One-Bit Clustering for Two Component Sub-Gaussian Mixture Models

2026-06-20

AI Summary

This work addresses the challenging problem of clustering a two-component sub-Gaussian mixture under extreme one-bit quantization per dimension per sample. It proposes a novel approach combining dithered quantization, Haar-distributed random rotation, and a refined Lloyd algorithm, which automatically satisfies required theoretical conditions under mild non-spikiness assumptions. The method achieves provably near-optimal performance: its separation requirement exceeds the information-theoretic threshold for the unquantized setting only by a logarithmic factor, and it yields an exponentially decaying misclustering rate that permits exact recovery with high probability. Furthermore, the authors establish a minimax lower bound demonstrating that the proposed procedure is nearly optimal in a fundamental sense. Numerical experiments confirm its efficiency and robustness in high-dimensional regimes.

Abstract

Clustering is a fundamental problem in statistics and machine learning. We propose the first one-bit clustering method for two-component sub-Gaussian mixture models. The method uses only one bit per entry of each sample obtained via a dithered quantizer. Under a mild non-spikiness condition on the cluster centers, we show that a variant of Lloyd's algorithm achieves a misclassification rate that decays exponentially with a signal-to-noise ratio comparable to that in the unquantized setting. This result further implies exact recovery under an explicit separation condition, which exceeds the optimal threshold for unquantized data by only a logarithmic factor. When the dimension $p$ is sufficiently large, the non-spikiness condition can be enforced by applying a random rotation using a Haar distributed matrix prior to quantization. In particular, it holds with high probability when $p \gtrsim 1$ for partial recovery and $p \gtrsim \log n \log\log n$ for exact recovery, where $n$ is the sample size. We also establish a minimax lower bound, showing that the misclassification rate and separation condition exhibit sharp constants in general. Numerical results are provided to corroborate the theory and demonstrate the efficacy of the proposed method.

Problem

• one-bit clustering
• sub-Gaussian mixture models
• quantization
• misclassification rate
• exact recovery

Innovation

• one-bit clustering
• sub-Gaussian mixture models
• dithered quantization
• Lloyd's algorithm
• minimax lower bound