Multivariate Low-Rank State-Space Model with SPDE Approach for High-Dimensional Data

2025-09-16

AI Summary

To address the poor scalability and high computational cost of state-space models for high-dimensional multivariate spatiotemporal data, this paper proposes a multivariate low-rank state-space model grounded in stochastic partial differential equations (SPDEs). Methodologically, latent processes are represented via finite-element discretization on sparse grids, and inter-variable dependencies are encoded through a weight matrix. Theoretically, we establish convergence and approximation accuracy guarantees for the finite-element solution under fixed-domain asymptotics. Parameter estimation integrates component-wise SPDE modeling, sparse matrix operations, and an EM algorithm enabling closed-form updates and efficient computation. Empirical evaluation on air quality data demonstrates a 93% reduction in computational time with only a 15% increase in prediction error, significantly enhancing modeling efficiency and practicality for high-dimensional settings.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper proposes a novel low-rank approximation to the multivariate State-Space Model. The Stochastic Partial Differential Equation (SPDE) approach is applied component-wise to the independent-in-time Matérn Gaussian innovation term in the latent equation, assuming component independence. This results in a sparse representation of the latent process on a finite element mesh, allowing for scalable inference through sparse matrix operations. Dependencies among observed components are introduced through a matrix of weights applied to the latent process. Model parameters are estimated using the Expectation-Maximisation algorithm, which features closed-form updates for most parameters and efficient numerical routines for the remaining parameters. We prove theoretical results regarding the accuracy and convergence of the SPDE-based approximation under fixed-domain asymptotics. Simulation studies show our theoretical results. We include an empirical application on air quality to demonstrate the practical usefulness of the proposed model, which maintains computational efficiency in high-dimensional settings. In this application, we reduce computation time by about 93%, with only a 15% increase in the validation error.

Problem

• Develops scalable multivariate state-space model for high-dimensional data
• Uses SPDE approach for sparse low-rank latent process representation
• Introduces dependencies through weight matrix for efficient inference

Innovation

• Low-rank state-space model with SPDE
• Sparse finite element mesh representation
• Expectation-Maximization with closed-form updates