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Learning Cartesian Product Graphs with Laplacian Constraints

Changhao Shi, Gal Mishne

TL;DR

This work tackles learning Laplacian-structured Cartesian product graphs from multi-way data. It formulates a penalized maximum likelihood objective with a log-determinant term and an off-diagonal sparsity penalty, exploiting the Kronecker-sum structure $\mathbf{L} = \mathbf{L}_1 \oplus \mathbf{L}_2$ for efficient optimization. The authors establish existence, uniqueness, and improved high-dimensional consistency for the estimator, and propose MWGL, an efficient spectral algorithm, together with MWGL-Missing for joint graph learning and imputation. Empirical results on synthetic data and real-world datasets (Molene and COIL-20) show that MWGL consistently outperforms competing GSP and GM methods, validating both the theoretical guarantees and practical utility of Laplacian-constrained Cartesian-product learning.

Abstract

Graph Laplacian learning, also known as network topology inference, is a problem of great interest to multiple communities. In Gaussian graphical models (GM), graph learning amounts to endowing covariance selection with the Laplacian structure. In graph signal processing (GSP), it is essential to infer the unobserved graph from the outputs of a filtering system. In this paper, we study the problem of learning Cartesian product graphs under Laplacian constraints. The Cartesian graph product is a natural way for modeling higher-order conditional dependencies and is also the key for generalizing GSP to multi-way tensors. We establish statistical consistency for the penalized maximum likelihood estimation (MLE) of a Cartesian product Laplacian, and propose an efficient algorithm to solve the problem. We also extend our method for efficient joint graph learning and imputation in the presence of structural missing values. Experiments on synthetic and real-world datasets demonstrate that our method is superior to previous GSP and GM methods.

Learning Cartesian Product Graphs with Laplacian Constraints

TL;DR

This work tackles learning Laplacian-structured Cartesian product graphs from multi-way data. It formulates a penalized maximum likelihood objective with a log-determinant term and an off-diagonal sparsity penalty, exploiting the Kronecker-sum structure for efficient optimization. The authors establish existence, uniqueness, and improved high-dimensional consistency for the estimator, and propose MWGL, an efficient spectral algorithm, together with MWGL-Missing for joint graph learning and imputation. Empirical results on synthetic data and real-world datasets (Molene and COIL-20) show that MWGL consistently outperforms competing GSP and GM methods, validating both the theoretical guarantees and practical utility of Laplacian-constrained Cartesian-product learning.

Abstract

Graph Laplacian learning, also known as network topology inference, is a problem of great interest to multiple communities. In Gaussian graphical models (GM), graph learning amounts to endowing covariance selection with the Laplacian structure. In graph signal processing (GSP), it is essential to infer the unobserved graph from the outputs of a filtering system. In this paper, we study the problem of learning Cartesian product graphs under Laplacian constraints. The Cartesian graph product is a natural way for modeling higher-order conditional dependencies and is also the key for generalizing GSP to multi-way tensors. We establish statistical consistency for the penalized maximum likelihood estimation (MLE) of a Cartesian product Laplacian, and propose an efficient algorithm to solve the problem. We also extend our method for efficient joint graph learning and imputation in the presence of structural missing values. Experiments on synthetic and real-world datasets demonstrate that our method is superior to previous GSP and GM methods.
Paper Structure (30 sections, 5 theorems, 78 equations, 7 figures, 2 algorithms)

This paper contains 30 sections, 5 theorems, 78 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Preliminaries
  5. Smoothness Prior
  6. Product Graph Learning
  7. Penalized MLE
  8. GSP Interpretation:
  9. GM Interpretation:
  10. Multi-Way Graph Learning
  11. Structural Missing Values
  12. Theoretical Results
  13. Experiments
  14. Synthetic Graphs
  15. Molene Weather Data
  16. ...and 15 more sections

Key Result

Lemma 1

The $\mathbf{H}_1$ and $\mathbf{H}_2$ matrices defined in eq:h1h2_costly can be efficiently computed as where $\mathbf{L}_1=\mathbf{U}_1\boldsymbol\Lambda_1\mathbf{U}_1^T$ and $\mathbf{L}_2=\mathbf{U}_2\boldsymbol\Lambda_2\mathbf{U}_2^T$ are the eigendecompositions of factor Laplacian matrices.

Figures (7)

  • Figure 1: An example of the Cartesian graph product.
  • Figure 2: Comparison of different methods on various synthetic data. Each sub-figure shows the trend of Rel-Err of the product or factor Laplacian matrices as $n$ increases. Black dash lines fit the theory in \ref{['ieq:convergence_rate']} to our results.
  • Figure 3: The inferred factor graphs of Molene. Stations are placed according to their real coordinates.
  • Figure 4: (a) The inferred object graph and (b) selected imputations of the COIL-20 dataset. The first column is observed images and other columns are reconstructions across missing angles.
  • Figure 5: Comparison of different methods on synthetic data in various scenarios. Each sub-figure shows the PR-AUC of edge estimation as $n$ increases.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Lemma 1: Efficient Computation
  • Theorem 2: Existence of MLE
  • Theorem 3: Uniqueness of MLE
  • Theorem 4: High-dimensional consistency
  • Lemma 5: Eigen-decomposition of Cartesian Product
  • proof
  • proof
  • proof
  • ...and 1 more