[Paper Review] Stable rank one matrix completion is solved by two rounds of semidefinite programming relaxation

Augustin Cosse, Laurent Demanet|arXiv (Cornell University)|Dec 31, 2017

Sparse and Compressive Sensing Techniques45 references3 citations

TL;DR

This paper proposes a two-round semidefinite programming relaxation using trace norm minimization to provably recover a unique rank-one matrix from O(m + n) observed entries, even under noise. The method ensures Lipschitz stability and constructs a dual certificate via sum-of-squares polynomials from the polynomial ideal of constraints, achieving exact recovery when the bipartite graph of observed entries is connected.

ABSTRACT

This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefinite programming relaxation, involving minimization of the nuclear norm, does not in general return the solution for this problem. In this paper, we show that in every instance where the problem has a unique solution, one can provably recover the original matrix through two rounds of semidefinite programming relaxation with minimization of the trace norm. We further show that the solution of the proposed semidefinite program is Lipschitz-stable with respect to perturbations of the observed entries, unlike more basic algorithms such as nonlinear propagation or ridge regression. Our proof is based on recursively building a certificate of optimality corresponding to a dual sum-of-squares (SOS) polynomial. This SOS polynomial is built from the polynomial ideal generated by the completion constraints and the monomials provided by the minimization of the trace. The proposed relaxation fits in the framework of the Lasserre hierarchy, albeit with the key addition of the trace objective function. We further show how to represent and manipulate the moment tensor in favorable complexity by means of a hierarchical low-rank decomposition.

Motivation & Objective

• To address the lack of deterministic stability estimates in rank-one matrix completion under noise.
• To develop a provably stable and exact recovery algorithm for rank-one matrix completion when the solution is unique.
• To overcome the failure of nuclear norm minimization in recovering the true rank-one matrix in deterministic settings.
• To establish a framework based on sum-of-squares (SOS) polynomials and moment tensor decomposition for certifying optimality.

Proposed method

• Proposes a two-round semidefinite programming relaxation with trace norm minimization to recover the original rank-one matrix.
• Constructs a dual certificate using sum-of-squares (SOS) polynomials derived from the polynomial ideal of completion constraints and trace minimization monomials.
• Employs hierarchical low-rank decomposition of the moment tensor to maintain favorable computational complexity.
• Uses the Lasserre hierarchy framework with the addition of a trace objective function to enhance recovery guarantees.
• Derives gradient updates via chain rule on low-rank factorizations of moment matrices, enabling efficient computation.
• Applies higher-order affine constraints by multiplying original constraints with monomials up to degree t−deg(hj) for relaxation order t.

Experimental results

Research questions

• RQ1 Can a two-round semidefinite programming relaxation with trace norm minimization provably recover a unique rank-one matrix from O(m + n) observed entries?
• RQ2 Does the proposed method ensure Lipschitz stability with respect to perturbations in observed entries, unlike nonlinear propagation or ridge regression?
• RQ3 Can a sum-of-squares (SOS) polynomial be constructed as a dual certificate to verify optimality in rank-one matrix completion?
• RQ4 How can the moment tensor be represented and manipulated efficiently in high-dimensional settings using low-rank structure?
• RQ5 What conditions on the observation pattern (e.g., bipartite graph connectivity) guarantee unique and stable recovery?

Key findings

• The proposed two-round SDP relaxation with trace norm minimization provably recovers the unique rank-one matrix whenever the solution exists and is unique.
• The solution is Lipschitz-stable with respect to perturbations in the observed entries, ensuring robustness to noise.
• A dual certificate is constructed via a sum-of-squares (SOS) polynomial derived from the polynomial ideal of constraints and trace minimization terms.
• The method achieves exact recovery when the bipartite graph of observed entries is connected, as guaranteed by Lemma 1.
• The moment tensor is efficiently represented and manipulated using hierarchical low-rank decomposition, reducing computational complexity.
• The approach fits within the Lasserre hierarchy framework while incorporating a trace objective function to improve recovery performance.