Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes

TL;DR

This paper develops a topological LMS algorithm for processing dynamic signals over simplicial complexes, incorporating stability analysis, adaptive topology inference, and distributed implementation, with demonstrated superior performance on traffic data.

Contribution

It introduces a novel topological LMS framework over simplicial complexes, including stability analysis, adaptive topology inference, and a distributed version, advancing high-order topological signal processing.

Findings

Outperforms existing graph-based LMS methods on traffic data

Provides stability and convergence analysis for the proposed algorithms

Demonstrates effectiveness of adaptive topology inference in dynamic settings

Abstract

This paper introduces a novel adaptive framework for processing dynamic flow signals over simplicial complexes, extending classical least-mean-squares (LMS) methods to high-order topological domains. Building on discrete Hodge theory, we present a topological LMS algorithm that efficiently processes streaming signals observed over time-varying edge subsets. We provide a detailed stochastic analysis of the algorithm, deriving its stability conditions, steady-state mean-square-error, and convergence speed, while exploring the impact of edge sampling on performance. We also propose strategies to design optimal edge sampling probabilities, minimizing rate while ensuring desired estimation accuracy. Assuming partial knowledge of the complex structure (e.g., the underlying graph), we introduce an adaptive topology inference method that integrates with the proposed LMS framework. Additionally, we propose a distributed version of the algorithm and analyze its stability and mean-square-error properties. Empirical results on synthetic and real-world traffic data demonstrate that our approach, in both centralized and distributed settings, outperforms graph-based LMS methods by leveraging higher-order topological features.