^-2 at sampling density p = .1 and rank k = 2.Running time is compared on a logarithmic scale.
  • Matrix Completion: SVP scales well as rank k increases and remains much faster than other matrix-completion methods.The comparison fixes n = 1000 and targets RMSE 10^-2.
  • Noise Robustness: Under approximately 5% Gaussian noise, SVT is particularly sensitive and incurs high RMSE.The experiment compares error and runtime as matrix size increases from 1000 to 5000.
  • Real-World Data: On Movie-Lens, SVP obtains RMSE 1.01 in 64.85 seconds, whereas ALS obtains RMSE 0.90 in 195.34 seconds.The ratings matrix is nonuniformly sampled, violating a crucial assumption of SVP and SVT.

7 Conclusion and Future Work

The paper concludes that SVP combines simple iterative hard-thresholding with geometric convergence guarantees and favorable empirical speed and robustness. It also identifies proving incoherence of SVP iterates in matrix completion as an open problem.

  • Conclusion: SVP uses iterative hard-thresholding with a simple RIP-based analysis and geometric convergence guarantees under noise.Its intermediate steps are less computationally demanding than those of current state-of-the-art methods.
  • Conclusion: Experiments indicate that SVP is significantly faster and more robust to uniformly bounded and outlier noise than most existing methods.
  • Future Work: A central open question is proving the hypothesis that SVP iterates remain sufficiently incoherent for low-rank matrix completion.The authors also suggest applying the method to low-rank-plus-sparse decomposition and related completion problems.
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