Structured Adaptive Tensor Prediction for Streaming Data

2026-06-08

AI Summary

Existing methods struggle to effectively address the challenge of adaptive forecasting for streaming matrix-valued time series in time-varying environments. This work proposes a novel adaptive tensor regression framework that extends adaptive filtering to matrix-valued streaming data for the first time, encompassing both Matrix-to-Matrix (MoM) and Tensor-to-Matrix (ToM) modeling paradigms. The framework leverages high-order tensor representations and incorporates low-dimensional structural priors—such as sparsity, low-rankness, and their joint structures—and employs stochastic gradient descent for efficient online learning. Theoretical analysis provides finite-time recovery guarantees, while experiments demonstrate that the ToM model achieves lower steady-state error, stronger denoising capability, and superior tracking performance in dynamic environments compared to MoM.

Abstract

Matrix-valued time series arise in a wide range of applications, such as spatio-temporal data from medical imaging and geophysics. Existing methods are mainly designed for static settings and lack adaptability to streaming and time-varying environments. Adaptive filtering techniques have also been largely limited to data with scalar or vector values, leaving adaptive forecasting for matrix-valued time series inadequately understood. To bridge these gaps, we develop an adaptive tensor regression framework that includes Matrix-on-Matrix (MoM) and Tensor-on-Matrix (ToM) formulations for streaming matrix-valued prediction. The two formulations differ in whether to directly model matrix-valued outputs or to exploit temporal structure via higher-order tensor representations. For the proposed tensor regression framework, we develop stochastic gradient descent (SGD) algorithms for online learning. We show that stacking multiple responses across time into higher-order tensors improves performance; in particular, the ToM achieves lower steady-state error and stronger denoising capability than MoM, motivating our focus on the ToM model. We further characterize the tracking behavior of SGD under time-varying dynamics. From a statistical perspective, we establish fixed-time recovery guarantees for ToM under general low-dimensional structures, including sparsity, low-rankness, and their joint sparselow-rank models.

Problem

• matrix-valued time series
• adaptive forecasting
• streaming data
• tensor regression
• time-varying dynamics

Innovation

• adaptive tensor regression
• streaming matrix-valued time series
• Tensor-on-Matrix (ToM)
• online learning
• low-dimensional structure