Thiernithi Variddhisai

Alexander Jenkins, Thiernithi Variddhisai, Ahmed El-Medany et al.

Graph Signal Processing (GSP) provides a powerful framework for analysing complex, interconnected systems by modelling data as signals on graphs. While recent advances have enabled graph topology learning from observed signals, existing methods often struggle with time-varying systems and real-time applications. To address this gap, we introduce AdaCGP, a sparsity-aware adaptive algorithm for dynamic graph topology estimation from multivariate time series. AdaCGP estimates the Graph Shift Operator (GSO) through recursive update formulae designed to address sparsity, shift-invariance, and bias. Through comprehensive simulations, we demonstrate that AdaCGP consistently outperforms multiple baselines across diverse graph topologies, achieving improvements exceeding 83% in GSO estimation compared to state-of-the-art methods while maintaining favourable computational scaling properties. Our variable splitting approach enables reliable identification of causal connections with near-zero false alarm rates and minimal missed edges. Applied to cardiac fibrillation recordings, AdaCGP tracks dynamic changes in propagation patterns more effectively than established methods like Granger causality, capturing temporal variations in graph topology that static approaches miss. The algorithm successfully identifies stability characteristics in conduction patterns that may maintain arrhythmias, demonstrating potential for clinical applications in diagnosis and treatment of complex biomedical systems.

Thiernithi Variddhisai, Danilo Mandic

The concept of a random process has been recently extended to graph signals, whereby random graph processes are a class of multivariate stochastic processes whose coefficients are matrices with a \textit{graph-topological} structure. The system identification problem of a random graph process therefore revolves around determining its underlying topology, or mathematically, the graph shift operators (GSOs) i.e. an adjacency matrix or a Laplacian matrix. In the same work that introduced random graph processes, a \textit{batch} optimization method to solve for the GSO was also proposed for the random graph process based on a \textit{causal} vertex-time autoregressive model. To this end, the online version of this optimization problem was proposed via the framework of adaptive filtering. The modified stochastic gradient projection method was employed on the regularized least squares objective to create the filter. The recursion is divided into 3 regularized sub-problems to address issues like multi-convexity, sparsity, commutativity and bias. A discussion on convergence analysis is also included. Finally, experiments are conducted to illustrate the performance of the proposed algorithm, from traditional MSE measure to successful recovery rate regardless correct values, all of which to shed light on the potential, the limit and the possible research attempt of this work.

Thiernithi Variddhisai, Danilo Mandic

A method for online tensor dictionary learning is proposed. With the assumption of separable dictionaries, tensor contraction is used to diminish a $N$-way model of $\mathcal{O}\left(L^N\right)$ into a simple matrix equation of $\mathcal{O}\left(NL^2\right)$ with a real-time capability. To avoid numerical instability due to inversion of sparse matrix, a class of stochastic gradient with memory is formulated via a least-square solution to guarantee convergence and robustness. Both gradient descent with exact line search and Newton's method are discussed and realized. Extensions onto how to deal with bad initialization and outliers are also explained in detail. Experiments on two synthetic signals confirms an impressive performance of our proposed method.