The EM Algorithm gives Sample-Optimality for Learning Mixtures of Well-Separated Gaussians
We consider the problem of spherical Gaussian Mixture models with $k \geq 3$ components when the components are well separated. A fundamental previous result established that separation of $\Omega(\sqrt{\log k})$ is necessary and sufficient for identifiability of the parameters with polynomial sample complexity (Regev and Vijayaraghavan, 2017). In the same context, we show that $\tilde{O} (kd/\epsilon^2)$ samples suffice for any $\epsilon \lesssim 1/k$, closing the gap from polynomial to linear, and thus giving the first optimal sample upper bound for the parameter estimation of well-separated Gaussian mixtures. We accomplish this by proving a new result for the Expectation-Maximization (EM) algorithm: we show that EM converges locally, under separation $\Omega(\sqrt{\log k})$. The previous best-known guarantee required $\Omega(\sqrt{k})$ separation (Yan, et al., 2017). Unlike prior work, our results do not assume or use prior knowledge of the (potentially different) mixing weights or variances of the Gaussian components. Furthermore, our results show that the finite-sample error of EM does not depend on non-universal quantities such as pairwise distances between means of Gaussian components.
Code (0)
등록된 구현이 없습니다.
Tasks
parameter estimationSimilar Papers 제목 키워드 기반
Clustering a Mixture of Gaussians with Unknown Covariance
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-si…
ClusteringDimension Reduction via Sum-of-Squares and Improved Clustering Algorithms for Non-Spherical Mixtures
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-preserv…
ClusteringDimensionality ReductionSource Identification for Mixtures of Product Distributions
We give an algorithm for source identification of a mixture of $k$ product distributions on $n$ bits. This is a fundamental problem in machine learning with many applications. Our algorithm identifies the source paramete…
Tensor Belief Propagation
We propose a new approximate inference algorithm for graphical models, tensor belief propagation, based on approximating the messages passed in the junction tree algorithm. Our algorithm represents the potential fun…
Efficient Density Estimation via Piecewise Polynomial Approximation
We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an in…
Density Estimation