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Spectral Clustering with Likelihood Refinement for High-dimensional Latent Class Recovery

Zhongyuan Lyu, Yuqi Gu

TL;DR

This work tackles latent class recovery for high-dimensional binary-response data by introducing SOLA, a two-stage method that first uses spectral clustering to obtain an initial partition and then refines the labels with a one-step likelihood update. The authors establish that SOLA achieves minimax-optimal mis-clustering rates and, under mild conditions, exact recovery, while also providing a consistent estimator for the number of latent classes $K$. Through simulations and a real Senate voting dataset, SOLA demonstrates strong accuracy, stability, and computational efficiency, often outperforming EM-based and tensor-based alternatives, especially when $J$ is large. Overall, SOLA offers a scalable, statistically near-optimal solution for latent class recovery in modern high-dimensional settings, with potential extensions to polytomous responses and statistical inference for item parameters.

Abstract

Latent class models are widely used for identifying unobserved subgroups from multivariate categorical data in social sciences, with binary data as a particularly popular example. However, accurately recovering individual latent class memberships remains challenging, especially when handling high-dimensional datasets with many items. This work proposes a novel two-stage algorithm for latent class models suited for high-dimensional binary responses. Our method first initializes latent class assignments by an easy-to-implement spectral clustering algorithm, and then refines these assignments with a one-step likelihood-based update. This approach combines the computational efficiency of spectral clustering with the improved statistical accuracy of likelihood-based estimation. We establish theoretical guarantees showing that this method is minimax-optimal for latent class recovery in the statistical decision theory sense. The method also leads to exact clustering of subjects with high probability under mild conditions. As a byproduct, we propose a computationally efficient consistent estimator for the number of latent classes. Extensive experiments on both simulated data and real data validate our theoretical results and demonstrate our method's superior performance over alternative methods.

Spectral Clustering with Likelihood Refinement for High-dimensional Latent Class Recovery

TL;DR

This work tackles latent class recovery for high-dimensional binary-response data by introducing SOLA, a two-stage method that first uses spectral clustering to obtain an initial partition and then refines the labels with a one-step likelihood update. The authors establish that SOLA achieves minimax-optimal mis-clustering rates and, under mild conditions, exact recovery, while also providing a consistent estimator for the number of latent classes . Through simulations and a real Senate voting dataset, SOLA demonstrates strong accuracy, stability, and computational efficiency, often outperforming EM-based and tensor-based alternatives, especially when is large. Overall, SOLA offers a scalable, statistically near-optimal solution for latent class recovery in modern high-dimensional settings, with potential extensions to polytomous responses and statistical inference for item parameters.

Abstract

Latent class models are widely used for identifying unobserved subgroups from multivariate categorical data in social sciences, with binary data as a particularly popular example. However, accurately recovering individual latent class memberships remains challenging, especially when handling high-dimensional datasets with many items. This work proposes a novel two-stage algorithm for latent class models suited for high-dimensional binary responses. Our method first initializes latent class assignments by an easy-to-implement spectral clustering algorithm, and then refines these assignments with a one-step likelihood-based update. This approach combines the computational efficiency of spectral clustering with the improved statistical accuracy of likelihood-based estimation. We establish theoretical guarantees showing that this method is minimax-optimal for latent class recovery in the statistical decision theory sense. The method also leads to exact clustering of subjects with high probability under mild conditions. As a byproduct, we propose a computationally efficient consistent estimator for the number of latent classes. Extensive experiments on both simulated data and real data validate our theoretical results and demonstrate our method's superior performance over alternative methods.

Paper Structure

This paper contains 40 sections, 10 theorems, 111 equations, 12 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation.
  3. Latent Class Model with Binary Responses
  4. Two-stage Clustering Algorithm
  5. Spectral clustering for the latent class model
  6. One-step Likelihood Refinement with Spectral Initialization
  7. Extension: Classification EM refinement
  8. Theoretical Guarantees
  9. Theoretical Guarantee of Spectral Clustering
  10. Fundamental Statistical Limit for LCMs
  11. Theoretical Guarantee of SOLA
  12. Comparison of Spectral Clustering and SOLA
  13. Estimation of the Number of Latent Classes $K$
  14. Numerical Experiments
  15. Simulation Studies
  16. ...and 25 more sections

Key Result

Proposition 1

Define $\sigma^2_{\theta}:=\left(1-2\theta\right)/\left[2\log\left(\left(1-\theta\right)\theta^{-1}\right)\right]$ for $\theta\in(0,1)\backslash\left\{1/2\right\}$ and $\sigma^2_{\theta}:=1/4$ for $\theta=1/2$, and let $\bar{\sigma}:=\max_{j,k}\sigma_{\theta_{j,k}}$. Assume $\text{rank}\left(\mathbf Then for the spectral clustering estimator $\widetilde{{\boldsymbol s}}$ in alg:Spec we have where

Figures (12)

  • Figure 1: An illustration for spectral clustering: row vectors of $\mathbf{U}\mathbf{\Sigma}$ (left) and $\widehat{\mathbf{U}}\widehat{\mathbf{\Sigma}}$ (right). Setting: $N=500$, $J=250$, $K=3$.
  • Figure 2: Simulation 1: Mis-clustering proportions v.s. number of items $J$. Entries of $\mathbf{\Theta}$ are independently generated from $\text{Beta}\left(5,5\right)$.
  • Figure 3: Simulation 2: Mis-clustering proportions v.s. number of items $J$. Entries of $\mathbf{\Theta}$ are independently generated from $\text{Beta}\left(1,8\right)$.
  • Figure 4: Simulation 3: Failure rate versus the number of items $J$. Entries of $\mathbf{\Theta}$ are independently generated from $\text{Beta}\left(1,8\right)$.
  • Figure 5: Simulation 4: Running time (seconds) of different methods.
  • ...and 7 more figures

Theorems & Definitions (13)

  • Remark 1
  • Proposition 1: Adapted from Theorem 3.1 in zhang2022leave
  • Proposition 2
  • Theorem 1: Minimax Lower Bound
  • Theorem 2
  • Remark 2
  • Corollary 1
  • Proposition 3
  • Lemma 1
  • Remark 3: EM implementations: polca vs. LatentGOLD
  • ...and 3 more