Fixed-Point Stability and Convergence of Deep Noncommutative Convolution Operators via Lyapunov-Divergence Operators

Fixed point theory is a rapidly developing area of mathematics that provides a unifying perspective across topology, optimization, and analysis. Classical theorems such as Brower’s, Banach’s contraction principle, Schauder’s topological theorems, and Tarski’s order-theoretic formulation have had such a wide effect that they effectively cover whole branches of mathematics. Yet their usefulness is limited because the remaining structural assumptions of metric contractivity or compactness are rarely satisfied in modern problems, such as machine learning. We present the Lyapunov Divergence Fixed Point (LDFP) setup that extends fixed point theory in three dimensions: (i) substituting a single metric by a distinguishing collection of divergences; (ii) replacing globe contraction by a Lyapunov descent requirement, and (iii) generalizing from one-valued to multivalued mappings. Applying the straightforward technique of the calculus of variations, we show the existence of fixed points, i.e., minimizers, for Lyapunov functions on uniqueness, compact under a Meir-Keeler shornlage with respect to the combined measure $D^{\ast}$. We show that the Lyapunov Divergence Fixed Point directly proximal maps the Wasserstein and Bregman geometries, and we summarize the outlook for fuzzy and stochastic cases.