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On spectral clustering under non-isotropic Gaussian mixture models

Kohei Kawamoto, Yuichi Goto, Koji Tsukuda

TL;DR

The paper addresses misclustering probability for a sign-based spectral clustering method applied to a two-component Gaussian mixture with general covariances. It develops a non-asymptotic bound on misclassification probabilities, expressed in terms of the signal-to-noise quantities $A$ and $\eta$, and leverages concentration of eigenvectors and Gaussian norms to derive the bound. The results extend prior isotropic analyses by handling non-isotropic covariances and provide a high-dimensional consistency corollary showing correct clustering with high probability as $n$ grows under suitable growth of $m$. The work yields practical guarantees for spectral clustering in settings with heterogeneous covariance structures and informs the required sample sizes for reliable binary clustering in high dimensions.

Abstract

We evaluate the misclustering probability of a spectral clustering algorithm under a Gaussian mixture model with a general covariance structure. The algorithm partitions the data into two groups based on the sign of the first principal component score. As a corollary of the main result, the clustering procedure is shown to be consistent in a high-dimensional regime.

On spectral clustering under non-isotropic Gaussian mixture models

TL;DR

The paper addresses misclustering probability for a sign-based spectral clustering method applied to a two-component Gaussian mixture with general covariances. It develops a non-asymptotic bound on misclassification probabilities, expressed in terms of the signal-to-noise quantities and , and leverages concentration of eigenvectors and Gaussian norms to derive the bound. The results extend prior isotropic analyses by handling non-isotropic covariances and provide a high-dimensional consistency corollary showing correct clustering with high probability as grows under suitable growth of . The work yields practical guarantees for spectral clustering in settings with heterogeneous covariance structures and informs the required sample sizes for reliable binary clustering in high dimensions.

Abstract

We evaluate the misclustering probability of a spectral clustering algorithm under a Gaussian mixture model with a general covariance structure. The algorithm partitions the data into two groups based on the sign of the first principal component score. As a corollary of the main result, the clustering procedure is shown to be consistent in a high-dimensional regime.
Paper Structure (5 sections, 3 theorems, 35 equations)

This paper contains 5 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Model and spectral clustering algorithm
  3. Main result
  4. Proofs
  5. Proof of Theorem \ref{['mthm3']}

Key Result

Theorem 3.1

Let $c$, $C$, $K$, and $K_g$ be absolute constants independent of $m$, $n$, $\bm{\mu}$, $\bm{\Sigma}_1$, and $\bm{\Sigma}_{-1}$. Define where $c_1 = 1+K_g^2/\sqrt{c}$. Suppose that there exists $\alpha \in (0,1)$ such that Then, for any $j=-1,1$, it holds that where $\Phi(\cdot)$ and $\phi(\cdot)$ denote the distribution function and density function of $\mathcal{N}(0,1)$, respectively.

Theorems & Definitions (8)

  • Theorem 3.1
  • Remark 1
  • Remark 2
  • Corollary 3.2
  • Remark 3
  • proof : Proof of Theorem \ref{['mthm3']}
  • Lemma 4.1
  • proof