Efficient Clustering for Stretched Mixtures: Landscape and Optimality

Kaizheng Wang; Yuling Yan; Mateo Díaz

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arXiv · 2003.09960

Efficient Clustering for Stretched Mixtures: Landscape and Optimality

Kaizheng Wang·Yuling Yan·Mateo Díaz

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Abstract

This paper considers a canonical clustering problem where one receives unlabeled samples drawn from a balanced mixture of two elliptical distributions and aims for a classifier to estimate the labels. Many popular methods including PCA and k-means require individual components of the mixture to be somewhat spherical, and perform poorly when they are stretched. To overcome this issue, we propose a non-convex program seeking for an affine transform to turn the data into a one-dimensional point cloud concentrating around $-1$ and $1$, after which clustering becomes easy. Our theoretical contributions are two-fold: (1) we show that the non-convex loss function exhibits desirable geometric properties when the sample size exceeds some constant multiple of the dimension, and (2) we leverage this to prove that an efficient first-order algorithm achieves near-optimal statistical precision without good initialization. We also propose a general methodology for clustering with flexible choices of feature transforms and loss objectives.

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BibTeX RIS

Kaizheng Wang, Yuling Yan, Mateo Díaz. 2020-03-22. Efficient Clustering for Stretched Mixtures: Landscape and Optimality. https://arxiv.org/abs/2003.09960

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