Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Theory of Random Graph Shift in Truncated-Spectrum vRKHS

Zhang Wan, Tingting Mu, Samuel Kaski

Abstract

This paper develops a theory of graph classification under domain shift through a random-graph generative lens, where we consider intra-class graphs sharing the same random graph model (RGM) and the domain shift induced by changes in RGM components. While classic domain adaptation (DA) theories have well-underpinned existing techniques to handle graph distribution shift, the information of graph samples, which are itself structured objects, is less explored. The non-Euclidean nature of graphs and specialized architectures for graph learning further complicate a fine-grained analysis of graph distribution shifts. In this paper, we propose a theory that assumes RGM as the data generative process, exploiting its connection to hypothesis complexity in function space perspective for such fine-grained analysis. Building on a vector-valued reproducing kernel Hilbert space (vRKHS) formulation, we derive a generalization bound whose shift penalty admits a factorization into (i) a domain discrepancy term, (ii) a spectral-geometry term summarized by the accessible truncated spectrum, and (iii) an amplitude term that aggregates convergence and construction-stability effects. We empirically verify the insights on these terms in both real data and simulations.

A Theory of Random Graph Shift in Truncated-Spectrum vRKHS

Abstract

This paper develops a theory of graph classification under domain shift through a random-graph generative lens, where we consider intra-class graphs sharing the same random graph model (RGM) and the domain shift induced by changes in RGM components. While classic domain adaptation (DA) theories have well-underpinned existing techniques to handle graph distribution shift, the information of graph samples, which are itself structured objects, is less explored. The non-Euclidean nature of graphs and specialized architectures for graph learning further complicate a fine-grained analysis of graph distribution shifts. In this paper, we propose a theory that assumes RGM as the data generative process, exploiting its connection to hypothesis complexity in function space perspective for such fine-grained analysis. Building on a vector-valued reproducing kernel Hilbert space (vRKHS) formulation, we derive a generalization bound whose shift penalty admits a factorization into (i) a domain discrepancy term, (ii) a spectral-geometry term summarized by the accessible truncated spectrum, and (iii) an amplitude term that aggregates convergence and construction-stability effects. We empirically verify the insights on these terms in both real data and simulations.
Paper Structure (74 sections, 28 theorems, 256 equations, 21 figures, 10 tables)

This paper contains 74 sections, 28 theorems, 256 equations, 21 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Pullback Domain Divergence
  5. Spectrum-Amplitude Factorization
  6. Hypothesis Disagreement Analysis
  7. Experiments
  8. Exp 1: Latent Wasserstein Distance as Domain Divergence ($\Delta_D$).
  9. Exp 2: Truncated-Spectrum and Geometry ($\lambda_r$).
  10. Exp 3: Amplitude ($K'\Xi+K"$).
  11. Conclusion
  12. Notations
  13. Definitions
  14. On RGM and Its Deformation
  15. On Neural Network Families and Perturbation
  16. ...and 59 more sections

Key Result

Proposition 2.2

Given a source domain $D_S=(\mu_S, g_{D})$, a target domain $D_T=(\mu_T, g_{D})$, a hypothesis $h$, and the source and target error risks $\epsilon_S(h,g_D)$ and $\epsilon_T(h,g_D)$ assessed by Eq. (error_function). Then, the following holds: where $\mathcal{H}_{K_\ell}$ is the vRKHS of the hypothesis function $h$, the labeling function $g_D$, and the loss mapping function $\ell_{h,g_D}$.

Figures (21)

  • Figure 1: Estimated latent WD correlates with test losses.
  • Figure 2: Layerwise perturbation sensitivity. Early-layer sensitivity (conv0) is consistently the largest. Both non-uniform schemes reduce early-layer fragility relative to L2.
  • Figure 3: Estimated latent WD correlates with test losses.
  • Figure 4: Eigenvalues on IMDB-MULTI (1k), NCI1 (2k), PROTEINS (1k).
  • Figure 5: $d_{\mathrm{eff}}$ on IMDB-MULTI (1k), NCI1 (2k), PROTEINS (1k).
  • ...and 16 more figures

Theorems & Definitions (64)

  • Definition 2.1: Random Graph Model
  • Proposition 2.2: Domain Adaptation Generalization Error
  • Theorem 3.1: Main Theorem
  • Corollary 3.2
  • Proposition 3.3
  • Proposition 3.5
  • Remark 3.6
  • Theorem 3.7: Class-conditional convergence maskey2022generalization
  • Theorem 3.9: Optimization term bound via construction stability
  • Definition B.1: Function Sampling Operator
  • ...and 54 more