Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interpretable Dynamic Network Modeling of Tensor Time Series via Kronecker Time-Varying Graphical Lasso

Shingo Higashiguchi, Koki Kawabata, Yasuko Matsubara, Yasushi Sakurai

TL;DR

The paper addresses the challenge of inferring dynamic networks from tensor time series by introducing Kronecker Time-Varying Graphical Lasso (KTVGL), which estimates mode-specific sparse precision matrices and represents the full tensor dependency via a Kronecker product. This tensor-aware, multi-network approach yields interpretable models and scalable computation, and it can be extended to streaming data through KTVGL-Stream. Across synthetic experiments, KTVGL achieves higher edge-estimation accuracy and substantially faster computation (up to 60.5x) than existing methods, while also enabling mode-specific change-point detection. Real-world case studies on Google Trends demonstrate the practical value of interpretable multi-network dynamics across modes, with open-source code and datasets available for reproduction.

Abstract

With the rapid development of web services, large amounts of time series data are generated and accumulated across various domains such as finance, healthcare, and online platforms. As such data often co-evolves with multiple variables interacting with each other, estimating the time-varying dependencies between variables (i.e., the dynamic network structure) has become crucial for accurate modeling. However, real-world data is often represented as tensor time series with multiple modes, resulting in large, entangled networks that are hard to interpret and computationally intensive to estimate. In this paper, we propose Kronecker Time-Varying Graphical Lasso (KTVGL), a method designed for modeling tensor time series. Our approach estimates mode-specific dynamic networks in a Kronecker product form, thereby avoiding overly complex entangled structures and producing interpretable modeling results. Moreover, the partitioned network structure prevents the exponential growth of computational time with data dimension. In addition, our method can be extended to stream algorithms, making the computational time independent of the sequence length. Experiments on synthetic data show that the proposed method achieves higher edge estimation accuracy than existing methods while requiring less computation time. To further demonstrate its practical value, we also present a case study using real-world data. Our source code and datasets are available at https://github.com/Higashiguchi-Shingo/KTVGL.

Interpretable Dynamic Network Modeling of Tensor Time Series via Kronecker Time-Varying Graphical Lasso

TL;DR

The paper addresses the challenge of inferring dynamic networks from tensor time series by introducing Kronecker Time-Varying Graphical Lasso (KTVGL), which estimates mode-specific sparse precision matrices and represents the full tensor dependency via a Kronecker product. This tensor-aware, multi-network approach yields interpretable models and scalable computation, and it can be extended to streaming data through KTVGL-Stream. Across synthetic experiments, KTVGL achieves higher edge-estimation accuracy and substantially faster computation (up to 60.5x) than existing methods, while also enabling mode-specific change-point detection. Real-world case studies on Google Trends demonstrate the practical value of interpretable multi-network dynamics across modes, with open-source code and datasets available for reproduction.

Abstract

With the rapid development of web services, large amounts of time series data are generated and accumulated across various domains such as finance, healthcare, and online platforms. As such data often co-evolves with multiple variables interacting with each other, estimating the time-varying dependencies between variables (i.e., the dynamic network structure) has become crucial for accurate modeling. However, real-world data is often represented as tensor time series with multiple modes, resulting in large, entangled networks that are hard to interpret and computationally intensive to estimate. In this paper, we propose Kronecker Time-Varying Graphical Lasso (KTVGL), a method designed for modeling tensor time series. Our approach estimates mode-specific dynamic networks in a Kronecker product form, thereby avoiding overly complex entangled structures and producing interpretable modeling results. Moreover, the partitioned network structure prevents the exponential growth of computational time with data dimension. In addition, our method can be extended to stream algorithms, making the computational time independent of the sequence length. Experiments on synthetic data show that the proposed method achieves higher edge estimation accuracy than existing methods while requiring less computation time. To further demonstrate its practical value, we also present a case study using real-world data. Our source code and datasets are available at https://github.com/Higashiguchi-Shingo/KTVGL.
Paper Structure (31 sections, 2 theorems, 9 equations, 6 figures, 5 tables, 1 algorithm)

This paper contains 31 sections, 2 theorems, 9 equations, 6 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed Method
  4. Problem definition and notation
  5. Multi-mode network
  6. Optimization algorithm
  7. Empirical covariance
  8. Scalability
  9. Extension to stream processing
  10. Experiments
  11. Synthetic datasets
  12. Experimental setting
  13. Evaluation metrics
  14. Baselines
  15. Results
  16. ...and 16 more sections

Key Result

Lemma 1

We define where $G_{t}^{(m)} = \otimes_{l \neq m} \Theta_{t}^{(l)} \in \mathbb{R}^{D_{\setminus m} \times D_{\setminus m}}$ and $A_{t,n_t}^{(m)} = \text{unfold}(X_{t,n_t}, m) \in \mathbb{R}^{d_m \times D_{\setminus m}}$. Then, for any $m$, the following equation holds:

Figures (6)

  • Figure 1: Modeling tensor time series (left) versus multivariate time series obtained by flattening tensors (right). Dynamic network inference on flattened data results in large and entangled networks with reduced accuracy and interpretability, whereas estimating mode-specific networks yields clearer and more interpretable results.
  • Figure 2: Temporal deviation of the estimated networks for datasets with different values of $M$, $d_m$, and ground truth network change points. The gray vertical lines indicate the true change points. Our method captures these change points across various settings and additionally identifies mode-specific changes.
  • Figure 3: Scalability of KTVGL : (a, b) Wall clock time vs. number of nodes per mode ($d_m$) for number of mode $M = 2,3$. While the computational time of TVGL increases rapidly with $d_m$ and $M$, KTVGL avoids the exponential growth in computational complexity. (c) Wall clock time vs. sequence length $T$. KTVGL scales linearly with sequence length.
  • Figure 4: Scalability of KTVGL-Stream: Wall clock time vs. sequence length. The computational time at each step is independent of the sequence length. We set $d_m=5$, and $M=2$ (left) and $M=3$ (right) respectively.
  • Figure 5: Snapshots of the time-varying networks estimated by KTVGL at three different time points for GoogleTrends country data. The upper panels represent keyword networks, while the lower panels represent country networks. Our multi-network structure provides interpretable modeling results for each non-temporal mode.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2