Modeling high-dimensional data: technical perspective
Communications of the ACM
Learning poisson binomial distributions
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Pathset graphs: a novel approach for comprehensive utilization of paired reads in genome assembly
RECOMB'12 Proceedings of the 16th Annual international conference on Research in Computational Molecular Biology
Effective principal component analysis
SISAP'12 Proceedings of the 5th international conference on Similarity Search and Applications
Learning mixtures of spherical gaussians: moment methods and spectral decompositions
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Clustering under approximation stability
Journal of the ACM (JACM)
Learning mixtures of arbitrary distributions over large discrete domains
Proceedings of the 5th conference on Innovations in theoretical computer science
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Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We give an algorithm for this problem that has running time and data requirements polynomial in the dimension and the inverse of the desired accuracy, with provably minimal assumptions on the Gaussians. As a simple consequence of our learning algorithm, we we give the first polynomial time algorithm for proper density estimation for mixtures of k Gaussians that needs no assumptions on the mixture. It was open whether proper density estimation was even statistically possible (with no assumptions) given only polynomially many samples, let alone whether it could be computationally efficient. The building blocks of our algorithm are based on the work (Kalai \emph{et al}, STOC 2010) that gives an efficient algorithm for learning mixtures of two Gaussians by considering a series of projections down to one dimension, and applying the method of moments to each univariate projection. A major technical hurdle in the previous work is showing that one can efficiently learn univariate mixtures of two Gaussians. In contrast, because pathological scenarios can arise when considering projections of mixtures of more than two Gaussians, the bulk of the work in this paper concerns how to leverage a weaker algorithm for learning univariate mixtures (of many Gaussians) to learn in high dimensions. Our algorithm employs hierarchical clustering and rescaling, together with methods for backtracking and recovering from the failures that can occur in our univariate algorithm. Finally, while the running time and data requirements of our algorithm depend exponentially on the number of Gaussians in the mixture, we prove that such a dependence is necessary.