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Many areas of machine learning and science involve large linear algebraproblems, such as eigendecompositions, solving linear systems, computing matrixexponentials, and trace estimation. The matrices involved often have Kronecker,convolutional, block diagonal, sum, or product structure. In this paper, wepropose a simple but general framework for large-scale linear algebra problemsin machine learning, named CoLA (Compositional Linear Algebra). By combining alinear operator abstraction with compositional dispatch rules, CoLAautomatically constructs memory and runtime efficient numerical algorithms.Moreover, CoLA provides memory efficient automatic differentiation, lowprecision computation, and GPU acceleration in both JAX and PyTorch, while alsoaccommodating new objects, operations, and rules in downstream packages viamultiple dispatch. CoLA can accelerate many algebraic operations, while makingit easy to prototype matrix structures and algorithms, providing an appealingdrop-in tool for virtually any computational effort that requires linearalgebra. We showcase its efficacy across a broad range of applications,including partial differential equations, Gaussian processes, equivariant modelconstruction, and unsupervised learning.

CoLA: Exploiting Compositional Structure for Automatic and Efficient Numerical Linear Algebra