Individual-heterogeneous sub-Gaussian Mixture Models

Huan Qing

stat.MLApr 7, 2026

Individual-heterogeneous sub-Gaussian Mixture Models

Authors: Huan Qing

Organizations: School of Economics and Finance, Chongqing University of Technology, Chongqing, 400054, China

Abstract

The classical Gaussian mixture model assumes homogeneity within clusters, an assumption that often fails in real-world data where observations naturally exhibit varying scales or intensities. To address this, we introduce the individual-heterogeneous sub-Gaussian mixture model, a flexible framework that assigns each observation its own heterogeneity parameter, thereby explicitly capturing the heterogeneity inherent in practical applications. Built upon this model, we propose an efficient spectral method that provably achieves exact recovery of the true cluster labels under mild separation conditions, even in high-dimensional settings where the number of features far exceeds the number of samples. Numerical experiments on both synthetic and real data demonstrate that our method consistently outperforms existing clustering algorithms, including those designed for classical Gaussian mixture models.

Figures & tables

Work	Model	Algorithm	Allowed growth of $K$	Exact recovery condition
[ 33 ]	Classical GMM	Spectral clustering	$K$ can grow	$\Delta\gg\max\bigl\{K^{10.5}\sqrt{p/n},\ \sqrt{\log n}\bigr\}$
[ 8 ]	Classical GMM	SDP relaxation of $K$ -means	$K\ll\frac{\log n}{\log\log n}$	$\Delta\gg\Bigl(1+\sqrt{1+\frac{Kp}{n\log n}}\Bigr)^{1/2}\sqrt{\log n}$
[ 38 ]	Classical GMM	Spectral + Lloyd’s (hollowed Gram)	$K=2$	$\Delta\gg\Bigl(1+\sqrt{1+\frac{2p}{n\log n}}\Bigr)^{1/2}\sqrt{\log n}$ (iff)
[ 49 ]	Sub-Gaussian MM	Spectral clustering	$K\ll\sqrt{n}$	$\Delta\gg\max\bigl\{K\sqrt{p/n},\ \sqrt{\log n}\bigr\}$
This paper	Ih-GMM	Spectral clustering (hollowed Gram)	$K\ll\begin{cases}n~\text{if }p\gg n,\\ \frac{n}{\log^{2}n}~\mathrm{othwerwise}\end{cases}$	$\Delta\gg\sqrt{K\log d}\;\max\Bigl\{1,\;(\frac{p}{n})^{\frac{1}{4}}\Bigr\}$

Table 1: Comparison of exact recovery conditions for Gaussian mixture models. Here

K

is the number of clusters,

n

the sample size,

p

the dimension,

\Delta

the minimum center separation, and

d=\max\{n,p\}

. Ih-GMM is short for our individual-heterogeneous sub-Gaussian mixture models.

Panel of Figure 1 — Figure 1: Numerical results of Experiment 1.

Panel of Figure 2 — Figure 2: Numerical results of Experiment 2.

Panel of Figure 3 — Figure 3: Numerical results of Experiment 3.

Panel of Figure 4 — Figure 4: Numerical results of Experiment 4.

Method	Iris	Wine	Seeds	Dermatology	DNA	segment	satimage	usps	pendigits
IhSC	7/150	9/178	20/210	20/358	353/2000	735/2310	1381/4435	2027/7291	1922/7494
PSC	16/150	7/178	16/210	30/358	580/2000	770/2310	1468/4435	2360/7291	2443/7494
k-means	16/150	6/178	17/210	95/358	566/2000	773/2310	1471/4435	2316/7291	2262/7494

Table 2: Misclassification rates of IhSC, k-means, and PSC on the nine real datasets.

Clustering

Clustering groups data points by shared structure, developing methods to discover, validate, and interpret groups across diverse data and constraints.

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Mixed Membership sub-Gaussian Models

Huan Qing

School of Economics and Finance, Chongqing University of Technology, Chongqing, 400054, China

Latent Variable Models Gaussian Mixture Models

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The Gaussian mixture model is widely used in unsupervised learning, owing to its simplicity and interpretability. However, a fundamental limitation of the classical Gaussian mixture model is that it forces each observation to belong to exactly one component. In many practical applications, such as genetics, social network analysis, and text mining, an observation may naturally belong to multiple components or exhibit partial membership in several latent components. To overcome this limitation, we propose the mixed membership sub-Gaussian model, which extends the classical Gaussian mixture framework by allowing each observation to belong to multiple components. This model inherits the interpretability of the classical Gaussian mixture model while offering greater flexibility for capturing complex overlapping structures. We develop an efficient spectral algorithm to estimate the mixed membership of each individual observation, and under mild separation conditions on the component centres, we prove that the estimation error of the per-individual membership vector can be made arbitrarily small with high probability. To our knowledge, this is the first work to provide a computationally efficient estimator with such a vanishing-error guarantee for a mixed-membership extension of the Gaussian mixture model. Extensive experimental studies demonstrate that our method outperforms existing approaches that ignore mixed memberships.

Abstract

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