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SpectralGap: Graph-Level Out-of-Distribution Detection via Laplacian Eigenvalue Gaps

Jiawei Gu, Ziyue Qiao, Zechao Li

TL;DR

SpecGap adjusts features by subtracting the component associated with the second-largest eigenvalue, scaled by the spectral gap, from the high-level features of the Laplacian matrix, and achieves state-of-the-art performance across multiple benchmark datasets.

Abstract

The task of graph-level out-of-distribution (OOD) detection is crucial for deploying graph neural networks in real-world settings. In this paper, we observe a significant difference in the relationship between the largest and second-largest eigenvalues of the Laplacian matrix for in-distribution (ID) and OOD graph samples: \textit{OOD samples often exhibit anomalous spectral gaps (the difference between the largest and second-largest eigenvalues)}. This observation motivates us to propose SpecGap, an effective post-hoc approach for OOD detection on graphs. SpecGap adjusts features by subtracting the component associated with the second-largest eigenvalue, scaled by the spectral gap, from the high-level features (i.e., $\mathbf{X}-\left(λ_n-λ_{n-1}\right) \mathbf{u}_{n-1} \mathbf{v}_{n-1}^T$). SpecGap achieves state-of-the-art performance across multiple benchmark datasets. We present extensive ablation studies and comprehensive theoretical analyses to support our empirical results. As a parameter-free post-hoc method, SpecGap can be easily integrated into existing graph neural network models without requiring any additional training or model modification.

SpectralGap: Graph-Level Out-of-Distribution Detection via Laplacian Eigenvalue Gaps

TL;DR

SpecGap adjusts features by subtracting the component associated with the second-largest eigenvalue, scaled by the spectral gap, from the high-level features of the Laplacian matrix, and achieves state-of-the-art performance across multiple benchmark datasets.

Abstract

The task of graph-level out-of-distribution (OOD) detection is crucial for deploying graph neural networks in real-world settings. In this paper, we observe a significant difference in the relationship between the largest and second-largest eigenvalues of the Laplacian matrix for in-distribution (ID) and OOD graph samples: \textit{OOD samples often exhibit anomalous spectral gaps (the difference between the largest and second-largest eigenvalues)}. This observation motivates us to propose SpecGap, an effective post-hoc approach for OOD detection on graphs. SpecGap adjusts features by subtracting the component associated with the second-largest eigenvalue, scaled by the spectral gap, from the high-level features (i.e., ). SpecGap achieves state-of-the-art performance across multiple benchmark datasets. We present extensive ablation studies and comprehensive theoretical analyses to support our empirical results. As a parameter-free post-hoc method, SpecGap can be easily integrated into existing graph neural network models without requiring any additional training or model modification.

Paper Structure

This paper contains 51 sections, 1 theorem, 29 equations, 7 figures, 9 tables.

Table of Contents

  1. Introduction
  2. SpecGap: Spectral Gap-based OOD Detection
  3. Preliminaries
  4. SpecGap Algorithm
  5. Feature Adjustment
  6. Integration into GNN Models
  7. Efficient Computation of Spectral Gap
  8. Lanczos Algorithm
  9. Implementation Details
  10. Experiments
  11. Experimental Setup
  12. Datasets
  13. Baselines and Our Method
  14. Evaluation Metrics
  15. Implementation Details
  16. ...and 36 more sections

Key Result

Theorem 1

Under assumptions (D1) and (D2) above, let $(\Delta\lambda_{\text{ID}},\mathbf{u}_{n-1,\text{ID}})$ and $(\Delta\lambda_{\text{OOD}},\mathbf{u}_{n-1,\text{OOD}})$ be drawn from their respective distributions. Then, on average, SpecGap produces a strictly larger separation: for some $\Gamma>0$ that depends on (i) the distribution difference in $\Delta\lambda$, and (ii) the typical angle between $\

Figures (7)

  • Figure 1: SpecGap: Spectral Gap-based OOD Detection. (a) Original Eigenvalue Distribution: OOD samples show larger and more varied spectral gaps compared to ID samples. (b) Spectral Gap Distribution: Clear separation between ID and OOD samples based on spectral gap. (c) Distribution After SpecGap: The method effectively brings OOD samples closer to the ID distribution.
  • Figure 2: Performance comparison of OOD detection methods before and after applying SpecGap. Each subplot represents a different dataset pair, with methods on the x-axis and AUC scores on the y-axis. Coral and slate blue bars indicate performance before and after SpecGap application, respectively.
  • Figure 3: Impact of the number of largest eigenvalues used in SpecGap on OOD detection performance (AUC) using the GCL$_S$ model on the ENZYMES-PROTEIN dataset.
  • Figure 4: Comparison of different feature projection methods in SpecGap on the IMDBM-IMDBB dataset using the JOAO$_S$ model.
  • Figure 5: An example ID vs. OOD spectral gap distribution from randomly perturbed graphs (Section \ref{['subsec:empirical_demo']}). The ID distribution (green) has a mild peak around $\Delta\lambda \approx 3.5$, while the OOD distribution (orange) is broader and shifted, with some overlap but still a noticeable difference. A threshold near $\tau\approx 3.0$ can separate many samples. This aligns with assumptions (D1)--(D2).
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1: Distribution-level Separation Gain