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
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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The question of polynomial learn ability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learn ability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learn ability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. Specifically, we show that parameters of a Gaussian mixture distribution with fixed number of components can be learned using a sample whose size is polynomial in dimension and all other parameters. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call “polynomial families” in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The result on learning polynomial families is quite general and is of independent interest. To estimate parameters of a Gaussian mixture distribution in high dimensions, we provide a deterministic algorithm for dimensionality reduction. This allows us to reduce learning a high-dimensional mixture to a polynomial number of parameter estimations in low dimension. Combining this reduction with the results on polynomial families yields our result on learning arbitrary Gaussian mixtures in high dimensions.