A mixture of factor analyzers is a semi-parametric density estimator that generalizes the well-known mixtures of Gaussians model by allowing each Gaussian in the mixture to be represented in a dierent lower-dimensional manifold. This paper presents a robust and parsimonious model selection algorithm for training a mixture of factor analyzers, carrying out simultaneous clustering and locally linear, globally nonlinear dimensionality reduction. Permitting dierent number of factors per mixture component, the algorithm adapts the model complexity to the data complexity. We compare the proposed algorithm with related automatic model selection algorithms on a number of benchmarks. The results indicate the eectiveness of this fast
Using the parsimonious factor analysis representation described in , it is possible to explore many models that are between full-covariance and diagonal Gaussian mixtures in their number of parameters. The resulting mixture of factor analysers (MoFA) can be considered as a noise-robust version of the mixtures of probabilistic principal component analysers (PPCA) approach @cite_16 . Figure summarizes the relations between the mixture representations in this area.
@cite_16
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In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. Gaussian mixture models are currently the dominant technique for modeling the emission distribution of hidden Markov models for speech recognition. We show that better phone recognition on the TIMIT dataset can be achieved by replacing Gaussian mixture models by deep neural networks that contain many layers of features and a very large number of parameters. These networks are first pre-trained as a multi-layer generative model of a window of spectral feature vectors without making use of any discriminative information. Once the generative pre-training has designed the features, we perform discriminative fine-tuning using backpropagation to adjust the features slightly to make them better at predicting a probability distribution over the states of monophone hidden Markov models. We address the problem of blind audio source separation in the under-determined and convolutive case. The contribution of each source to the mixture channels in the time-frequency domain is modeled by a zero-mean Gaussian random vector with a full rank covariance matrix composed of two terms: a variance which represents the spectral properties of the source and which is modeled by a nonnegative matrix factorization (NMF) model and another full rank covariance matrix which encodes the spatial properties of the source contribution in the mixture. We address the estimation of these parameters by maximizing the likelihood of the mixture using an expectation-maximization (EM) algorithm. Theoretical propositions are corroborated by experimental studies on stereo reverberant music mixtures. We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when a significant amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis. Our general analysis subsumes several existing ad-hoc convergence bounds and resolves a number of open problems in multiple applications: Streaming PCA. A recent work of (NIPS 2013) gives a space-efficient algorithm for PCA in a streaming model where samples are drawn from a gaussian spiked covariance model. We give a simpler and more general analysis that applies to arbitrary distributions confirming experimental evidence of Moreover, even in the spiked covariance model our result gives quantitative improvements in a natural parameter regime. It is also notably simpler and follows easily from our general convergence analysis of the noisy power method together with a matrix Chernoff bound. Private PCA. We provide the first nearly-linear time algorithm for the problem of differentially private principal component analysis that achieves nearly tight worst-case error bounds. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by an essentially tight dependence on the coherence of the matrix. This result resolves the main problem left open by Hardt and Roth (STOC 2013). The coherence is always bounded by the matrix dimension but often substantially smaller thus leading to strong average-case improvements over the optimal worst-case bound.