Sketching Low-Rank Plus Diagonal Matrices
This provides a scalable tool for high-fidelity approximations of large-scale operators like deep learning Hessians, addressing a bottleneck in machine learning and scientific computing.
The paper tackles the problem of approximating high-dimensional linear operators that are low-rank plus diagonal (LoRD) from few matrix-vector products, introducing SKETCHLORD to jointly estimate both components and showing it outperforms sequential methods in accuracy.
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.