[Paper Review] Missing Entries Matrix Approximation And Completion

Amir Averbuch, Gil Shabat+1 more|arXiv (Cornell University)|Feb 27, 2013

Sparse and Compressive Sensing Techniques12 references3 citations

TL;DR

This paper proposes a general, parameter-free matrix completion and approximation framework based on projected gradient descent that converges to the global solution for convex constraints such as nuclear norm, spectral norm, and orthogonality. It extends low-rank approximation to broader constraints and proves optimal step size is fixed at 1, enabling robust, tunable-free completion for applications in image reconstruction and PDE-based data recovery.

ABSTRACT

Publication in the conference proceedings of SampTA, Bremen, Germany, 2013

Motivation & Objective

• To develop a unified framework for matrix approximation and completion under various convex constraints beyond low-rank approximation.
• To address the limitations of existing matrix completion methods that require tuning parameters such as in Singular Value Thresholding (SVT).
• To prove global convergence of the proposed algorithm for convex constraints, including nuclear norm, spectral norm, and Ky-Fan norms.
• To enable robust matrix completion for real-world applications such as image reconstruction and PDE-based data recovery without parameter tuning.
• To demonstrate that the optimal step size in the projected gradient method is fixed at 1 for all iterations, simplifying implementation and ensuring stability.

Proposed method

• Formulates matrix completion as minimizing the Frobenius norm of observed entries subject to a convex constraint on the full matrix, e.g., ‖X‖_* ≤ λ or ‖X‖₂ ≤ λ.
• Applies projected gradient descent with orthogonal projection onto the constraint set, using a fixed optimal step size of μₙ = 1.
• Uses binary search over the constraint parameter λ to find the minimal λ such that the observed entries are preserved within tolerance.
• Leverages closed-form solutions for constrained approximation problems, such as X = (P M / ‖P M‖_F) * λ for Frobenius norm constraints.
• Extends the framework to non-low-rank constraints, including spectral norm (‖X‖₂), Ky-Fan norms (sum of k largest singular values), and orthogonal constraints (XᵀX = I).
• Employs a robust, parameter-free algorithm that does not require tuning of τ or δ, unlike SVT, and converges to the global solution under convexity.

Experimental results

Research questions

• RQ1 Can a unified, parameter-free matrix completion algorithm be developed that works for a wide range of convex constraints beyond low-rank approximation?
• RQ2 What is the optimal step size in the projected gradient descent for matrix approximation under convex constraints, and does it remain constant across iterations?
• RQ3 Does the proposed algorithm converge to the global solution for convex constraints such as nuclear norm, spectral norm, and orthogonality?
• RQ4 How does the algorithm perform in practical applications such as image reconstruction with missing blocks, compared to standard nuclear norm minimization?
• RQ5 Can the algorithm be applied to PDE-based data recovery, such as Helmholtz’s equation, without requiring problem-specific parameter tuning?

Key findings

• The projected gradient descent algorithm with fixed step size μₙ = 1 converges to the global solution for all convex constraints, including nuclear norm, spectral norm, and Ky-Fan norms.
• The algorithm is robust and requires no external parameters beyond convergence tolerance, unlike SVT which depends on τ and δ.
• For image reconstruction, the completed matrix achieved a nuclear norm of 50,418, close to the original (51,625), with singular values nearly identical to the original, indicating high fidelity.
• In a corrupted image experiment with 18% missing data in 3×3 blocks, the algorithm successfully reconstructed the image with minimal distortion, preserving the singular value decay pattern.
• Theoretical proofs show that the sum of singular values in the solution satisfies the constraint optimally, and the convergence is guaranteed due to the convexity and projection structure.
• The method generalizes to non-low-rank constraints such as XᵀX = I, enabling applications in orthogonal matrix approximation and related problems.