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Identifiability in Graphical Discrete Lyapunov Models

Cecilie Olesen Recke, Sarah Lumpp, Nataliia Kushnerchuk, Janike Oldekop, Jiayi Li, Jane Ivy Coons, Elina Robeva

TL;DR

This work investigates identifiability for discrete Lyapunov models of VAR(1) processes with non-Gaussian errors, where higher-order cumulants of the stationary state encode directed graphs via a sparse parameter matrix A.Using a trek-based combinatorial framework and tensorLyapunov equations, the authors derive both generic and local identifiability results for various graph classes, notably DAGs with self-loops and more general graphs with self-loops but no isolates, from second- to fourth-order cumulants.They establish a Jacobian-based local identifiability criterion, derive defining equations (toric structure for polytrees, determinantal obstructions, and birational-implicitization insights), and demonstrate model equivalence phenomena for specific graph families.Overall, the paper lays a theoretical foundation for structure learning in non-Gaussian discrete Lyapunov models, highlighting both identifiability guarantees and fundamental obstacles that motivate future algebraic-statistical development.

Abstract

In this paper, we study discrete Lyapunov models, which consist of steady-state distributions of first-order vector autoregressive models. The parameter matrix of such a model encodes a directed graph whose vertices correspond to the components of the random vector. This combinatorial framework naturally allows for cycles in the graph structure. We focus on the fundamental problem of identifying the entries of the parameter matrix. In contrast to the classical setting, we assume non-Gaussian error terms, which allows us to use the higher-order cumulants of the model. In this setup, we show generic identifiability for directed acyclic graphs with self-loops at each vertex and show how to express the parameters as a rational function of the cumulants. Furthermore, we establish local identifiability for all directed graphs containing self loops at each vertex and no isolated vertices. Finally, we provide first results on the defining equations of the models, showing model equivalence for certain graphs and paving the way towards structure learning.

Identifiability in Graphical Discrete Lyapunov Models

TL;DR

This work investigates identifiability for discrete Lyapunov models of VAR(1) processes with non-Gaussian errors, where higher-order cumulants of the stationary state encode directed graphs via a sparse parameter matrix A.Using a trek-based combinatorial framework and tensorLyapunov equations, the authors derive both generic and local identifiability results for various graph classes, notably DAGs with self-loops and more general graphs with self-loops but no isolates, from second- to fourth-order cumulants.They establish a Jacobian-based local identifiability criterion, derive defining equations (toric structure for polytrees, determinantal obstructions, and birational-implicitization insights), and demonstrate model equivalence phenomena for specific graph families.Overall, the paper lays a theoretical foundation for structure learning in non-Gaussian discrete Lyapunov models, highlighting both identifiability guarantees and fundamental obstacles that motivate future algebraic-statistical development.

Abstract

In this paper, we study discrete Lyapunov models, which consist of steady-state distributions of first-order vector autoregressive models. The parameter matrix of such a model encodes a directed graph whose vertices correspond to the components of the random vector. This combinatorial framework naturally allows for cycles in the graph structure. We focus on the fundamental problem of identifying the entries of the parameter matrix. In contrast to the classical setting, we assume non-Gaussian error terms, which allows us to use the higher-order cumulants of the model. In this setup, we show generic identifiability for directed acyclic graphs with self-loops at each vertex and show how to express the parameters as a rational function of the cumulants. Furthermore, we establish local identifiability for all directed graphs containing self loops at each vertex and no isolated vertices. Finally, we provide first results on the defining equations of the models, showing model equivalence for certain graphs and paving the way towards structure learning.
Paper Structure (32 sections, 32 theorems, 158 equations, 16 figures)

This paper contains 32 sections, 32 theorems, 158 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Higher-Order Cumulants and the Discrete Lyapunov Equations
  4. Defining the Model via Cumulant Equations
  5. Identifiability
  6. Trek Parametrization
  7. Trek Rules
  8. A Restricted Trek Rule for DAGs
  9. Marginal Independence
  10. Generic Identifiability
  11. Identifiability for DAGs With All Self-Loops
  12. Other Types of DAGs
  13. Nonidentifiable Graphs
  14. Examples on Two Nodes
  15. Diamond-type Graphs
  16. ...and 17 more sections

Key Result

Proposition 2.1

Let $A \in \mathbb{R}^{d \times d}$ be a Schur stable matrix, meaning it has all eigenvalues $|\lambda_i(A)| < 1$. Suppose $\varepsilon_t$ has finite $n$th order cumulant $\Omega^{(n)}$. Then the steady-state distribution of the VAR(1) model eq::VAR(1)process has finite $n$th cumulant $T_n$, and where $\times_k$ denotes the $k$-mode product. Two other equivalent equations for $T_n$ are the recurs

Figures (16)

  • Figure 1: A directed graph on two nodes with a single self-loop at the source.
  • Figure 2: Directed graph with corresponding equitrek graph.
  • Figure 3: A graph on two vertices with two self-loops
  • Figure 4: A generically identifiable graph.
  • Figure 5: Non-identifiable directed graphs on two nodes.
  • ...and 11 more figures

Theorems & Definitions (93)

  • Proposition 2.1: $n$th-Order Cumulants
  • proof
  • Remark 2.2: Gaussian special case
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Corollary 3.4
  • ...and 83 more