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arXiv · 1306.5825

Fourier PCA and Robust Tensor Decomposition

Navin Goyal·Santosh Vempala·Ying Xiao
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Abstract

Fourier PCA is Principal Component Analysis of a matrix obtained from higher order derivatives of the logarithm of the Fourier transform of a distribution.We make this method algorithmic by developing a tensor decomposition method for a pair of tensors sharing the same vectors in rank-$1$ decompositions. Our main application is the first provably polynomial-time algorithm for underdetermined ICA, i.e., learning an $n \times m$ matrix $A$ from observations $y=Ax$ where $x$ is drawn from an unknown product distribution with arbitrary non-Gaussian components. The number of component distributions $m$ can be arbitrarily higher than the dimension $n$ and the columns of $A$ only need to satisfy a natural and efficiently verifiable nondegeneracy condition. As a second application, we give an alternative algorithm for learning mixtures of spherical Gaussians with linearly independent means. These results also hold in the presence of Gaussian noise.

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Navin Goyal, Santosh Vempala, Ying Xiao. 2013-06-25. Fourier PCA and Robust Tensor Decomposition. https://arxiv.org/abs/1306.5825

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