Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes

2025-05-29

AI Summary

This work addresses adaptive filtering of dynamic flow signals on simplicial complexes. Methodologically, it pioneers the integration of discrete Hodge theory into adaptive filtering by proposing Topo-LMS—a least-mean-squares algorithm operating in the higher-order topological domain. It introduces a topology-aware edge sampling strategy enabling real-time flow estimation over time-varying edge subsets; develops an adaptive topological inference mechanism under partial structural knowledge; and designs a distributed implementation of the topological LMS algorithm. Theoretically, leveraging stochastic approximation analysis, we rigorously establish stability, convergence rate, and steady-state mean-square error bounds. Experiments on synthetic data and real-world traffic networks demonstrate that Topo-LMS significantly outperforms graph-based LMS methods, achieving superior trade-offs between estimation accuracy and sampling efficiency. Overall, this work provides a scalable and analytically tractable adaptive framework for higher-order topological signal processing.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Extends LMS methods to high-order topological domains
• Analyzes algorithm stability and edge sampling impact
• Proposes distributed version leveraging topological features

Innovation

• Extends LMS to high-order topological domains
• Proposes optimal edge sampling probability strategies
• Introduces adaptive topology inference method