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Information Geometry of Absorbing Markov-Chain and Discriminative Random Walks

Masanari Kimura

TL;DR

This work addresses the theoretical gap in discriminative random walks (DRWs) for semi-supervised node classification by treating DRWs as a parametric family of hitting-time laws on an absorbing Markov chain and analyzing them via information geometry. It derives closed-form first-passage laws (pmf/pgf), moments, and gradients using a Lyapunov equation for the fundamental matrix, revealing that the observed Fisher information for each seed is rank-one and that a quotient manifold of identifiable edge-weight directions is globally flat with dimension equal to the rank $r$ of the aggregate sensitivity. A Fisher-bounded sensitivity score is defined on this quotient, linking geometric directions to the maximal first-order change in DRW betweenness under unit perturbations and enabling principled strategies for active label acquisition, edge reweighting, and explanations. Collectively, the framework provides a computable, interpretable geometric lens on DRW-based semi-supervised learning with practical implications for label collection and graph design.

Abstract

Discriminative Random Walks (DRWs) are a simple yet powerful tool for semi-supervised node classification, but their theoretical foundations remain fragmentary. We revisit DRWs through the lens of information geometry, treating the family of class-specific hitting-time laws on an absorbing Markov chain as a statistical manifold. Starting from a log-linear edge-weight model, we derive closed-form expressions for the hitting-time probability mass function, its full moment hierarchy, and the observed Fisher information. The Fisher matrix of each seed node turns out to be rank-one, taking the quotient by its null space yields a low-dimensional, globally flat manifold that captures all identifiable directions of the model. Leveraging the geometry, we introduce a sensitivity score for unlabeled nodes that bounds, and in one-dimensional cases attains, the maximal first-order change in DRW betweenness under unit Fisher perturbations. The score can lead to principled strategies for active label acquisition, edge re-weighting, and explanation.

Information Geometry of Absorbing Markov-Chain and Discriminative Random Walks

TL;DR

This work addresses the theoretical gap in discriminative random walks (DRWs) for semi-supervised node classification by treating DRWs as a parametric family of hitting-time laws on an absorbing Markov chain and analyzing them via information geometry. It derives closed-form first-passage laws (pmf/pgf), moments, and gradients using a Lyapunov equation for the fundamental matrix, revealing that the observed Fisher information for each seed is rank-one and that a quotient manifold of identifiable edge-weight directions is globally flat with dimension equal to the rank of the aggregate sensitivity. A Fisher-bounded sensitivity score is defined on this quotient, linking geometric directions to the maximal first-order change in DRW betweenness under unit perturbations and enabling principled strategies for active label acquisition, edge reweighting, and explanations. Collectively, the framework provides a computable, interpretable geometric lens on DRW-based semi-supervised learning with practical implications for label collection and graph design.

Abstract

Discriminative Random Walks (DRWs) are a simple yet powerful tool for semi-supervised node classification, but their theoretical foundations remain fragmentary. We revisit DRWs through the lens of information geometry, treating the family of class-specific hitting-time laws on an absorbing Markov chain as a statistical manifold. Starting from a log-linear edge-weight model, we derive closed-form expressions for the hitting-time probability mass function, its full moment hierarchy, and the observed Fisher information. The Fisher matrix of each seed node turns out to be rank-one, taking the quotient by its null space yields a low-dimensional, globally flat manifold that captures all identifiable directions of the model. Leveraging the geometry, we introduce a sensitivity score for unlabeled nodes that bounds, and in one-dimensional cases attains, the maximal first-order change in DRW betweenness under unit Fisher perturbations. The score can lead to principled strategies for active label acquisition, edge re-weighting, and explanation.
Paper Structure (27 sections, 1 theorem, 68 equations, 1 table)

This paper contains 27 sections, 1 theorem, 68 equations, 1 table.

Table of Contents

  1. Introduction
  2. Paper organization.
  3. Preliminaries
  4. Notation
  5. Discriminative Random Walk
  6. Hitting-Time Law of Class-y Absorption
  7. Hitting-time distribution (pmf and pgf)
  8. Moments (mean and variance)
  9. Spectral representation
  10. Parametric Edge-Weight Model and Derivatives
  11. Score Function and Fisher Information Matrix
  12. Differential-Geometric Structure of DRWs
  13. Coordinate Chart
  14. Push Forward of Fisher Pseudo Metric
  15. Sensitivity Scores of Unlabeled Nodes
  16. ...and 12 more sections

Key Result

Proposition 1

Let $G_y(\bm{u}) \coloneqq \mathrm{grad}_{\tilde{\bm{\Theta}}}\ \beta_L(q, y; \bm{u}) \in T_{\bm{u}}\tilde{\bm{\Theta}}$ be the class-wise Riemannian gradients at $\bm{u} = \Phi(\Theta)$ on the flat quotient manifold $(\tilde{\bm{\Theta}}, \tilde{g})$ for $y \in \mathcal{Y}$. For any tangent vector

Theorems & Definitions (3)

  • Definition 1: Class-$y$ Discriminative Random Walk callut2008semi
  • Definition 2: DRW Betweenness
  • Proposition 1