Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fast Semi-supervised Learning on Large Graphs: An Improved Green-function Method

Feiping Nie, Yitao Song, Wei Chang, Rong Wang, Xuelong Li

TL;DR

This work proposes a workable approach to improve the Green's function method, which can also apply two accelerating techniques, Gaussian Elimination, and Anchored Graphs to become more efficient on large graphs.

Abstract

In the graph-based semi-supervised learning, the Green-function method is a classical method that works by computing the Green's function in the graph space. However, when applied to large graphs, especially those sparse ones, this method performs unstably and unsatisfactorily. We make a detailed analysis on it and propose a novel method from the perspective of optimization. On fully connected graphs, the method is equivalent to the Green-function method and can be seen as another interpretation with physical meanings, while on non-fully connected graphs, it helps to explain why the Green-function method causes a mess on large sparse graphs. To solve this dilemma, we propose a workable approach to improve our proposed method. Unlike the original method, our improved method can also apply two accelerating techniques, Gaussian Elimination, and Anchored Graphs to become more efficient on large graphs. Finally, the extensive experiments prove our conclusions and the efficiency, accuracy, and stability of our improved Green's function method.

Fast Semi-supervised Learning on Large Graphs: An Improved Green-function Method

TL;DR

This work proposes a workable approach to improve the Green's function method, which can also apply two accelerating techniques, Gaussian Elimination, and Anchored Graphs to become more efficient on large graphs.

Abstract

In the graph-based semi-supervised learning, the Green-function method is a classical method that works by computing the Green's function in the graph space. However, when applied to large graphs, especially those sparse ones, this method performs unstably and unsatisfactorily. We make a detailed analysis on it and propose a novel method from the perspective of optimization. On fully connected graphs, the method is equivalent to the Green-function method and can be seen as another interpretation with physical meanings, while on non-fully connected graphs, it helps to explain why the Green-function method causes a mess on large sparse graphs. To solve this dilemma, we propose a workable approach to improve our proposed method. Unlike the original method, our improved method can also apply two accelerating techniques, Gaussian Elimination, and Anchored Graphs to become more efficient on large graphs. Finally, the extensive experiments prove our conclusions and the efficiency, accuracy, and stability of our improved Green's function method.

Paper Structure

This paper contains 22 sections, 5 theorems, 61 equations, 7 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Overview of Graph-based Semi-supervised Learning
  4. Green-function Method
  5. Learning With Local and Global Consistency
  6. Proposed Method and Theoretical Derivation
  7. Rules for Derivation
  8. Theorems About Solving the Problem (\ref{['eq:min_without']})
  9. Our Proposed Method and its Physical Meanings
  10. Equivalence Between Coding Ways
  11. When It Comes to Non-fully Connected Graphs
  12. Accelerating Techniques on Large Graphs
  13. Accelerating by Gauss Elimination
  14. Accelerating by Anchored Graph
  15. Relationship With LLGC
  16. ...and 7 more sections

Key Result

Lemma 1

Let's say $\mathbf{x}$ is an unknown $n$-dimensional vector. There are solutions to $\mathbf{A} \mathbf{x} = \mathbf{b}$ if and only if ${\mathbf{U}_\perp} ^{\mathrm{T}} \mathbf{b} = \mathbf{0}$, where $\mathbf{0}$ denotes a zero vector.

Figures (7)

  • Figure 1: The schematic diagram of the paper.
  • Figure 2: An example of $\mathbf{U}_\perp$ on 2-connected graph. It can be validated that $\mathbf{U}_\perp$ spans the null space of $\mathbf{L}$.
  • Figure 3: Average classification accuracy versus the number of labeled samples per class.
  • Figure 4: Two-Ring Dataset Which is Non-fully Connected.
  • Figure 5: Two-Ring Dataset Which is Fully Connected.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Corollary 2