Convex Analysis in Spectral Decomposition Systems
Hòa T. Bùi, Minh N. Bùi, Christian Clason
TL;DR
This work introduces spectral decomposition systems as an abstract framework that unifies convex analysis of spectral functions across Hermitian eigenvalue problems, singular-value decompositions, and Euclidean Jordan algebras. It proves that core convex-analytic objects—convexity, lower semicontinuity, Fenchel conjugates, subdifferentials, and Bregman proximity operators—can be reduced to corresponding properties of invariant functions on spectra via the spectral mapping $\gamma$, together with a generalization of Ky Fan majorization. A central contribution is the reduced minimization principle, enabling efficient computation of conjugates and subdifferentials for spectral functions, with broad implications for optimization algorithms. The results extend to Bregman proximities, providing explicit calculus rules for spectral proximal operators and distances to spectral sets, thereby facilitating proximal-point and splitting methods in matrix- and operator-based optimization. Overall, the paper offers a cohesive, general toolkit for spectral-function analysis with potential impact in robust matrix estimation, structured low-rank problems, and operator learning.
Abstract
This work is concerned with convex analysis of so-called spectral functions of matrices that only depend on eigenvalues of the matrix. An abstract framework of spectral decomposition systems is proposed that covers a wide range of previously studied settings, including eigenvalue decomposition of Hermitian matrices and singular value decomposition of rectangular matrices and allows deriving new results in more general settings such as Euclidean Jordan algebras. The main results characterize convexity, lower semicontinuity, Fenchel conjugates, convex subdifferentials, and Bregman proximity operators of spectral functions in terms of the reduced functions. As a byproduct, a generalization of the Ky Fan majorization theorem is obtained.