QuatIca: Advanced Numerical Linear Algebra and Optimization for Quaternionic Matrices in Python

TL;DR

QuatIca is a comprehensive open-source Python library that enables advanced numerical linear algebra and optimization for quaternion matrices, facilitating research and practical applications in multi-channel signal processing.

Contribution

It introduces a robust, structure-preserving quaternion linear algebra library with diverse algorithms and optimization routines, filling a gap in software support for quaternion computations.

Findings

Provides core quaternion matrix operations and decompositions

Includes iterative solvers like quaternion GMRES and Newton-Schulz

Demonstrates applications in image deblurring and signal filtering

Abstract

Quaternion-valued representations provide a convenient way to model coupled multi-channel signals (e.g., RGB imagery, polarization data, vector fields, and multi-detector time series). Yet practical and numerically reliable software support remains far less mature than those based on the real/complex setting. Here, we present QuatIca, an open-source Python library for quaternion numerical linear algebra and optimization, designed for both research prototyping and reproducible experimentation. QuatIca provides core quaternion matrix operations and norms; dense decompositions and reductions (QR, LU, Q-SVD, eigendecomposition, Hessenberg/tridiagonal reduction, Cholesky decomposition, and Schur helpers); iterative solvers including quaternion GMRES (with preconditioning) and Newton-Schulz pseudoinverse schemes; and domain-focused routines for signal and image processing such as quaternion Tikhonov restoration. The library also includes OptiQ, which solves quaternion Hermitian semidefinite programs using log-det barrier Newton methods with $\mu$-continuation. We highlight design choices that preserve quaternion structure, and we provide end-to-end demonstrations including quaternion image deblurring, Lorenz-attractor filtering, and quaternion image completion. QuatIca is distributed via PyPI and accompanied by open-source development on GitHub and continuously deployed documentation with runnable tutorials.