Clustering a Mixture of Gaussians with Unknown Covariance
This addresses clustering challenges in high-dimensional statistics with unknown covariance, though it is incremental as it builds on existing mixture model frameworks.
The paper tackles clustering data from Gaussian mixtures with unknown, potentially ill-conditioned covariance by developing a Max-Cut integer program that achieves optimal misclassification rates with linear sample complexity, and an efficient spectral algorithm that achieves optimal rates but requires quadratic samples, with evidence suggesting a statistical-computational gap.
We investigate a clustering problem with data from a mixture of Gaussians that share a common but unknown, and potentially ill-conditioned, covariance matrix. We start by considering Gaussian mixtures with two equally-sized components and derive a Max-Cut integer program based on maximum likelihood estimation. We prove its solutions achieve the optimal misclassification rate when the number of samples grows linearly in the dimension, up to a logarithmic factor. However, solving the Max-cut problem appears to be computationally intractable. To overcome this, we develop an efficient spectral algorithm that attains the optimal rate but requires a quadratic sample size. Although this sample complexity is worse than that of the Max-cut problem, we conjecture that no polynomial-time method can perform better. Furthermore, we gather numerical and theoretical evidence that supports the existence of a statistical-computational gap. Finally, we generalize the Max-Cut program to a $k$-means program that handles multi-component mixtures with possibly unequal weights. It enjoys similar optimality guarantees for mixtures of distributions that satisfy a transportation-cost inequality, encompassing Gaussian and strongly log-concave distributions.