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Information-geometry-driven graph sequential growth

Harry T. Bond, Bertrand Gauthier, Kirstin Strokorb

TL;DR

We address structure learning for sparse Gaussian graphical models using regularisation-free optimization by leveraging information geometry. The method builds graphs via information-optimal sequential growth, which maps to fully-corrective coordinate descent with diagonal initialisation and BFCI selection, and is augmented with efficient approximations (Algorithm 2) and relaxed rules (BBI, GSL). Empirical results on synthetic and real data show reliable edge recovery with few false detections, and activation ranks provide insight into edge significance, offering a diagnostic alternative to regularization. The approach is tuning-parameter-free and has complexity comparable to standard coordinate-descent methods, making it scalable and practically appealing, particularly when used in conjunction with subsampling for stability-like analysis. Overall, the work contributes a principled, geometry-driven framework for sequential growth of Gaussian graphical models and demonstrates its practical utility for sparse graph recovery and exploratory data analysis.

Abstract

We investigate the properties of a class of regularisation-free approaches for Gaussian graphical inference based on the information-geometry-driven sequential growth of initially edgeless graphs. Relating the growth of a graph to a coordinate descent process, we characterise the fully-corrective descents corresponding to information-optimal growths, and propose numerically efficient strategies for their approximation. We demonstrate the ability of the proposed procedures to reliably extract sparse graphical models while limiting the number of false detections, and illustrate how activation ranks can provide insight into the informational relevance of edge sets. The considered approaches are tuning-parameter-free and have complexities akin to coordinate descents.

Information-geometry-driven graph sequential growth

TL;DR

We address structure learning for sparse Gaussian graphical models using regularisation-free optimization by leveraging information geometry. The method builds graphs via information-optimal sequential growth, which maps to fully-corrective coordinate descent with diagonal initialisation and BFCI selection, and is augmented with efficient approximations (Algorithm 2) and relaxed rules (BBI, GSL). Empirical results on synthetic and real data show reliable edge recovery with few false detections, and activation ranks provide insight into edge significance, offering a diagnostic alternative to regularization. The approach is tuning-parameter-free and has complexity comparable to standard coordinate-descent methods, making it scalable and practically appealing, particularly when used in conjunction with subsampling for stability-like analysis. Overall, the work contributes a principled, geometry-driven framework for sequential growth of Gaussian graphical models and demonstrates its practical utility for sparse graph recovery and exploratory data analysis.

Abstract

We investigate the properties of a class of regularisation-free approaches for Gaussian graphical inference based on the information-geometry-driven sequential growth of initially edgeless graphs. Relating the growth of a graph to a coordinate descent process, we characterise the fully-corrective descents corresponding to information-optimal growths, and propose numerically efficient strategies for their approximation. We demonstrate the ability of the proposed procedures to reliably extract sparse graphical models while limiting the number of false detections, and illustrate how activation ranks can provide insight into the informational relevance of edge sets. The considered approaches are tuning-parameter-free and have complexities akin to coordinate descents.
Paper Structure (18 sections, 6 theorems, 43 equations, 10 figures)

This paper contains 18 sections, 6 theorems, 43 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Framework and notations
  3. Gaussian graphical loss
  4. Coordinate directions and graph likelihood
  5. Information-optimal graph sequential growth
  6. Coordinate descent and graph optimality
  7. Exact block update
  8. Graph-optimal matrix approximation
  9. Relaxed graph sequential growth
  10. Best block improvement (BBI).
  11. Gauss-Southwell-Lipschitz (GSL).
  12. Experiments
  13. Synthetic example based on 494_bus
  14. Fully-synthetic simulation study
  15. Recovering a gene network: the Riboflavin data
  16. ...and 3 more sections

Key Result

Lemma 2.1

For all $\mathbf{S}\in\mathcal{S}_{\succcurlyeq 0}^{d}$, the map $f_{\mathbf{S}}$ is strictly convex on $\mathcal{S}_{\succ 0}^{d}$. If $\mathbf{S}$ is invertible, then $f_{\mathbf{S}}$ is coercive over $\mathcal{S}_{\succ 0}^{d}$ and is minimised at $\mathbf{Q}=\mathbf{S}^{-1}$. If $\mathbf{S}$ is

Figures (10)

  • Figure 1: For the 494_bus example, ROC curves (top row) and precision-recall curves (bottom row) for the graphs for the sequential growths induced by the GSL, BBI and BFCI selection rules with approximate full correction. For each sample size, $100$ repetitions are performed for GSL and BBI, and $10$ for BFCI. The curves indicate the median accuracies, and the coloured regions the pointwise interdecile ranges. See Section \ref{['sec:494bus']}.
  • Figure 2: For the 494_bus example, ROC curves (top row) and precision-recall curves (bottom row) for the graphs recovered via GSL, Glasso, Prec and Pcorr. Sample sizes correspond to columns in the figure. The curves indicate the median accuracies over $100$ repetitions and the coloured regions the pointwise interdecile ranges. See Section \ref{['sec:494bus']}
  • Figure 3: Graphical representation of the true partial correlations and precisions of the $30$ true edges of the 494_bus example; the edges are ordered by decreasing absolute precision (top-left). For the sequential growth procedures GSL, Prec and PCorr, with $k=30$ (that is, $30$ edges are extracted), empirical frequency of detection of the true edges over $500$ repetitions; the sample size is $n=30$ (bottom-left). The empirical distribution of the false positives for GSL is also presented (right). See Section \ref{['sec:494bus']}
  • Figure 4: Distribution of the edge activation ranks under the full GSL growth procedure ($k=1{,}225$) for the 494_bus example with $n=90$ and $500$ repetitions. The edges are ordered according to their median activation position (top horizontal axis). The $55$ foremost edges are displayed (left), together with the remaining true edges (right). See Section \ref{['sec:494bus']}.
  • Figure 5: For a sample ($n=90$) drawn in the setting of the 494_bus example, evolution of the number of inner iterations of the GSL variant of Algorithm \ref{['algo:RelaxSeqGrowth']} as a function of the number of outer iterations, for three different values of the threshold parameter $\tau$, and with $\alpha=1$ and $\beta=10$ (see Remark \ref{['rem:StopAlgo2']}). The evolution of the accuracy of the underlying graph-recovery process is also presented. See Section \ref{['sec:494bus']}.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Lemma 2.3
  • Definition 3.1
  • Lemma 4.1: exact order-$m$ update
  • Theorem 4.1
  • Remark 4.1
  • Remark 6.1
  • Remark 6.2: naive graph sequential growths
  • ...and 2 more