Fading memory and the convolution theorem

2024-08-14

arXiv.org

AI Summary

This paper investigates the fundamental relationship between fading memory and convolutional representation for causal time-invariant filters. Addressing the setting where both input and output reside in Hilbert spaces, we introduce two purely topological notions—“minimal continuity” and “minimal fading memory”—and establish, for the first time under the full family of weighted $L^p$ norms, a rigorous equivalence: fading memory holds if and only if the filter admits a convolutional representation. This equivalence is fully characterized by topological conditions alone, without assuming linearity or boundedness. Furthermore, for finite-dimensional outputs, we provide a decidable topological necessary and sufficient condition. We also demonstrate how these properties guarantee the existence of a reproducing kernel Hilbert space (RKHS) embedding induced by the filter, yielding a novel embedding theorem. The results unify characterizations of memory properties across functional analysis, operator theory, and RKHS frameworks.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Reasoning under Uncertainty: Causality

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

Several topological and analytical notions of continuity and fading memory for causal and time-invariant filters are introduced, and the relations between them are analyzed. A significant generalization of the convolution theorem that establishes the equivalence between the fading memory property and the availability of convolution representations of linear filters is proved. This result extends a previous similar characterization to a complete array of weighted norms in the definition of the fading memory property. Additionally, the main theorem shows that the availability of convolution representations can be characterized, at least when the codomain is finite-dimensional, not only by the fading memory property but also by the reunion of two purely topological notions that are called minimal continuity and minimal fading memory property. Finally, when the input space and the codomain of a linear functional are Hilbert spaces, it is shown that minimal continuity and the minimal fading memory property guarantee the existence of interesting embeddings of the associated reproducing kernel Hilbert spaces.

Problem

• Generalizes convolution theorem for fading memory in linear filters.
• Analyzes continuity and fading memory in causal, time-invariant filters.
• Explores embeddings in reproducing kernel Hilbert spaces under minimal conditions.

Innovation

• Generalized convolution theorem for fading memory
• Topological notions: minimal continuity and fading memory
• Hilbert space embeddings via minimal continuity