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Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization
Shiqian Ma, Donald Goldfarb, Lifeng Chen
TL;DR
The paper seeks scalable methods for linearly constrained matrix rank minimization because semidefinite programming formulations of its nuclear-norm relaxation are computationally expensive for large matrices. It develops fixed-point and Bregman algorithms with approximate-SVD FPCA, establishes convergence for the fixed-point scheme, and reports superior speed and recoverability on matrix completion problems.
Problem
The paper seeks scalable methods for linearly constrained matrix rank minimization because semidefinite programming formulations of its nuclear-norm relaxation are computationally expensive for large matrices.
Method
The authors develop fixed-point and Bregman iterative algorithms for nuclear norm minimization and use approximate SVD within FPCA.
Results
FPCA greatly outperforms SDP solvers such as SDPT3 in speed and recoverability on matrix completion problems.
Takeaways & Limitations
FPCA can recover large low-rank matrices efficiently in matrix-completion settings.
Takeaways & Limitations
Further study is needed to prove convergence of FPCA, and the DNA microarray matrices tested are not low-rank without low-rank preprocessing.
Abstract
from arXiv · showhide
The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is computationally expensive to solve when the matrices are large. In this paper, we propose fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 x 1000 matrices of rank 50 with a relative error of 1e-5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms.
1 Introduction
The paper studies affinely constrained matrix rank minimization and its nuclear-norm relaxation, motivated by applications and the computational limits of semidefinite programming. It introduces FPCA, an approximate-SVD fixed-point method designed to improve scalability, speed, and recoverability.
- Applications: Matrix completion models missing entries in applications including collaborative filtering, where observed user ratings are used to predict unobserved ratings.The completion problem is a special case of the affine model.
- Problem formulation: Matrix rank minimization is NP-hard, so the paper uses nuclear norm minimization as a convex, computationally tractable approximation.The nuclear norm is the convex envelope of rank on the unit operator-norm ball.
- Recovery motivation: Under suitable random measurement conditions, nuclear norm minimization can recover most rank-r m × n matrices when p ≥ Cr(m+n)log(mn).The cited result concerns suitably random entries of the linear measurement operator.
- Proposed approach: FPCA combines fixed-point continuation with a Monte Carlo approximate SVD and can recover 1000 × 1000 rank-50 matrices with relative error 10^-5 in about 3 minutes using 20% of entries.The paper reports faster performance than SDPT3 and recovery of matrices that some competing methods cannot recover with the same samples.
2 Fixed point iterative algorithm
The fixed-point method solves nuclear norm minimization through operator splitting and matrix shrinkage. Its iterations are linked to optimality conditions, and the generated sequence converges under a stated step-size condition.
- Connection to ℓ1 methods: The method is motivated by the analogous fixed-point iterative algorithm for ℓ1-regularized optimization.The connection carries over the operator-splitting idea from compressed sensing.
- Algorithm construction: The algorithm derives a fixed-point iteration from the convex problem’s optimality conditions and an operator-splitting technique.The resulting scheme is presented as a simple two-line algorithm.
- Shrinkage operator: Matrix shrinkage applies scalar nonnegative shrinkage to the singular values of a matrix while retaining its singular vectors.The paper defines both vector and matrix shrinkage operators through the SVD.
- Optimality characterization: An optimal solution X* satisfies the fixed-point relation X* = Sτµ(h(X*)), where h(·) = I(·) − τg(·).This is stated as an if-and-only-if characterization.
3 Convergence results
The paper establishes non-expansiveness properties for the shrinkage and gradient-related operators, then uses them to prove convergence of the fixed-point iterations to an optimal solution.
- The matrix shrinkage operator Sν is non-expansive.
- The equality conditions for the shrinkage operator are characterized through singular-value decompositions and trace maximization over orthogonal matrices.
- For τ ∈ (0,2/λmax(A⊤A)), the operator h(X)=I(X)−τg(X) is non-expansive.
- If the fixed-point condition holds, the resulting X is an optimal solution of problem (1.8).
- The fixed-point sequence converges to some X* in the set of optimal solutions when τ ∈ (0,2/λmax(A⊤A)).
4 Fixed point continuation
Fixed point continuation accelerates solution of the regularized subproblems by decreasing µ through a homotopy sequence, while practical stopping rules control inner iterations.
- The continuation technique accelerates convergence of the fixed-point iteration through a homotopy approach.
- FPC solves a sequence of problems with decreasing parameters µ1 > µ2 > ··· > µL = µ̄ > 0.
- 4.2 Stopping criteria for inner iterations: The optimality-based stopping rule uses a small tolerance gtol but is avoided for large matrices because estimating the largest singular value is expensive.
- 4.2 Stopping criteria for inner iterations: For large matrices, the algorithm instead stops when consecutive iterates become sufficiently close, using a small xtol threshold.
- Debiasing fixes the singular vectors of Xk and solves a least squares problem for the singular values, triggered when iterate changes become relatively small.
5 Bregman iterative algorithm
The paper extends Bregman iterative regularization to nuclear norm minimization, solving a limited number of unconstrained subproblems and updating their data vectors or subgradients.
- Embedding FPC in a Bregman iteration solves nuclear norm minimization through a limited number of instances of problem (1.8) with different b.
- The Bregman distance is defined for a convex function J using a subgradient p ∈ ∂J(v).
- The nuclear norm algorithm repeatedly solves the unconstrained problem (1.8), with J(X)=µ∥X∥* and an updated subgradient.
- The equivalent iterative scheme initializes b0 and X0 at zero and updates bk+1 using b+(bk−A(Xk)).
- The subproblem in the Bregman iterative algorithm can be solved by Algorithm FPC.
6 An approximate SVD based FPC algorithm: FPCA
FPCA replaces full singular value decompositions in FPC with randomized low-rank approximations, substantially reducing computation while retaining robust low-rank recovery performance.
- FPCA computes only a rank-r approximation to each matrix Y instead of a full singular value decomposition, greatly reducing computational effort.
- The linear-time approximate SVD samples columns and returns approximations to the largest ks singular values and corresponding left singular vectors.
- With high probability, the randomized approximation Aks approximates the best rank-ks approximation to A.
- The exact SVD is computed on the smaller matrix C⊤C, so cs trades computation time against approximation quality.
- FPCA adjusts ks upward when the shrinkage operator's non-expansive property is violated repeatedly, supporting robustness.
- Numerical results show that FPCA is very fast, robust, and significantly outperforms solvers such as SDPT3 in recovering low-rank matrices.
7 Numerical results
The experiments evaluate FPC, FPCA, Bregman, and related solvers on random, large-scale, and real matrix-completion problems. FPCA generally combines stronger recoverability with lower computational cost, including on difficult and large instances.
- Experimental setup: The experiments compare FPC variants, FPCA, Bregman, and SDPT3 using recovery counts, relative errors, and average solution times.Recovery is declared when relative error is below 10^-3; NS counts successful recoveries and AT records average time in seconds.
- Random matrix completion: FPC1 and FPC2 achieve almost the same recoverability and relative error, while FPC2 costs more time and FPC3 costs more time for debiasing.SDPT3 seems to obtain more accurate solutions than the three FPC variants on the small problems.
- Random matrix completion: FPCA recovered all 50 small examples for ranks r ≤ 8 and all 50 medium examples for ranks r ≤ 6 at a 20% measurement rate.For small matrices with r = 9, FPCA failed on one example; for r = 10, it recovered more than 60% of examples.
- Comparison with SVT: FPCA was much faster and usually more accurate than SVT on easy problems, while on hard problems SVT diverged, timed out, or produced inaccurate solutions and FPCA remained efficient.The hard-problem comparisons used averaged relative errors and times over 5 runs.
- Large-scale problems: FPCA efficiently solved 1000 × 1000 problems that SDPT3 could not solve because of matrix size and the large number of constraints.This extends the comparison beyond the small and medium random matrix tests.
- Real data matrices: On DNA microarray data, the original matrices were not low-rank or approximately low-rank, so rank-2 and rank-5 approximations were used before FPCA recovery.With 60% of entries observed, FPCA achieved relative errors as low as 10^-5 and 10^-6 on the resulting low-rank matrices.
8 Conclusions and discussions
The paper develops fixed-point and Bregman methods for nuclear norm minimization, with FPCA delivering faster and more recoverable matrix completion than SDP solvers. FPCA convergence remains unproved.
- FPCA greatly outperforms SDP solvers such as SDPT3 in speed and recoverability on matrix completion problems.
- The methods target nuclear norm minimization as a convex relaxation of the NP-hard linearly constrained matrix rank minimization problem.
- The fixed-point iterative scheme is proven convergent, whereas further study is needed to prove convergence of FPCA.