We consider the mixed regression problem with two components, under adversarial and stochastic noise. We give a convex optimization formulation that provably recovers the true solution, and provide upper bounds on the recovery errors for both arbitrary noise and stochastic noise settings. We also give matching minimax lower bounds (up to log factors), showing that under certain assumptions, our algorithm is information-theoretically optimal. Our results represent the first tractable algorithm guaranteeing successful recovery with tight bounds on recovery errors and sample complexity.
Mixture models and latent variable modeling are very broadly used in a wide array of contexts far beyond regression. Subspace clustering @cite_20 @cite_12 @cite_15 , Gaussian mixture models @cite_3 @cite_24 and @math -means clustering are popular examples of unsupervised learning for mixture models. The most popular and broadly implemented approach to mixture problems, including mixed regression, is the so-called Expectation-Maximization (EM) algorithm @cite_27 @cite_17 . In fact, EM has been used for mixed regression for various application domains @cite_25 @cite_30 . Despite its wide use, still little is known about its performance beyond local convergence @cite_28 @cite_1 .
@cite_30 @cite_28 @cite_1 @cite_3 @cite_24 @cite_27 @cite_15 @cite_20 @cite_25 @cite_12 @cite_17
2015245929 2184753682 2149655761 2150260301
This paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data. The adjective "unsupervised" is justified by two properties of the algorithm: 1) it is capable of selecting the number of components and 2) unlike the standard expectation-maximization (EM) algorithm, it does not require careful initialization. The proposed method also avoids another drawback of EM for mixture fitting: the possibility of convergence toward a singular estimate at the boundary of the parameter space. The novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models; instead, we seamlessly integrate estimation and model selection in a single algorithm. Our technique can be applied to any type of parametric mixture model for which it is possible to write an EM algorithm; in this paper, we illustrate it with experiments involving Gaussian mixtures. These experiments testify for the good performance of our approach. Latent variable models are a fundamental modeling tool in machine learning applications, but they present significant computational and analytical challenges. The popular EM algorithm and its variants, is a much used algorithmic tool; yet our rigorous understanding of its performance is highly incomplete. Recently, work in (2014) has demonstrated that for an important class of problems, EM exhibits linear local convergence. In the high-dimensional setting, however, the M-step may not be well defined. We address precisely this setting through a unified treatment using regularization. While regularization for high-dimensional problems is by now well understood, the iterative EM algorithm requires a careful balancing of making progress towards the solution while identifying the right structure (e.g., sparsity or low-rank). In particular, regularizing the M-step using the state-of-the-art high-dimensional prescriptions (e.g., Wainwright (2014)) is not guaranteed to provide this balance. Our algorithm and analysis are linked in a way that reveals the balance between optimization and statistical errors. We specialize our general framework to sparse gaussian mixture models, high-dimensional mixed regression, and regression with missing variables, obtaining statistical guarantees for each of these examples. Mixture models are a fundamental tool in applied statistics and machine learning for treating data taken from multiple subpopulations. The current practice for estimating the parameters of such models relies on local search heuristics (e.g., the EM algorithm) which are prone to failure, and existing consistent methods are unfavorable due to their high computational and sample complexity which typically scale exponentially with the number of mixture components. This work develops an efficient method of moments approach to parameter estimation for a broad class of high-dimensional mixture models with many components, including multi-view mixtures of Gaussians (such as mixtures of axis-aligned Gaussians) and hidden Markov models. The new method leads to rigorous unsupervised learning results for mixture models that were not achieved by previous works; and, because of its simplicity, it offers a viable alternative to EM for practical deployment. This paper proposes a generative model-based speaker clustering algorithm in the maximum a posteriori adapted Gaussian mixture model (GMM) mean supervector space. The algorithm can be viewed as an extension of the standard expectation maximization algorithm for fitting a mixture model to the data, which iterates between two steps - a sample re-assignment step (E-step) and a model re-estimation step (M-step) - until it converges. The directional scattering patterns of GMM mean supervectors suggest that we employ a mixture of von Mises-Fisher distributions in the model re-estimation step. In the sample re-assignment step, four sample-to-mixture assignment strategies, namely soft, hard, stochastic, and deterministic annealing assignments, are used. Our experiments on the GALE Mandarin dataset show that the use of a mixture of von Mises-Fisher distributions as the underlying model yields significantly higher speaker clustering accuracies than the use of a mixture of Gaussian distributions. It is further shown that deterministic annealing assignment outperforms soft assignment, that soft assignment is comparable to stochastic assignment, and that both soft and stochastic assignments outperform hard assignment.