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Regression graphs and sparsity-inducing reparametrizations

Jakub Rybak, Heather Battey, Karthik Bharath

TL;DR

The paper shows that population-level sparsity can arise from sparsity-inducing reparametrizations of covariance structures that respect positive definiteness, linking four parametrizations to the Iwasawa decomposition and chain-graph models. It develops exact and approximate-zero characterizations across $ ext{Sigma}_{pd}$, $ ext{Sigma}_{o}$, $ ext{Sigma}_{lt}$, and $ ext{Sigma}_{ltu}$, and provides a unified, graph-based interpretation via causal ordering and path weights. It then devises estimation strategies, proving spectral-norm consistency for sparse-logarithmic estimators under high-dimensional scaling, and demonstrates practical benefits through simulations and a leukemia data example, where reparametrization improves classification performance. Overall, the work clarifies when and why sparsity on transformed scales yields improved interpretability and estimation accuracy in covariance modeling, with implications for chain graphs and high-dimensional inference.$

Abstract

That parametrization and sparsity are inherently linked raises the possibility that relevant models, not obviously sparse in their natural formulation, exhibit a population-level sparsity after reparametrization. In covariance models, positive-definiteness enforces additional constraints on how sparsity can legitimately manifest. It is therefore natural to consider reparametrization maps in which sparsity respects positive definiteness. The main purpose of this paper is to provide insight into structures on the physically-natural scale that induce and are induced by sparsity after reparametrization. The richest of the four structures initially uncovered can be generated, under a causal ordering, by the joint-response graphs studied by Wermuth & Cox (2004), while the most restrictive is that induced by sparsity on the scale of the matrix logarithm, studied by Battey (2017). The Iwasawa decomposition of the general linear group, combined with the graphical-models interpretation, points to a class of reparametrizations for the chain-graph models (Andersson et al. 2001), with undirected and directed acyclic graphs as special cases. An important insight is the interpretation of approximate zeros, explaining the modelling implications of enforcing sparsity after reparameterization: in effect, the relation between two variables would be declared null if relatively direct regression effects were negligible and others manifested through long paths. The insights have a bearing on methodology, some aspects of which are presented. A detailed simulation uses the theoretical insights to further explore regimes under which reparametrization is beneficial.

Regression graphs and sparsity-inducing reparametrizations

TL;DR

The paper shows that population-level sparsity can arise from sparsity-inducing reparametrizations of covariance structures that respect positive definiteness, linking four parametrizations to the Iwasawa decomposition and chain-graph models. It develops exact and approximate-zero characterizations across , , , and , and provides a unified, graph-based interpretation via causal ordering and path weights. It then devises estimation strategies, proving spectral-norm consistency for sparse-logarithmic estimators under high-dimensional scaling, and demonstrates practical benefits through simulations and a leukemia data example, where reparametrization improves classification performance. Overall, the work clarifies when and why sparsity on transformed scales yields improved interpretability and estimation accuracy in covariance modeling, with implications for chain graphs and high-dimensional inference.$

Abstract

That parametrization and sparsity are inherently linked raises the possibility that relevant models, not obviously sparse in their natural formulation, exhibit a population-level sparsity after reparametrization. In covariance models, positive-definiteness enforces additional constraints on how sparsity can legitimately manifest. It is therefore natural to consider reparametrization maps in which sparsity respects positive definiteness. The main purpose of this paper is to provide insight into structures on the physically-natural scale that induce and are induced by sparsity after reparametrization. The richest of the four structures initially uncovered can be generated, under a causal ordering, by the joint-response graphs studied by Wermuth & Cox (2004), while the most restrictive is that induced by sparsity on the scale of the matrix logarithm, studied by Battey (2017). The Iwasawa decomposition of the general linear group, combined with the graphical-models interpretation, points to a class of reparametrizations for the chain-graph models (Andersson et al. 2001), with undirected and directed acyclic graphs as special cases. An important insight is the interpretation of approximate zeros, explaining the modelling implications of enforcing sparsity after reparameterization: in effect, the relation between two variables would be declared null if relatively direct regression effects were negligible and others manifested through long paths. The insights have a bearing on methodology, some aspects of which are presented. A detailed simulation uses the theoretical insights to further explore regimes under which reparametrization is beneficial.
Paper Structure (45 sections, 33 theorems, 110 equations, 11 figures, 6 tables)

This paper contains 45 sections, 33 theorems, 110 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Reparametrization
  4. Structure of $\Sigma(\alpha)$ induced by, and inducing, exact zeros in $\alpha$
  5. Sparsity under the $\Sigma_{ltu}$ parametrization
  6. Causal ordering
  7. The matrix logarithm and weighted causal paths
  8. Exact zeros
  9. Approximate zeros
  10. Sparsity under the $\Sigma_{pd}$ and $\Sigma_{o}$ parametrizations
  11. Sparsity-induced structures
  12. Interpretation of zeros under the $\Sigma_{pd}$ parametrization
  13. Chain graphs and the Iwasawa decomposition
  14. A unifying parametrization for chain graphs
  15. Connection to the Iwasawa decomposition of the general linear group
  16. ...and 30 more sections

Key Result

Theorem 4.1

Consider $M=e^L\in\textsf{M}(p)$ where $L\in\textsf{V}(p)$, a vector space with canonical basis $\mathcal{B}$ of dimension $m$. The matrix $M$ is logarithmically sparse in the sense that $L=L(\alpha)=\alpha_1 B_1 + \cdots + \alpha_m B_m$, $B_j\in \mathcal{B}$ with $\|\alpha\|_0=s^*$ if and only if $

Figures (11)

  • Figure 1: Example of a structure of $M$ as established in Theorem \ref{['prop2.3']} with $p = 10$, $d^{*}_{r} = 7$, $d^{*}_{c} = 8$ and $d^{*} = 9$. The entries that are zero by Theorem \ref{['prop2.3']} are light blue, those equal to one are medium blue, and the remaining entries, whose values are unconstrained, are dark blue.
  • Figure 2: Structure of $\Sigma(\alpha)$ induced by sparsity of $\alpha$. Zero entries are depicted by light blue, unit entries by medium blue, and the unrestricted entries by dark blue.
  • Figure 3: Directed acyclic graph with edge weights corresponding to regression coefficients.
  • Figure 4: Directed acyclic graph corresponding to $U$ and $L$ satisfying $U_{42} = 0$ and $L_{42} = 0$.
  • Figure 5: Directed acyclic graphs corresponding to the example of Figure \ref{['fig:sigma_example']}. Arrows indicate directed edges and nodes correspond to random variables.
  • ...and 6 more figures

Theorems & Definitions (46)

  • Theorem 4.1
  • Corollary 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Corollary 4.4
  • Lemma 4.1
  • Example 5.1
  • Proposition 5.1
  • Corollary 5.1
  • Proposition 5.2
  • ...and 36 more