Tensor Decompositions for Learning Latent Variable Models
CALIFORNIA UNIV IRVINE
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This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models--including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation--which exploits a certain tensor structure in their low-order observable moments typically, of second- and third-order. Specifically, parameter estimation is reduced to the problem of extracting a certain orthogonal decomposition of a symmetric tensor derived from the moments this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices. Although tensor decompositions are generally intractable to compute, the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches, including power iterations and maximization approaches similar to the case of matrices. A detailed analysis of a robust tensor power method is provided, establishing an analogue of Wedins perturbation theorem for the singular vectors of matrices. This implies a robust and computationally tractable estimation approach for several popular latent variable models.
- Theoretical Mathematics
- Statistics and Probability