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Supervised low-rank semi-nonnegative matrix factorization with frequency regularization for forecasting spatio-temporal data

Keunsu Kim, Hanbaek Lyu, Jinsu Kim, Jae-Hun Jung

TL;DR

This work addresses forecasting spatio-temporal data by formulating a supervised semi-nonnegative matrix factorization (SSNMF) that jointly learns a small set of latent spatial patterns and a nonnegative, low-rank temporal representation shared across the primary and auxiliary data. It introduces frequency-domain regularization to the temporal factors, offering soft ($\lambda \| \widehat{\mathbf{H}} \|_{1,M}$) and hard frequency constraints, with theoretical convergence guarantees for the proposed block-coordinate descent algorithms. The approach enables interpretable temporal patterns by promoting sparsity or controlled support in the Fourier domain, and provides a practical encoding strategy to forecast missing data. Applied to GRACE geophysical data, the method achieves competitive forecasting performance while enhancing interpretability through explicit frequency-selective temporal structure.

Abstract

We propose a novel methodology for forecasting spatio-temporal data using supervised semi-nonnegative matrix factorization (SSNMF) with frequency regularization. Matrix factorization is employed to decompose spatio-temporal data into spatial and temporal components. To improve clarity in the temporal patterns, we introduce a nonnegativity constraint on the time domain along with regularization in the frequency domain. Specifically, regularization in the frequency domain involves selecting features in the frequency space, making an interpretation in the frequency domain more convenient. We propose two methods in the frequency domain: soft and hard regularizations, and provide convergence guarantees to first-order stationary points of the corresponding constrained optimization problem. While our primary motivation stems from geophysical data analysis based on GRACE (Gravity Recovery and Climate Experiment) data, our methodology has the potential for wider application. Consequently, when applying our methodology to GRACE data, we find that the results with the proposed methodology are comparable to previous research in the field of geophysical sciences but offer clearer interpretability.

Supervised low-rank semi-nonnegative matrix factorization with frequency regularization for forecasting spatio-temporal data

TL;DR

This work addresses forecasting spatio-temporal data by formulating a supervised semi-nonnegative matrix factorization (SSNMF) that jointly learns a small set of latent spatial patterns and a nonnegative, low-rank temporal representation shared across the primary and auxiliary data. It introduces frequency-domain regularization to the temporal factors, offering soft () and hard frequency constraints, with theoretical convergence guarantees for the proposed block-coordinate descent algorithms. The approach enables interpretable temporal patterns by promoting sparsity or controlled support in the Fourier domain, and provides a practical encoding strategy to forecast missing data. Applied to GRACE geophysical data, the method achieves competitive forecasting performance while enhancing interpretability through explicit frequency-selective temporal structure.

Abstract

We propose a novel methodology for forecasting spatio-temporal data using supervised semi-nonnegative matrix factorization (SSNMF) with frequency regularization. Matrix factorization is employed to decompose spatio-temporal data into spatial and temporal components. To improve clarity in the temporal patterns, we introduce a nonnegativity constraint on the time domain along with regularization in the frequency domain. Specifically, regularization in the frequency domain involves selecting features in the frequency space, making an interpretation in the frequency domain more convenient. We propose two methods in the frequency domain: soft and hard regularizations, and provide convergence guarantees to first-order stationary points of the corresponding constrained optimization problem. While our primary motivation stems from geophysical data analysis based on GRACE (Gravity Recovery and Climate Experiment) data, our methodology has the potential for wider application. Consequently, when applying our methodology to GRACE data, we find that the results with the proposed methodology are comparable to previous research in the field of geophysical sciences but offer clearer interpretability.
Paper Structure (8 sections, 13 equations, 1 figure, 3 algorithms)

This paper contains 8 sections, 13 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Methods
  3. Problem setup and low-rank spatio-temporal model
  4. Infering the missing spatio-temporal data given the latent spatial patterns
  5. Spatio-temporal matrix factorization with frequency regularization
  6. Algorithms
  7. Soft frequency regularization
  8. Hard constraints in the frequency domain

Figures (1)

  • Figure 1: (a) Schematic illustration of the low-rank spatio-temporal model. (b) From the joint spatio-temporal data during the period $[0,T)$, we learn the latent spatial patterns $(\mathcal{W},\mathcal{W}')$ and low-rank time-series $\mathbf{H}$, where the temporal structure of $\mathbf{H}$ is regularized in the frequency domain. (c) Using the auxiliary data $\mathcal{Y}$ and the latent dictionary $\mathcal{W}'$, we infer the low-rank time-series $\mathbf{H}_{\textup{new}}$ during the missing period $[T,T_{\textup{tot}})$. (d) The missing data $\mathcal{X}[:,:,T:]$ is then inferred as $\mathcal{W}\times_{3} \mathbf{H}_{\textup{new}}$.

Theorems & Definitions (3)

  • remark thmcounterremark
  • remark thmcounterremark
  • definition thmcounterdefinition: Fourier transform