Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of uncertainty in thefield.

Many areas of machine learning and science involve large linear algebra problems, such as eigendecompositions, solving linear systems, computing matrix exponentials, and trace estimation. The matrices involved often have Kronecker, convolutional, block diagonal, sum, or product structure. In this paper, we propose a simple but general framework for large-scale linear algebra problems in machine learning, named CoLA (Compositional Linear Algebra).

We present spd-metrics-id, a Python package for computing distances and divergences between symmetric positive-definite (SPD) matrices. Unlike traditional toolkits that focus on specific applications, spd-metrics-id provides a unified, extensible, and reproducible framework for SPD distance computation. The package supports a wide variety of geometry-aware metrics, including Alpha-z Bures-Wasserstein, Alpha-Procrustes, affine-invariant Riemannian, log-Euclidean, and others, and is accessible both via a command-line interface and a Python API. Reproducibility is ensured through Docker images and Zenodo archiving. We illustrate usage through a connectome fingerprinting example, but the package is broadly applicable to covariance analysis, diffusion tensor imaging, and other domains requiring SPD matrix comparison. The package is openly available at https://pypi.org/project/spd-metrics-id/.

In this section we state some elementary results that we will use for our main proofs. The next Lemma is part of the proof of [44, Lemma 4.2], which we state here as a separate result to save some space from the longer proofs that follow later. This is part of the proof of [44, Lemma 4.2]. In this section we specialize the definitions to the case of Gaussian matrices. Lemma 7. Let n 1 be an integer, and δ (0, 1/2) .

In this Section, we visualize four "hidden graphs" for each of the

Moreover, CoLA provides memory efficient automatic differentiation, low precision computation, and GPU acceleration in both JAX and PyTorch, while also accommodating new objects, operations, and rules in downstream packages via multiple dispatch. CoLA can accelerate many algebraic operations, while making it easy to prototype matrix structures and algorithms, providing an appealing drop-in tool for virtually any computational effort that requires linear algebra. We showcase its efficacy across a broad range of applications, including partial differential equations, Gaussian processes, equivariant model construction, and unsupervised learning.

Moreover, CoLA provides memory efficient automatic differentiation, low precision computation, and GPU acceleration in both JAX and PyTorch, while also accommodating new objects, operations, and rules in downstream packages via multiple dispatch. CoLA can accelerate many algebraic operations, while making it easy to prototype matrix structures and algorithms, providing an appealing drop-in tool for virtually any computational effort that requires linear algebra. We showcase its efficacy across a broad range of applications, including partial differential equations, Gaussian processes, equivariant model construction, and unsupervised learning.