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Information-Theoretic Limits and Strong Consistency on Binary Non-uniform Hypergraph Stochastic Block Models

Hai-Xiao Wang

TL;DR

The paper tackles exact recovery in binary non-uniform hypergraph SBMs with two equal-size communities by linking strong-consistency thresholds to a Generalized Hellinger distance D_GH. It shows that strong recovery is impossible when D_GH < 1 and achievable when D_GH > 1, under appropriate growth of q_N, and it derives a universal IT lower bound on misclassification. A novel two-stage approach using aggregation, spectral initialization, and power-iteration or majority-voting refinements achieves strong consistency above the threshold and attains the IT lower bound below it, proving optimality. Aggregation across multiple uniform layers is shown to strictly improve or at least non-negatively contribute to clustering performance, contrasting with constant-degree regimes where subset selection is sometimes necessary. The results extend the understanding of phase transitions and optimal algorithms in hypergraph SBMs, providing practical refinements for clustering with only aggregated information.

Abstract

We investigate the unsupervised node classification problem on random hypergraphs under the non-uniform Hypergraph Stochastic Block Model (HSBM) with two equal-sized communities. In this model, edges appear independently with probabilities depending only on the labels of their vertices. We identify the threshold for strong consistency, expressed in terms of the Generalized Hellinger distance. Below this threshold, strong consistency is impossible, and we derive the Information-Theoretic (IT) lower bound on the expected mismatch ratio. Above the threshold, the parameter space is typically divided into two disjoint regions. When only the aggregated adjacency matrices are accessible, while one-stage algorithms accomplish strong consistency with high probability in the region far from the threshold, they fail in the region closer to the threshold. We propose a new refinement algorithm which, in conjunction with the initial estimation, provably achieves strong consistency throughout the entire region above the threshold, and attains the IT lower bound when below the threshold, proving its optimality. This novel refinement algorithm applies the power iteration method to a weighted adjacency matrix, where the weights are determined by hyperedge sizes and the initial label estimate. Unlike the constant degree regime where a subset selection of uniform layers is necessary to enhance clustering accuracy, in the scenario with diverging degrees, each uniform layer contributes non-negatively to clustering accuracy. Therefore, aggregating information across all uniform layers yields better performance than using any single layer alone.

Information-Theoretic Limits and Strong Consistency on Binary Non-uniform Hypergraph Stochastic Block Models

TL;DR

The paper tackles exact recovery in binary non-uniform hypergraph SBMs with two equal-size communities by linking strong-consistency thresholds to a Generalized Hellinger distance D_GH. It shows that strong recovery is impossible when D_GH < 1 and achievable when D_GH > 1, under appropriate growth of q_N, and it derives a universal IT lower bound on misclassification. A novel two-stage approach using aggregation, spectral initialization, and power-iteration or majority-voting refinements achieves strong consistency above the threshold and attains the IT lower bound below it, proving optimality. Aggregation across multiple uniform layers is shown to strictly improve or at least non-negatively contribute to clustering performance, contrasting with constant-degree regimes where subset selection is sometimes necessary. The results extend the understanding of phase transitions and optimal algorithms in hypergraph SBMs, providing practical refinements for clustering with only aggregated information.

Abstract

We investigate the unsupervised node classification problem on random hypergraphs under the non-uniform Hypergraph Stochastic Block Model (HSBM) with two equal-sized communities. In this model, edges appear independently with probabilities depending only on the labels of their vertices. We identify the threshold for strong consistency, expressed in terms of the Generalized Hellinger distance. Below this threshold, strong consistency is impossible, and we derive the Information-Theoretic (IT) lower bound on the expected mismatch ratio. Above the threshold, the parameter space is typically divided into two disjoint regions. When only the aggregated adjacency matrices are accessible, while one-stage algorithms accomplish strong consistency with high probability in the region far from the threshold, they fail in the region closer to the threshold. We propose a new refinement algorithm which, in conjunction with the initial estimation, provably achieves strong consistency throughout the entire region above the threshold, and attains the IT lower bound when below the threshold, proving its optimality. This novel refinement algorithm applies the power iteration method to a weighted adjacency matrix, where the weights are determined by hyperedge sizes and the initial label estimate. Unlike the constant degree regime where a subset selection of uniform layers is necessary to enhance clustering accuracy, in the scenario with diverging degrees, each uniform layer contributes non-negatively to clustering accuracy. Therefore, aggregating information across all uniform layers yields better performance than using any single layer alone.
Paper Structure (42 sections, 44 theorems, 271 equations, 1 figure, 4 algorithms)

This paper contains 42 sections, 44 theorems, 271 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. The model
  3. Formulation of the problems
  4. Phase transition threshold for strong consistency
  5. Information-Theoretic lower bound
  6. Contributions
  7. Organization
  8. Main Results
  9. Information-Theoretic limits
  10. Spectral partition using purely the adjacency matrix
  11. Refinement algorithms
  12. Refinement via power iteration
  13. Refinement via majority voting
  14. Strong consistency after refinement
  15. Benefits of aggregation
  16. ...and 27 more sections

Key Result

Theorem 2.1

For model def:non_uniform_HSBM_binary under Assumption ass:asymptotics_binary, suppose then with probability tending to $1$, every algorithm will misclassify at least two vertices.

Figures (1)

  • Figure 1: Plots for areas $\mathds{D}_{\mathrm{GH}} < 1$ (red), $\mathds{D}_{\rm{AM}} < 1 < \mathds{D}_{\mathrm{GH}}$ (green) and $\mathds{D}_{\rm{AM}} > 1$ (blue)

Theorems & Definitions (88)

  • Definition 1.1: Hypergraph
  • Definition 1.2: Symmetric binary uniform HSBM
  • Definition 1.3: Non-uniform HSBM
  • Remark 1.4
  • Definition 1.5: Aggregated adjacency matrix
  • Theorem 2.1: Impossibility of exact recovery
  • Remark 2.2
  • Theorem 2.3: Lower bound on the expected mismatch ratio
  • Theorem 2.4: Strong consistency
  • Theorem 2.5: Optimality
  • ...and 78 more