# Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes

arXiv: https://arxiv.org/abs/2505.23160 · HTML: https://mlsanity.com/paper/2505.23160/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 53 references that are in the map.
- First posted on arXiv 29 May 2025.

## Builds on (most cited first, 22 of 22)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Simplicial Attention Networks](https://mlsanity.com/paper/2204.09455/) (2022)
- [Signal Processing on Cell Complexes](https://mlsanity.com/paper/2110.05614/) (2021)
- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Dif-MAML: Decentralized Multi-Agent Meta-Learning](https://mlsanity.com/paper/2010.02870/) (2020)
- [From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Module](https://mlsanity.com/paper/2305.16174/) (2023)
- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Adaptive Least Mean Squares Estimation of Graph Signals](https://mlsanity.com/paper/1602.05703/) (2016)
- [Topological Signal Processing and Learning: Recent Advances and Future Challenges](https://mlsanity.com/paper/2412.01576/) (2024)
- [Adaptive Graph Signal Processing: Algorithms and Optimal Sampling Strategies](https://mlsanity.com/paper/1709.03726/) (2017)
- [Topological Slepians: Maximally Localized Representations of Signals over Simplicial Complexes](https://mlsanity.com/paper/2210.14758/) (2022)
- [Signal processing on simplicial complexes](https://mlsanity.com/paper/2004.02392/) (2020)
- [Topological Dictionary Learning](https://mlsanity.com/paper/2503.11470/) (2025)
- [Distributed Adaptive Learning of Graph Signals](https://mlsanity.com/paper/1609.06100/) (2016)
- [Online Distributed Learning over Graphs with Multitask Graph-Filter Models](https://mlsanity.com/paper/1912.05805/) (2019)
- [Adaptive Joint Estimation of Temporal Vertex and Edge Signals](https://mlsanity.com/paper/2211.06533/) (2022)

## Built on by (most cited first, 2 of 2)

- [Simplicial Gaussian Models: Representation and Inference](https://mlsanity.com/paper/2510.12983/) (2025)
- [Topological Kalman Filtering on Cell Complexes](https://mlsanity.com/paper/2605.15955/) (2026)

## Abstract (arXiv)

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.