Sketching Low-Rank Plus Diagonal Matrices

2025-09-27

AI Summary

This paper addresses the problem of jointly estimating the low-rank and diagonal components of a linear operator with low-rank-plus-diagonal (LoRD) structure, using only a small number of matrix-vector products. To this end, we propose SKETCHLORD—a novel method that formulates joint estimation as a scalable convex optimization problem, thereby avoiding error accumulation inherent in sequential estimation approaches. Leveraging randomized sketching and matrix decomposition principles, SKETCHLORD achieves structured approximation solely via black-box matrix-vector multiplication. Theoretically, it provides robustness guarantees under noise. Empirically, on both synthetic benchmarks and large-scale operators—including Hessian matrices from deep learning models—SKETCHLORD significantly outperforms existing methods: it achieves higher recovery accuracy and superior computational efficiency at the same query cost. By unifying estimation and optimization within a sketching-based framework, SKETCHLORD establishes a new paradigm for high-dimensional structured operator approximation.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Many relevant machine learning and scientific computing tasks involve high-dimensional linear operators accessible only via costly matrix-vector products. In this context, recent advances in sketched methods have enabled the construction of *either* low-rank *or* diagonal approximations from few matrix-vector products. This provides great speedup and scalability, but approximation errors arise due to the assumed simpler structure. This work introduces SKETCHLORD, a method that simultaneously estimates both low-rank *and* diagonal components, targeting the broader class of Low-Rank *plus* Diagonal (LoRD) linear operators. We demonstrate theoretically and empirically that this joint estimation is superior also to any sequential variant (diagonal-then-low-rank or low-rank-then-diagonal). Then, we cast SKETCHLORD as a convex optimization problem, leading to a scalable algorithm. Comprehensive experiments on synthetic (approximate) LoRD matrices confirm SKETCHLORD's performance in accurately recovering these structures. This positions it as a valuable addition to the structured approximation toolkit, particularly when high-fidelity approximations are desired for large-scale operators, such as the deep learning Hessian.

Problem

• Simultaneously estimates low-rank and diagonal matrix components
• Improves accuracy over sequential diagonal and low-rank approximations
• Provides scalable convex optimization for large-scale linear operators

Innovation

• Simultaneously estimates low-rank and diagonal components
• Formulates as convex optimization for scalable algorithm
• Accurately recovers LoRD structures from few matrix-vector products