Singular Value Decomposition

Singular value decomposition expresses a matrix through orthogonal directions ordered by their importance, making it a standard tool for finding low-dimensional structure in data and models. Work spans randomized and incremental algorithms for faster decompositions, tensor and functional variants, and applications such as clustering, imaging, reduced-order models, and language-model compression. In neural networks, researchers use SVD to compress weights and attention caches, guide parameter-efficient adaptation, and study how spectral structure relates to optimization and model behavior. A central challenge is choosing ranks and components that preserve downstream performance, since minimizing matrix reconstruction error alone may not preserve a system’s task performance, while more accurate or adaptive methods can cost additional computation.