# Dimension Reduction via Sum-of-Squares and Improved Clustering Algorithms for Non-Spherical Mixtures

arXiv: https://arxiv.org/abs/2411.12438 · HTML: https://mlsanity.com/paper/2411.12438/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Nov 2024.

## Built on by (most cited first, 1 of 1)

- [Learning Mixture Models via Efficient High-dimensional Sparse Fourier Transforms](https://mlsanity.com/paper/2601.05157/) (2026)

## Abstract (arXiv)

We develop a new approach for clustering non-spherical (i.e., arbitrary component covariances) Gaussian mixture models via a subroutine, based on the sum-of-squares method, that finds a low-dimensional separation-preserving projection of the input data. Our method gives a non-spherical analog of the classical dimension reduction, based on singular value decomposition, that, among several other applications, forms a key component of the celebrated spherical clustering algorithm of Vempala and Wang [VW04]. As applications, we obtain an algorithm to (1) cluster an arbitrary total-variation separated mixture of k centered (i.e., zero-mean) Gaussians with n≥ operatornamepoly(d) f(wmin-1) samples and operatornamepoly(n) time, and (2) cluster an arbitrary total-variation separated mixture of k Gaussians with identical but arbitrary unknown covariance with n ≥ d^O(log wmin-1) f(wmin-1) samples and n^O(log wmin-1) time. Here, wmin is the minimum mixing weight of the input mixture, and f does not depend on the dimension d. Our algorithms naturally extend to tolerating a dimension-independent fraction of arbitrary outliers. Before this work, the techniques in the state-of-the-art non-spherical clustering algorithms needed dO(k) f(wmin-1) samples and time for clustering such mixtures. Our results may come as a surprise in the context of the dOmega(k) statistical query and sum-of-squares lower bounds [DKS17, DKPP24] for clustering non-spherical Gaussian mixtures. While these results are usually thought to rule out do(k) cost algorithms for the problem, our results show that the lower bounds can in fact be circumvented for a remarkably general class of Gaussian mixtures.