Factorized Schatten-p Norm Technique Speeds Up Low-Rank Matrix Completion

Researchers have introduced a novel factorized reformulation of the Schatten-p norm for low-rank matrix completion that avoids repeated singular value decomposition operations, substantially reducing computational complexity while improving accuracy and computational efficiency over existing methods.

In low-rank matrix completion, Schatten-p norm optimization functions as an effective non-convex relaxation. However, the primary computational bottleneck in this approach stems from repeated singular value decomposition operations required throughout optimization, which imposes a high computational cost.

To overcome this limitation, the authors developed a factorization strategy that decomposes the Schatten-p norm into combinations of nuclear and Frobenius norms. The study examined three distinct decomposition schemes: the sum of two nuclear norms, the sum of two Frobenius norms, and the combination of a nuclear norm and a Frobenius norm.

Using these reformulations, the research team developed an alternating gradient descent algorithm supported by rigorous convergence guarantees. By eliminating the necessity of singular value decomposition, the proposed optimization method significantly reduces overall computational complexity.

The investigators evaluated their approach through extensive experiments on synthetic datasets, image inpainting tasks, and recommender systems. Across these benchmarks, the factorized framework consistently outperformed several state-of-the-art matrix completion methods in terms of both reconstruction accuracy and computational efficiency.

The factorized reformulation addresses the primary computational bottleneck in low-rank matrix completion, which stems from the repeated singular value decomposition operations required during Schatten-p norm optimization. While Schatten-p norm optimization has emerged as an effective non-convex relaxation for completing low-rank matrices, executing singular value decomposition repeatedly throughout the optimization process imposes substantial computational cost. By reformulating the Schatten-p norm through factorization, the optimization process avoids singular value decomposition operations entirely, significantly reducing overall computational complexity while maintaining the non-convex relaxation properties of the objective.

The underlying decomposition strategy investigates three distinct mathematical schemes that decouple the Schatten-p norm into combinations of nuclear and Frobenius norms. Under the first investigated scheme, the Schatten-p norm is reformulated as the sum of two nuclear norms. Under the second scheme, the formulation relies on the sum of two Frobenius norms. The third scheme examines a hybrid structure combining a nuclear norm and a Frobenius norm. Each of these three decomposition schemes establishes an explicit factorized structure, enabling the optimization to bypass singular value decomposition without departing from the targeted Schatten-p norm formulation.

Based on these three factorized formulations, the research develops an alternating gradient descent algorithm designed for the resulting bilinear factorization framework. The algorithm updates the factorized matrix components through alternating gradient optimization steps rather than relying on full matrix decompositions. Crucially, the authors provide rigorous convergence guarantees for this alternating gradient descent approach, ensuring theoretical stability during optimization while simultaneously eliminating the singular value decomposition operations that historically caused high computational overhead in low-rank matrix completion.

The empirical performance of the alternating gradient descent algorithm and its factorized formulations was evaluated through extensive experiments across three separate experimental settings: synthetic datasets, image inpainting tasks, and recommender systems. Across all three benchmark domains, comparative testing demonstrated that the proposed method consistently outperforms several state-of-the-art matrix completion methods. These experimental comparisons confirmed concurrent improvements in both completion accuracy and computational efficiency across the evaluated synthetic and applied tasks.

Published as an original research article in Frontiers in Artificial Intelligence within the Machine Learning section, the study establishes both theoretical foundations and empirical evidence for composite Schatten-p norm factorization. The scope of the work centers on providing rigorous convergence guarantees for alternating gradient optimization while removing the singular value decomposition bottleneck that previously constrained non-convex low-rank matrix completion in large-scale computational applications.

Subject of Research: Efficient low-rank matrix completion via factorized Schatten-p norm optimization

Article Title: Composite Schatten-p norm factorization for efficient matrix completion

Article References: Zhao, J., & Yi, S. (2026). Composite Schatten-p norm factorization for efficient matrix completion. Frontiers in Artificial Intelligence, 9, Article 1892198. https://doi.org/10.3389/frai.2026.1892198

Image Credits: AI Generated

DOI: 10.3389/frai.2026.1892198

Keywords: matrix completion, Schatten-p norm, alternating gradient descent, Frobenius norm, nuclear norm, low-rank optimization, singular value decomposition, image inpainting, Composite, Schatten-p, norm, factorization