This paper presents a computationally and statistically efficient method for parameter estimation in a wide class of latent variable models, including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation. The method exploits a certain tensor structure in their low-order observable moments (typically second- and third-order). 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 Wedin's perturbation theorem for the singular vectors of matrices. This implies a robust and computationally tractable estimation approach for several popular latent variable models. The method of moments is a classical parameter estimation technique from statistics which has proved invaluable in a number of application domains. The basic paradigm is simple and intuitive: (i) compute certain statistics of the data—often empirical moments such as means and correlations—and (ii) find model parameters that give rise to (nearly) the same corresponding population quantities. In a number of cases, the method of moments leads to consistent estimators which can be efficiently computed; this is especially relevant in the context of latent variable models, where standard maximum likelihood approaches are typically computationally prohibitive, and heuristic methods can be unreliable and difficult to validate with high-dimensional data. Furthermore, the method of moments can be viewed as complementary to the maximum likelihood approach; simply taking a single step of Newton-Raphson on the likelihood function starting from the moment based estimator (Le Cam, 1986) often leads to the best of both worlds: a computationally efficient estimator that is (asymptotically) statistically optimal. The primary difficulty in learning latent variable models is that the latent (hidden) state of the data is not directly observed; rather only observed variables correlated with the hidden state are observed. As such, it is not evident the method of moments should fare any better than maximum likelihood in terms of computational performance: matching the model parameters to the observed moments may involve solving computationally intractable systems of multivariate polynomial equations. Fortunately, for many classes of latent variable models, there is rich structure in low-order moments (typically second- and third-order) which allow for this inverse moment problem to be solved efficiently (Cattell, 1944; Cardoso, 1991; Chang, 1996; Mossel and Roch, 2006; Hsu et al., 2012b; Anandkumar et al., 2012c,a; Hsu and Kakade, 2013). WhatThis paper presents a computationally and statistically efficient method for parameter estimation in a wide class of latent variable models, including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation. The method exploits a certain tensor structure in their low-order observable moments (typically second- and third-order). 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 Wedin's perturbation theorem for the singular vectors of matrices. This implies a robust and computationally tractable estimation approach for several popular latent variable models. The method of moments is a classical parameter estimation technique from statistics which has proved invaluable in a number of application domains. The basic paradigm is simple and intuitive: (i) compute certain statistics of the data—often empirical moments such as means and correlations—and (ii) find model parameters that give rise to (nearly) the same corresponding population quantities. In a number of cases, the method of moments leads to consistent estimators which can be efficiently computed; this is especially relevant in the context of latent variable models, where standard maximum likelihood approaches are typically computationally prohibitive, and heuristic methods can be unreliable and difficult to validate with high-dimensional data. Furthermore, the method of moments can be viewed as complementary to the maximum likelihood approach; simply taking a single step of Newton-Raphson on the likelihood function starting from the moment based estimator (Le Cam, 1986) often leads to the best of both worlds: a computationally efficient estimator that is (asymptotically) statistically optimal. The primary difficulty in learning latent variable models is that the latent (hidden) state of the data is not directly observed; rather only observed variables correlated with the hidden state are observed. As such, it is not evident the method of moments should fare any better than maximum likelihood in terms of computational performance: matching the model parameters to the observed moments may involve solving computationally intractable systems of multivariate polynomial equations. Fortunately, for many classes of latent variable models, there is rich structure in low-order moments (typically second- and third-order) which allow for this inverse moment problem to be solved efficiently (Cattell, 1944; Cardoso, 1991; Chang, 1996; Mossel and Roch, 2006; Hsu et al., 2012b; Anandkumar et al., 2012c,a; Hsu and Kakade, 2013). What