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arXiv · 2509.23587

Sketching Low-Rank Plus Diagonal Matrices

Andres Fernandez·Felix Dangel·Philipp Hennig·Frank Schneider
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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.

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Andres Fernandez, Felix Dangel, Philipp Hennig, Frank Schneider. 2025-10-02. Sketching Low-Rank Plus Diagonal Matrices. https://arxiv.org/abs/2509.23587

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