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Online Graph Topology Learning from Matrix-valued Time Series

Yiye Jiang, Jérémie Bigot, Sofian Maabout

TL;DR

This work develops online graph learning for matrix-valued time series by modeling the data with a matrix-variate VAR(1) that uses a Cartesian (KS) product structure: $\mathbf{X}_t = A_N \mathbf{X}_{t-1} + \mathbf{X}_{t-1} A_F^\top + \mathbf{Z}_t$, with $A_N = A_N^T$, $A_F = A_F^T$, and diag constraints ensuring identifiability. The approach combines a low-dimensional projected-OLS Wald-testing path and a high-dimensional structured matrix-variate Lasso with homotopy updates, plus an augmented model to handle periodic trends via $\mathbf{x}_t = \mathbf{b}_t^0 + \mathbf{x}_t'$, $\mathbf{x}'_t = A \mathbf{x}'_{t-1} + \mathbf{z}_t$. Key contributions include explicit KS constraint projection using an orthonormal basis $\{U_k\}$, online procedures with adaptive regularization, and an extended framework for periodic trends with new GLS/OLS estimators and preserved asymptotics. Experiments on synthetic and climatology data demonstrate improved graph recovery and forecasting over KP-based MAR and VAR baselines, confirming the practical value of online KS MAR learning for real-time networked systems.

Abstract

The focus is on the statistical analysis of matrix-valued time series, where data is collected over a network of sensors, typically at spatial locations, over time. Each sensor records a vector of features at each time point, creating a vectorial time series for each sensor. The goal is to identify the dependency structure among these sensors and represent it with a graph. When only one feature per sensor is observed, vector auto-regressive (VAR) models are commonly used to infer Granger causality, resulting in a causal graph. The first contribution extends VAR models to matrix-variate models for the purpose of graph learning. Additionally, two online procedures are proposed for both low and high dimensions, enabling rapid updates of coefficient estimates as new samples arrive. In the high-dimensional setting, a novel Lasso-type approach is introduced, and homotopy algorithms are developed for online learning. An adaptive tuning procedure for the regularization parameter is also provided. Given that the application of auto-regressive models to data typically requires detrending, which is not feasible in an online context, the proposed AR models are augmented by incorporating trend as an additional parameter, with a particular focus on periodic trends. The online algorithms are adapted to these augmented data models, allowing for simultaneous learning of the graph and trend from streaming samples. Numerical experiments using both synthetic and real data demonstrate the effectiveness of the proposed methods.

Online Graph Topology Learning from Matrix-valued Time Series

TL;DR

This work develops online graph learning for matrix-valued time series by modeling the data with a matrix-variate VAR(1) that uses a Cartesian (KS) product structure: , with , , and diag constraints ensuring identifiability. The approach combines a low-dimensional projected-OLS Wald-testing path and a high-dimensional structured matrix-variate Lasso with homotopy updates, plus an augmented model to handle periodic trends via , . Key contributions include explicit KS constraint projection using an orthonormal basis , online procedures with adaptive regularization, and an extended framework for periodic trends with new GLS/OLS estimators and preserved asymptotics. Experiments on synthetic and climatology data demonstrate improved graph recovery and forecasting over KP-based MAR and VAR baselines, confirming the practical value of online KS MAR learning for real-time networked systems.

Abstract

The focus is on the statistical analysis of matrix-valued time series, where data is collected over a network of sensors, typically at spatial locations, over time. Each sensor records a vector of features at each time point, creating a vectorial time series for each sensor. The goal is to identify the dependency structure among these sensors and represent it with a graph. When only one feature per sensor is observed, vector auto-regressive (VAR) models are commonly used to infer Granger causality, resulting in a causal graph. The first contribution extends VAR models to matrix-variate models for the purpose of graph learning. Additionally, two online procedures are proposed for both low and high dimensions, enabling rapid updates of coefficient estimates as new samples arrive. In the high-dimensional setting, a novel Lasso-type approach is introduced, and homotopy algorithms are developed for online learning. An adaptive tuning procedure for the regularization parameter is also provided. Given that the application of auto-regressive models to data typically requires detrending, which is not feasible in an online context, the proposed AR models are augmented by incorporating trend as an additional parameter, with a particular focus on periodic trends. The online algorithms are adapted to these augmented data models, allowing for simultaneous learning of the graph and trend from streaming samples. Numerical experiments using both synthetic and real data demonstrate the effectiveness of the proposed methods.

Paper Structure

This paper contains 29 sections, 11 theorems, 130 equations, 27 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Causal Product Graphs and Matrix-variate AR(1) Models
  3. Online Graph Learning
  4. Orthonormal Basis and Projection Operator of ${\mathcal{K_G}}$
  5. Approach 1: Projected OLS Estimators and Wald Test
  6. Approach 2: Structured Matrix-variate Lasso and Homotopy Algorithms
  7. Optimality Conditions
  8. Homotopy from $\mathbf{A}(t, \lambda_1)$ to $\mathbf{A}(t, \lambda_{2})$
  9. Homotopy from $\mathbf{A}(t,\lambda)$ to $\mathbf{A}(t+1,\lambda)$
  10. Update from $\lambda_t$ to $\lambda_{t+1}$
  11. Augmented Model for Periodic Trends
  12. New OLS Estimators and Asymptotic Distributions
  13. Augmented Structured Matrix-variate Lasso and the Optimality Conditions
  14. Experiments
  15. Synthetic Data
  16. ...and 14 more sections

Key Result

Lemma 3.1

The set of matrices $U_k, \; k \in K := \{1, \ldots, NF + \frac{1}{2}F(F-1) + \frac{1}{2}N(N-1)\}$, defined below form an orthogonal basis of ${\mathcal{K_G}}$ where when $k \in K_{\mathrm{D}}$, $E_k \in {\rm I\!R}^{NF\times NF}, \hbox{with} [E_k]_{i,j} = 1, \hbox{if} i=j=k, \hbox{otherwise} 0$, when $k \in K_{\mathrm{F}}$, $E_k \in {\rm I\!R}^{F\times F}$ is almost a zero matrix except when $k \

Figures (27)

  • Figure 1: Monthly climatological records of weather stations in California. On the left is the network of weather stations in California. On the right are demonstrated the vectorial observations on a certain station $i$, where a vector $\mathbf{x}_{it} \in {\rm I\!R}^4$ of min/max/avg temperature and precipitation is recorded per time $t$ at $i$, leading to $4$ scalar time series. The interest is in learning a graph of weather dependency for the network.
  • Figure 2: Comparison of the Cartesian and the tensor products of graphs. The node set of both product graphs is the Cartesian product of the components' node sets, yet follows the different adjacencies. The example is based on sandryhaila2014big.
  • Figure 3: Matrices $(U_k)_k$ as entry locators, which characterise the structure of ${\mathcal{K_G}}$.
  • Figure 4: Top is the stationary time series from Model \ref{['eq: var1_matrix']} at one component, bottom is the time series from the augmented Model \ref{['eq: var1_vect_aug']} in the same realisation.
  • Figure 5: Initial spatial graph estimates which start the online procedures. True $A_{\mathrm{N}}$ (left), $\widehat{\bm A_{\mathrm{N}}}_{,91}$ of the low-dimensional procedure (middle), and $\mathbf{A}_{\mathrm{N}}(20, 0.05)$ of the high-dimensional procedure (right) are represented by heatmaps. Simulation settings: $N = 10$, $F = 4$, number of model parameters = $571$, significance level of $\chi^2$ test in Corollary \ref{['coro: Wald test']}$= 0.1$.
  • ...and 22 more figures

Theorems & Definitions (13)

  • Lemma 3.1
  • Proposition 3.2
  • Theorem 3.4
  • Corollary 3.4.1
  • Remark 1
  • Remark 2
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 4.1
  • Proposition 4.2
  • ...and 3 more