Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes

• URL: http://arxiv.org/abs/2505.23160v1
• Date: Thu, 29 May 2025 06:55:19 GMT
• Title: Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes
• Authors: Lorenzo Marinucci, Claudio Battiloro, Paolo Di Lorenzo,
• Abstract summary: This paper introduces a novel adaptive framework for processing dynamic flow signals over simplicial complexes.<n>We present a topological LMS algorithm that efficiently processes streaming signals observed over time-varying edge subsets.
• Score: 13.291627429657416
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Learning Overspecified Gaussian Mixtures Exponentially Fast with the EM Algorithm [5.625796693054093]
  We investigate the convergence properties of the EM algorithm when applied to overspecified Gaussian mixture models.<n>We demonstrate that the population EM algorithm converges exponentially fast in terms of the Kullback-Leibler (KL) distance.
  arXiv  Detail & Related papers  (2025-06-13T14:57:57Z)
• Stochastic Optimization with Optimal Importance Sampling [49.484190237840714]
  We propose an iterative-based algorithm that jointly updates the decision and the IS distribution without requiring time-scale separation between the two.<n>Our method achieves the lowest possible variable variance and guarantees global convergence under convexity of the objective and mild assumptions on the IS distribution family.
  arXiv  Detail & Related papers  (2025-04-04T16:10:18Z)
• The Distributionally Robust Optimization Model of Sparse Principal Component Analysis [7.695578200868269]
  We consider sparse principal component analysis (PCA) under a setting where the underlying probability distribution of the random parameter is uncertain.<n>This problem is formulated as a distributionally robust optimization (DRO) model based on a constructive approach to capturing uncertainty.<n>We prove that the inner problem admits a closed-form solution, reformulating the original DRO model into an equivalent minimization problem on the Stiefel manifold.
  arXiv  Detail & Related papers  (2025-03-04T11:00:08Z)
• Distributed Markov Chain Monte Carlo Sampling based on the Alternating Direction Method of Multipliers [143.6249073384419]
  In this paper, we propose a distributed sampling scheme based on the alternating direction method of multipliers. We provide both theoretical guarantees of our algorithm's convergence and experimental evidence of its superiority to the state-of-the-art. In simulation, we deploy our algorithm on linear and logistic regression tasks and illustrate its fast convergence compared to existing gradient-based methods.
  arXiv  Detail & Related papers  (2024-01-29T02:08:40Z)
• FastPart: Over-Parameterized Stochastic Gradient Descent for Sparse optimisation on Measures [1.9950682531209156]
  This paper presents a novel algorithm that leverages Gradient Descent strategies in conjunction with Random Features to augment the scalability of Conic Particle Gradient Descent (CPGD) We provide rigorous proofs demonstrating the following key findings: (i) The total variation norms of the solution measures along the descent trajectory remain bounded, ensuring stability and preventing undesirable divergence; (ii) We establish a global convergence guarantee with a convergence rate of $mathcalO(log(K)/sqrtK)$ over $K$, showcasing the efficiency and effectiveness of our algorithm; (iii) Additionally, we analyze and establish
  arXiv  Detail & Related papers  (2023-12-10T20:41:43Z)
• Ensemble Kalman Filtering Meets Gaussian Process SSM for Non-Mean-Field and Online Inference [47.460898983429374]
  We introduce an ensemble Kalman filter (EnKF) into the non-mean-field (NMF) variational inference framework to approximate the posterior distribution of the latent states. This novel marriage between EnKF and GPSSM not only eliminates the need for extensive parameterization in learning variational distributions, but also enables an interpretable, closed-form approximation of the evidence lower bound (ELBO) We demonstrate that the resulting EnKF-aided online algorithm embodies a principled objective function by ensuring data-fitting accuracy while incorporating model regularizations to mitigate overfitting.
  arXiv  Detail & Related papers  (2023-12-10T15:22:30Z)
• Federated Conditional Stochastic Optimization [110.513884892319]
  Conditional optimization has found in a wide range of machine learning tasks, such as in-variant learning tasks, AUPRC, andAML. This paper proposes algorithms for distributed federated learning.
  arXiv  Detail & Related papers  (2023-10-04T01:47:37Z)
• Distributed Bayesian Learning of Dynamic States [65.7870637855531]
  The proposed algorithm is a distributed Bayesian filtering task for finite-state hidden Markov models. It can be used for sequential state estimation, as well as for modeling opinion formation over social networks under dynamic environments.
  arXiv  Detail & Related papers  (2022-12-05T19:40:17Z)
• Structured Optimal Variational Inference for Dynamic Latent Space Models [16.531262817315696]
  We consider a latent space model for dynamic networks, where our objective is to estimate the pairwise inner products plus the intercept of the latent positions. To balance posterior inference and computational scalability, we consider a structured mean-field variational inference framework.
  arXiv  Detail & Related papers  (2022-09-29T22:10:42Z)
• Finite-Time Analysis of Entropy-Regularized Neural Natural Actor-Critic Algorithm [29.978816372127085]
  We present a finite-time analysis of Natural actor-critic (NAC) with neural network approximation. We identify the roles of neural networks, regularization and optimization techniques to achieve provably good performance.
  arXiv  Detail & Related papers  (2022-06-02T02:13:29Z)
• Optimal Sampling Density for Nonparametric Regression [5.3219212985943924]
  We propose a novel active learning strategy for regression, which is model-agnostic, robust against model mismatch, and interpretable. We adopt the mean integrated error (MISE) as a generalization criterion, and use the behavior of the MISE as well as thelocally optimal bandwidths. The almost model-free nature of our approach should encode raw properties of the target problem, and thus provide a robust and model-agnostic active learning strategy.
  arXiv  Detail & Related papers  (2021-05-25T14:52:17Z)
• Adaptive Sampling for Best Policy Identification in Markov Decision Processes [79.4957965474334]
  We investigate the problem of best-policy identification in discounted Markov Decision (MDPs) when the learner has access to a generative model. The advantages of state-of-the-art algorithms are discussed and illustrated.
  arXiv  Detail & Related papers  (2020-09-28T15:22:24Z)
• A Dynamical Systems Approach for Convergence of the Bayesian EM Algorithm [59.99439951055238]
  We show how (discrete-time) Lyapunov stability theory can serve as a powerful tool to aid, or even lead, in the analysis (and potential design) of optimization algorithms that are not necessarily gradient-based. The particular ML problem that this paper focuses on is that of parameter estimation in an incomplete-data Bayesian framework via the popular optimization algorithm known as maximum a posteriori expectation-maximization (MAP-EM) We show that fast convergence (linear or quadratic) is achieved, which could have been difficult to unveil without our adopted S&C approach.
  arXiv  Detail & Related papers  (2020-06-23T01:34:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.