We study compressed sensing (CS) signal reconstruction problems where an input signal is measured via matrix multiplication under additive white Gaussian noise. Our signals are assumed to be stationary and ergodic, but the input statistics are unknown; the goal is to provide reconstruction algorithms that are universal to the input statistics. We present a novel algorithm that combines: (i) the approximate message passing (AMP) CS reconstruction framework, which converts the matrix channel recovery problem into scalar channel denoising; (ii) a universal denoising scheme based on context quantization, which partitions the stationary ergodic signal denoising into independent and identically distributed (i.i.d.) subsequence denoising; and (iii) a density estimation approach that approximates the probability distribution of an i.i.d. sequence by fitting a Gaussian mixture (GM) model. In addition to the algorithmic framework, we provide three contributions: (i) numerical results showing that state evolution holds for non-separable Bayesian sliding-window denoisers; (ii) a universal denoiser that does not require the input signal to be bounded; and (iii) we modify the GM learning algorithm, and extend it to an i.i.d. denoiser. Our universal CS recovery algorithm compares favorably with existing reconstruction algorithms in terms of both reconstruction quality and runtime, despite not knowing the input statistics of the stationary ergodic signal.
Fitting Gaussian mixture models: Figueiredo and Jain @cite_3 propose an algorithm that fits a given data sequence with a GM model. The algorithm employs a cost function that resembles the minimum message length (MML) criterion, and the parameters are learned using a modified expectation-maximization (EM) algorithm.
@cite_3
2150260301 2770814335 2094248329 131996945
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. Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, which gives rise to a new algorithm. Specifically, we propose minimizing the sliced-Wasserstein distance between the mixture model and the data distribution with respect to the GMM parameters. In contrast to the KL-divergence, the energy landscape for the sliced-Wasserstein distance is more well-behaved and therefore more suitable for a stochastic gradient descent scheme to obtain the optimal GMM parameters. We show that our formulation results in parameter estimates that are more robust to random initializations and demonstrate that it can estimate high-dimensional data distributions more faithfully than the EM algorithm. 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. This paper describes a probabilistic generative model and its associated algorithm to jointly register multiple point sets. The vast majority of state-of-the-art registration techniques select one of the sets as the “model” and perform pairwise alignments between the other sets and this set. The main drawback of this mode of operation is that there is no guarantee that the model-set is free of noise and outliers, which contaminates the estimation of the registration parameters. Unlike previous work, the proposed method treats all the point sets on an equal footing: they are realizations of a Gaussian mixture (GMM) and the registration is cast into a clustering problem. We formally derive an EM algorithm that estimates both the GMM parameters and the rotations and translations that map each individual set onto the “central” model. The mixture means play the role of the registered set of points while the variances provide rich information about the quality of the registration. We thoroughly validate the proposed method with challenging datasets, we compare it with several state-of-the-art methods, and we show its potential for fusing real depth data.