This paper presents the recurrent estimation of distributions (RED) for modeling real-valued data in a semiparametric fashion. RED models make two novel uses of recurrent neural networks (RNNs) for density estimation of general real-valued data. First, RNNs are used to transform input covariates into a latent space to better capture conditional dependencies in inputs. After, an RNN is used to compute the conditional distributions of the latent covariates. The resulting model is efficient to train, compute, and sample from, whilst producing normalized pdfs. The effectiveness of RED is shown via several real-world data experiments. Our results show that RED models achieve a lower held-out negative log-likelihood than other neural network approaches across multiple dataset sizes and dimensionalities. Further context of the efficacy of RED is provided by considering anomaly detection tasks, where we also observe better performance over alternative models.
NICE models assume that data is drawn from a latent independent Gaussian space and transformed @cite_19 . The transformation uses several additive coupling'' shifting transformations on the second half of dimensions, using the first half of dimensions. The full transformation is the result of stacking several of these additive coupling layers together followed by a final rescaling operation. The resulting composition of these transformation yields a simple determinant of the Jacobian.
@cite_19
1583912456 2096198554 1907927509 2094248329
We propose a deep learning framework for modeling complex high-dimensional densities called Non-linear Independent Component Estimation (NICE). It is based on the idea that a good representation is one in which the data has a distribution that is easy to model. For this purpose, a non-linear deterministic transformation of the data is learned that maps it to a latent space so as to make the transformed data conform to a factorized distribution, i.e., resulting in independent latent variables. We parametrize this transformation so that computing the Jacobian determinant and inverse transform is trivial, yet we maintain the ability to learn complex non-linear transformations, via a composition of simple building blocks, each based on a deep neural network. The training criterion is simply the exact log-likelihood, which is tractable. Unbiased ancestral sampling is also easy. We show that this approach yields good generative models on four image datasets and can be used for inpainting. In previous work on "transformed mixtures of Gaussians" and "transformed hidden Markov models", we showed how the EM algorithm in a discrete latent variable model can be used to jointly normalize data (e.g., center images, pitch-normalize spectrograms) and learn a mixture model of the normalized data. The only input to the algorithm is the data, a list of possible transformations, and the number of clusters to find. The main criticism of this work was that the exhaustive computation of the posterior probabilities over transformations would make scaling up to large feature vectors and large sets of transformations intractable. Here, we describe how a tremendous speed-up is acheived through the use of a variational technique for decoupling transformations, and a fast Fourier transform method for computing posterior probabilities. For N × N images, learning C clusters under N rotations, N scales, N x-translations and N y-translations takes only (C + 2 log N)N2 scalar operations per iteration. In contrast, the original algorithm takes CN6 operations to account for these transformations. We give results on learning a 4-component mixture model from a video sequence with frames of size 320×240. The model accounts for 360 rotations and 76,800 translations. Each iteration of EM takes only 10 seconds per frame in MATLAB, which is over 5 million times faster than the original algorithm. In recent years analysis of complexity of learning Gaussian mixture models from sampled data has received significant attention in computational machine learning and theory communities. In this paper we present the first result showing that polynomial time learning of multidimensional Gaussian Mixture distributions is possible when the separation between the component means is arbitrarily small. Specifically, we present an algorithm for learning the parameters of a mixture of k identical spherical Gaussians in n-dimensional space with an arbitrarily small separation between the components, which is polynomial in dimension, inverse component separation and other input parameters for a fixed number of components k. The algorithm uses a projection to k dimensions and then a reduction to the 1-dimensional case. It relies on a theoretical analysis showing that two 1-dimensional mixtures whose densities are close in the L norm must have similar means and mixing coefficients. To produce the necessary lower bound for the L norm in terms of the distances between the corresponding means, we analyze the behavior of the Fourier transform of a mixture of Gaussians in one dimension around the origin, which turns out to be closely related to the properties of the Vandermonde matrix obtained from the component means. Analysis of minors of the Vandermonde matrix together with basic function approximation results allows us to provide a lower bound for the norm of the mixture in the Fourier domain and hence a bound in the original space. Additionally, we present a separate argument for reconstructing variance. A hierarchical Bayes Gaussian Mixture Model (HBGMM) with adaptive sparsity is proposed.We impose our hierarchical Bayes prior on the entries of precision matrices in GMM to encourage sparsity.Our adaptive sparsity can reduce the dimensionality of the parameters in GMM to be estimated.Our HBGMM method does not involve any hyperparameters to be estimated, i.e., it is hyperparameter-free. Gaussian Mixture Model (GMM) has been widely used in statistics for its great flexibility. However, parameter estimation for GMM with high dimensionality is a challenge because of the large number of parameters and the lack of observation data. In this paper, we propose an effective method named hierarchical Bayes based Adaptive Sparsity in Gaussian Mixture Model (ASGMM) to estimate the parameters in a GMM by incorporating a two-layer hierarchical Bayes based adaptive sparsity prior. The prior we impose on the precision matrices can encourage sparsity and hence reduce the dimensionality of the parameters to be estimated. In contrast to the l 1 -norm penalty or Laplace prior, our approach does not involve any hyperparameters that must be tuned, and the sparsity adapts to the observation data. The proposed method is achieved by three steps: first, we formulate an adaptive hierarchical Bayes model of the precision matrices in the GMM with a Jeffrey's noninformative hyperprior, which expresses scale-invariance and, more importantly, is hyperparameter-free and unbiased. Second, we perform a Cholesky decomposition on the precision matrices to impose the positive definite property. Finally, we exploit the expectation maximization (EM) algorithm to obtain the final estimated parameters in the GMM. Experimental results on synthetic and real-world datasets demonstrate that ASGMM cannot only adapt the sparsity of high-dimensional data with small estimated error, but also achieve better clustering performance comparing with several classical methods.