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Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes

Lorenzo Marinucci, Claudio Battiloro, Paolo Di Lorenzo

TL;DR

The paper develops Topo-LMS, an adaptive LMS framework for signals on simplicial complexes by leveraging discrete Hodge theory to incorporate higher-order topological features. It provides a thorough stochastic analysis, including mean convergence, steady-state MSD, and convergence rate, and introduces optimal edge-sampling strategies to balance rate and accuracy. An adaptive topology inference mechanism is proposed to learn latent higher-order relations jointly with learning, and a diffusion-based distributed version is analyzed with closed-form MSD results. Numerical experiments on synthetic and real traffic data demonstrate that Topo-LMS and its distributed variant outperform graph-only LMS baselines, validating the benefits of higher-order topology for dynamic signal processing in networks.

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.

Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes

TL;DR

The paper develops Topo-LMS, an adaptive LMS framework for signals on simplicial complexes by leveraging discrete Hodge theory to incorporate higher-order topological features. It provides a thorough stochastic analysis, including mean convergence, steady-state MSD, and convergence rate, and introduces optimal edge-sampling strategies to balance rate and accuracy. An adaptive topology inference mechanism is proposed to learn latent higher-order relations jointly with learning, and a diffusion-based distributed version is analyzed with closed-form MSD results. Numerical experiments on synthetic and real traffic data demonstrate that Topo-LMS and its distributed variant outperform graph-only LMS baselines, validating the benefits of higher-order topology for dynamic signal processing in networks.

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.

Paper Structure

This paper contains 20 sections, 4 theorems, 78 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Simplicial Complexes and Signals
  4. Algebraic Representation and Filters
  5. Topological Adaptive Least Mean Squares
  6. Algorithm Derivation
  7. Stochastic Behavior
  8. Optimal Sampling Strategies
  9. Adaptive Topology Inference
  10. Distributed Topological Least-mean Squares
  11. Stochastic Behavior
  12. Numerical Results
  13. Topological LMS
  14. Optimal sampling strategies
  15. Topology Inference and Tracking
  16. ...and 5 more sections

Key Result

Theorem 1

Under Assumptions 1-3, if condition (eq:step) holds, the MSD at steady state of the Topo-LMS algorithm can be written as: where Furthermore, letting $\delta=\lambda_{\text{min}}\left(\mathbf{C}_X(\mathbf{p})\right)$ and $\nu=\lambda_{\text{max}}\left(\mathbf{C}_X(\mathbf{p})\right)$, if $\mu \ll 2 \delta/\nu^2$, the convergence rate $\alpha$ of the Topo-LMS algorithm can be well approximated as:

Figures (9)

  • Figure 1: (Left) The edge $(1,4)$ is lower-adjacent to the blue edges (denoted with $\mathcal{N}^{(d)}_{(1,4)})$. (Right) The edge $(1,4)$ is upper-adjacent to the red edges (denoted with $\mathcal{N}^{(u)}_{(1,4)})$.
  • Figure 2: MSD vs iteration index, for different percentages of observed edges, compared with theoretical expressions.
  • Figure 3: (Top) Prediction error over training data. (Bottom) Performance comparison of different methods over test data.
  • Figure 4: Optimal sampling probabilities, obtained as solution of Problem \ref{['prob_2']} for different values of $\alpha$.
  • Figure 5: MSD's evolution over time with varying convergence speed of Algorithm (\ref{['alg:topolms']}).
  • ...and 4 more figures

Theorems & Definitions (11)

  • Remark 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Definition 1
  • Theorem 2
  • proof
  • Remark 2
  • Lemma 2
  • ...and 1 more