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24
NonParametric Bayesian Dictionary Learning for Sparse Image Representations
"... Nonparametric Bayesian techniques are considered for learning dictionaries for sparse image representations, with applications in denoising, inpainting and compressive sensing (CS). The beta process is employed as a prior for learning the dictionary, and this nonparametric method naturally infers ..."
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Cited by 92 (34 self)
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Nonparametric Bayesian techniques are considered for learning dictionaries for sparse image representations, with applications in denoising, inpainting and compressive sensing (CS). The beta process is employed as a prior for learning the dictionary, and this nonparametric method naturally infers an appropriate dictionary size. The Dirichlet process and a probit stickbreaking process are also considered to exploit structure within an image. The proposed method can learn a sparse dictionary in situ; training images may be exploited if available, but they are not required. Further, the noise variance need not be known, and can be nonstationary. Another virtue of the proposed method is that sequential inference can be readily employed, thereby allowing scaling to large images. Several example results are presented, using both Gibbs and variational Bayesian inference, with comparisons to other stateoftheart approaches.
Negative Binomial Process Count and Mixture Modeling
, 2013
"... The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization lead ..."
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Cited by 17 (10 self)
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The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process for count modeling. A draw from the NB process consists of a Poisson distributed finite number of distinct atoms, each of which is associated with a logarithmic distributed number of data samples. We reveal relationships between various count and mixturemodeling distributions and construct a Poissonlogarithmic bivariate distribution that connects the NB and Chinese restaurant table distributions. Fundamental properties of the models are developed, and we derive efficient Bayesian inference. It is shown that with augmentation and normalization, the NB process and gammaNB process can be reduced to the Dirichlet process and hierarchical Dirichlet process, respectively. These relationships highlight theoretical, structural and computational advantages of the NB process. A variety of NB processes, including the betageometric, betaNB, markedbetaNB, markedgammaNB and zeroinflatedNB processes, with distinct sharing mechanisms, are also constructed. These models are applied to topic modeling, with connections made to existing algorithms under Poisson factor analysis. Example results show the importance of inferring both the NB dispersion and probability parameters.
AugmentandConquer Negative Binomial Processes
"... By developing data augmentation methods unique to the negative binomial (NB) distribution, we unite seemingly disjoint count and mixture models under the NB process framework. We develop fundamental properties of the models and derive efficient Gibbs sampling inference. We show that the gammaNB pro ..."
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Cited by 10 (7 self)
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By developing data augmentation methods unique to the negative binomial (NB) distribution, we unite seemingly disjoint count and mixture models under the NB process framework. We develop fundamental properties of the models and derive efficient Gibbs sampling inference. We show that the gammaNB process can be reduced to the hierarchical Dirichlet process with normalization, highlighting its unique theoretical, structural and computational advantages. A variety of NB processes with distinct sharing mechanisms are constructed and applied to topic modeling, with connections to existing algorithms, showing the importance of inferring both the NB dispersion and probability parameters. 1
1 Coded Hyperspectral Imaging and Blind Compressive Sensing
"... Blind compressive sensing (CS) is considered for reconstruction of hyperspectral data imaged by a coded aperture camera. The measurements are manifested as a superposition of the coded wavelengthdependent data, with the ambient threedimensional hyperspectral datacube mapped to a twodimensional mea ..."
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Blind compressive sensing (CS) is considered for reconstruction of hyperspectral data imaged by a coded aperture camera. The measurements are manifested as a superposition of the coded wavelengthdependent data, with the ambient threedimensional hyperspectral datacube mapped to a twodimensional measurement. The hyperspectral datacube is recovered using a Bayesian implementation of blind CS. Several demonstration experiments are presented, including measurements performed using a coded aperture snapshot spectral imager (CASSI) camera. The proposed approach is capable of efficiently reconstructing large hyperspectral datacubes. Comparisons are made between the proposed algorithm and other techniques employed in compressive sensing, dictionary learning and matrix factorization. Index Terms hyperspectral images, image reconstruction, projective transformation, dictionary learning, nonparametric Bayesian, BetaBernoulli model, coded aperture snapshot spectral imager (CASSI). I.
Distance Dependent Infinite Latent Feature Models
, 2011
"... Latent feature models are widely used to decompose data into a small number of components. Bayesian nonparametric variants of these models, which use the Indian buffet process (IBP) as a prior over latent features, allow the number of features to be determined from the data. We present a generalizat ..."
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Latent feature models are widely used to decompose data into a small number of components. Bayesian nonparametric variants of these models, which use the Indian buffet process (IBP) as a prior over latent features, allow the number of features to be determined from the data. We present a generalization of the IBP, the distance dependent Indian buffet process (ddIBP), for modeling nonexchangeable data. It relies on a distance function defined between data points, biasing nearby data to share more features. The choice of distance function allows for many kinds of dependencies, including temporal or spatial. Further, the original IBP is a special case of the ddIBP. In this paper, we develop the ddIBP and theoretically characterize the distribution of how features are shared between data. We derive a Markov chain Monte Carlo sampler for a linear Gaussian model with a ddIBP prior and study its performance on several data sets for which exchangeability is not a reasonable assumption.
The Kernel Beta Process
"... A new Lévy process prior is proposed for an uncountable collection of covariatedependent featurelearning measures; the model is called the kernel beta process (KBP). Available covariates are handled efficiently via the kernel construction, with covariates assumed observed with each data sample (“cu ..."
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Cited by 6 (1 self)
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A new Lévy process prior is proposed for an uncountable collection of covariatedependent featurelearning measures; the model is called the kernel beta process (KBP). Available covariates are handled efficiently via the kernel construction, with covariates assumed observed with each data sample (“customer”), and latent covariates learned for each feature (“dish”). Each customer selects dishes from an infinite buffet, in a manner analogous to the beta process, with the added constraint that a customer first decides probabilistically whether to “consider ” a dish, based on the distance in covariate space between the customer and dish. If a customer does consider a particular dish, that dish is then selected probabilistically as in the beta process. The beta process is recovered as a limiting case of the KBP. An efficient Gibbs sampler is developed for computations, and stateoftheart results are presented for image processing and music analysis tasks. 1
A survey of nonexchangeable priors for Bayesian nonparametric models
, 2014
"... Dependent nonparametric processes extend distributions over measures, such as the Dirichlet process and the beta process, to give distributions over collections of measures, typically indexed by values in some covariate space. Such models are appropriate priors when exchangeability assumptions do ..."
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Cited by 3 (0 self)
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Dependent nonparametric processes extend distributions over measures, such as the Dirichlet process and the beta process, to give distributions over collections of measures, typically indexed by values in some covariate space. Such models are appropriate priors when exchangeability assumptions do not hold, and instead we want our model to vary fluidly with some set of covariates. Since the concept of dependent nonparametric processes was formalized by MacEachern [1], there have been a number of models proposed and used in the statistics and machine learning literatures. Many of these models exhibit underlying similarities, an understanding of which, we hope, will help in selecting an appropriate prior, developing new models, and leveraging inference techniques.
A unifying representation for a class of dependent random measures
, 1211
"... We present a general construction for dependent random measures based on thinning Poisson processes on an augmented space. The framework is not restricted to dependent versions of a specific nonparametric model, but can be applied to all models that can be represented using completely random measure ..."
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We present a general construction for dependent random measures based on thinning Poisson processes on an augmented space. The framework is not restricted to dependent versions of a specific nonparametric model, but can be applied to all models that can be represented using completely random measures. Several existing dependent random measures can be seen as specific cases of this framework. Interesting properties of the resulting measures are derived and the efficacy of the framework is demonstrated by constructing a covariatedependent latent feature model and topic model that obtain superior predictive performance. 1
1Hierarchical Infinite Divisibility for Multiscale Shrinkage
"... Abstract—A new shrinkagebased construction is developed for a compressible vector x ∈ Rn, for cases in which the components of x are naturally associated with a tree structure. Important examples are when x corresponds to the coefficients of a wavelet or blockDCT representation of data. The method ..."
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Abstract—A new shrinkagebased construction is developed for a compressible vector x ∈ Rn, for cases in which the components of x are naturally associated with a tree structure. Important examples are when x corresponds to the coefficients of a wavelet or blockDCT representation of data. The method we consider in detail, and for which numerical results are presented, is based on the gamma distribution. The gamma distribution is a heavytailed distribution that is infinitely divisible, and these characteristics are leveraged within the model. We further demonstrate that the general framework is appropriate for many other types of infinitelydivisible heavytailed distributions. Bayesian inference is carried out by approximating the posterior with samples from an MCMC algorithm, as well as by constructing a variational approximation to the posterior. We also consider expectationmaximization (EM) for a MAP (point) solution. Stateoftheart results are manifested for compressive sensing and denoising applications, the latter with spiky (nonGaussian) noise. I.
Nonparametric discovery of activity patterns from video collections
 In CVPR Workshop on Perceptual Organization in Computer Vision
, 2012
"... We propose a nonparametric framework based on the beta process for discovering temporal patterns within a heterogenous video collection. Starting from quantized local motion descriptors, we describe the longrange temporal dynamics of each video via transitions between a set of dynamical behaviors. ..."
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We propose a nonparametric framework based on the beta process for discovering temporal patterns within a heterogenous video collection. Starting from quantized local motion descriptors, we describe the longrange temporal dynamics of each video via transitions between a set of dynamical behaviors. Bayesian nonparametric statistical methods allow the number of such behaviors and the subset exhibited by each video to be learned without supervision. We extend the earlier beta process HMM in two ways: adding datadriven MCMC moves to improve inference on realistic datasets and allowing global sharing of behavior transition parameters. We illustrate discovery of intuitive and useful dynamical structure, at various temporal scales, from videos of simple exercises, recipe preparation, and Olympic sports. Segmentation and retrieval experiments show the benefits of our nonparametric approach. 1.