Exponential Speedup of the Janashia-Lagvilava Matrix Spectral Factorization Algorithm
Ying Wang, Lasha Ephremidze, Ronaldo Garcıa Reyes, Pedro Valdes-Sosa
TL;DR
The paper tackles the computational bottleneck of matrix spectral factorization for high-dimensional systems by extending the Janashia-Lagvilava framework to non-commutative, block-matrix coefficients and by doubling submatrix sizes rather than incrementally increasing them. It introduces a block-general boundary-value approach that yields a para-unitary factorization step, enabling exponential speedups in MSF. The authors provide a detailed comparison between the old and new algorithms and validate the approach with numerical simulations showing substantial runtime reductions while preserving accuracy, suggesting real-time MSF feasibility for massive datasets in neuroscience and machine learning. The work thus broadens the practical applicability of MSF to massive matrices by combining rigorous block-matrix theory with parallelizable algorithmic design.
Abstract
Spectral factorization is a powerful mathematical tool with diverse applications in signal processing and beyond. The Janashia-Lagvilava method has emerged as a leading approach for matrix spectral factorization. In this paper, we extend a central equation of the method to the non-commutative case, enabling polynomial coefficients to be represented in block matrix form while preserving the equation's fundamental structure. This generalization results in an exponential speedup for high-dimensional matrices. Our approach addresses challenges in factorizing massive-dimensional matrices encountered in neural data analysis and other practical applications.