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Encoder Embedding for General Graph and Node Classification

Cencheng Shen

TL;DR

This paper proves that the encoder embedding satisfies the law of large numbers and the central limit theorem on a per-observation basis, and achieves asymptotic normality on a per-class basis, enabling optimal classification through discriminant analysis.

Abstract

Graph encoder embedding, a recent technique for graph data, offers speed and scalability in producing vertex-level representations from binary graphs. In this paper, we extend the applicability of this method to a general graph model, which includes weighted graphs, distance matrices, and kernel matrices. We prove that the encoder embedding satisfies the law of large numbers and the central limit theorem on a per-observation basis. Under certain condition, it achieves asymptotic normality on a per-class basis, enabling optimal classification through discriminant analysis. These theoretical findings are validated through a series of experiments involving weighted graphs, as well as text and image data transformed into general graph representations using appropriate distance metrics.

Encoder Embedding for General Graph and Node Classification

TL;DR

This paper proves that the encoder embedding satisfies the law of large numbers and the central limit theorem on a per-observation basis, and achieves asymptotic normality on a per-class basis, enabling optimal classification through discriminant analysis.

Abstract

Graph encoder embedding, a recent technique for graph data, offers speed and scalability in producing vertex-level representations from binary graphs. In this paper, we extend the applicability of this method to a general graph model, which includes weighted graphs, distance matrices, and kernel matrices. We prove that the encoder embedding satisfies the law of large numbers and the central limit theorem on a per-observation basis. Under certain condition, it achieves asymptotic normality on a per-class basis, enabling optimal classification through discriminant analysis. These theoretical findings are validated through a series of experiments involving weighted graphs, as well as text and image data transformed into general graph representations using appropriate distance metrics.
Paper Structure (14 sections, 3 theorems, 39 equations, 2 figures, 1 table)

This paper contains 14 sections, 3 theorems, 39 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. General Graph Model and Encoder Transformation
  3. Model Definition
  4. Examples
  5. The Encoder Transformation
  6. Sample Method
  7. Asymptotic Theorems
  8. Assumptions
  9. Asymptotic Normality
  10. Discriminant Analysis
  11. Experiments
  12. Simulations
  13. Real Data
  14. Conclusion

Key Result

Theorem 1

As $n$ increases to infinity, the encoder embedding conditioned on $X=x$ satisfies the weak law of large number and central limit theorem: Here, $\mu_{x} \in \mathbb{R}^{K}$ is a conditional mean vector where each dimension satisfies for $k=1,\ldots,K$, and $\Sigma_{x} \in \mathbb{R}^{K \times K}$ is a diagonal matrix where each diagonal entry satisfies for $k=1,\ldots,K$.

Figures (2)

  • Figure 1: This figure provides visualizations of the original data, the embedded data, and the resulting 5-fold classification error for both multivariate Gaussian data (on the left) and a weighted stochastic block model (on the right). In the embedding visualization, different colors represent observations from different classes.
  • Figure 2: This figure visualizes the original data and the embedded data using three different graph transformations.

Theorems & Definitions (7)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof