We present a new estimation principle for parameterized statistical models. The idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise, using the model log-density function in the regression nonlinearity. We show that this leads to a consistent (convergent) estimator of the parameters, and analyze the asymptotic variance. In particular, the method is shown to directly work for unnormalized models, i.e. models where the density function does not integrate to one. The normalization constant can be estimated just like any other parameter. For a tractable ICA model, we compare the method with other estimation methods that can be used to learn unnormalized models, including score matching, contrastive divergence, and maximum-likelihood where the normalization constant is estimated with importance sampling. Simulations show that noise-contrastive estimation offers the best trade-off between computational and statistical efficiency. The method is then applied to the modeling of natural images: We show that the method can successfully estimate a large-scale two-layer model and a Markov random field.

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Noise-contrastive estimation: A new estimation principle for unnormalized statistical models

Learning good image priors is of utmost importance for the study of vision, computer vision and image processing applications. Learning priors and optimizing over whole images can lead to tremendous computational challenges. In contrast, when we work with small image patches, it is possible to learn priors and perform patch restoration very efficiently. This raises three questions - do priors that give high likelihood to the data also lead to good performance in restoration? Can we use such patch based priors to restore a full image? Can we learn better patch priors? In this work we answer these questions. We compare the likelihood of several patch models and show that priors that give high likelihood to data perform better in patch restoration. Motivated by this result, we propose a generic framework which allows for whole image restoration using any patch based prior for which a MAP (or approximate MAP) estimate can be calculated. We show how to derive an appropriate cost function, how to optimize it and how to use it to restore whole images. Finally, we present a generic, surprisingly simple Gaussian Mixture prior, learned from a set of natural images. When used with the proposed framework, this Gaussian Mixture Model outperforms all other generic prior methods for image denoising, deblurring and inpainting.

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From learning models of natural image patches to whole image restoration

Machine learning methods extract value from vast data sets quickly and with modest resources They are established tools in a wide range of industrial applications, including search engines, DNA sequencing, stock market analysis, and robot locomotion, and their use is spreading rapidly People who know the methods have their choice of rewarding jobs This hands-on text opens these opportunities to computer science students with modest mathematical backgrounds It is designed for final-year undergraduates and master's students with limited background in linear algebra and calculus Comprehensive and coherent, it develops everything from basic reasoning to advanced techniques within the framework of graphical models Students learn more than a menu of techniques, they develop analytical and problem-solving skills that equip them for the real world Numerous examples and exercises, both computer based and theoretical, are included in every chapter Resources for students and instructors, including a MATLAB toolbox, are available online

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Bayesian Reasoning and Machine Learning

We propose a definition of saliency by considering what the visual system is trying to optimize when directing attention. The resulting model is a Bayesian framework from which bottom-up saliency emerges naturally as the self-information of visual features, and overall saliency (incorporating top-down information with bottom-up saliency) emerges as the pointwise mutual information between the features and the target when searching for a target. An implementation of our framework demonstrates that our model’s bottom-up saliency maps perform as well as or better than existing algorithms in predicting people’s fixations in free viewing. Unlike existing saliency measures, which depend on the statistics of the particular image being viewed, our measure of saliency is derived from natural image statistics, obtained in advance from a collection of natural images. For this reason, we call our model SUN (Saliency Using Natural statistics). A measure of saliency based on natural image statistics, rather than based on a single test image, provides a straightforward explanation for many search asymmetries observed in humans; the statistics of a single test image lead to predictions that are not consistent with these asymmetries. In our model, saliency is computed locally, which is consistent with the neuroanatomy of the early visual system and results in an efficient algorithm with few free parameters.

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SUN: A Bayesian framework for saliency using natural statistics.

Motivated in part by the hierarchical organization of the cortex, a number of algorithms have recently been proposed that try to learn hierarchical, or "deep," structure from unlabeled data. While several authors have formally or informally compared their algorithms to computations performed in visual area V1 (and the cochlea), little attempt has been made thus far to evaluate these algorithms in terms of their fidelity for mimicking computations at deeper levels in the cortical hierarchy. This paper presents an unsupervised learning model that faithfully mimics certain properties of visual area V2. Specifically, we develop a sparse variant of the deep belief networks of Hinton et al. (2006). We learn two layers of nodes in the network, and demonstrate that the first layer, similar to prior work on sparse coding and ICA, results in localized, oriented, edge filters, similar to the Gabor functions known to model V1 cell receptive fields. Further, the second layer in our model encodes correlations of the first layer responses in the data. Specifically, it picks up both colinear ("contour") features as well as corners and junctions. More interestingly, in a quantitative comparison, the encoding of these more complex "corner" features matches well with the results from the Ito & Komatsu's study of biological V2 responses. This suggests that our sparse variant of deep belief networks holds promise for modeling more higher-order features.

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Sparse deep belief net model for visual area V2

A fundamental function of the visual system is to encode the building blocks of natural scenes-edges, textures and shapes-that subserve visual tasks such as object recognition and scene understanding. Essential to this process is the formation of abstract representations that generalize from specific instances of visual input. A common view holds that neurons in the early visual system signal conjunctions of image features, but how these produce invariant representations is poorly understood. Here we propose that to generalize over similar images, higher-level visual neurons encode statistical variations that characterize local image regions. We present a model in which neural activity encodes the probability distribution most consistent with a given image. Trained on natural images, the model generalizes by learning a compact set of dictionary elements for image distributions typically encountered in natural scenes. Model neurons show a diverse range of properties observed in cortical cells. These results provide a new functional explanation for nonlinear effects in complex cells and offer insight into coding strategies in primary visual cortex (V1) and higher visual areas.

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Emergence of complex cell properties by learning to generalize in natural scenes

Capturing statistical regularities in complex, high-dimensional data is an important problem in machine learning and signal processing. Models such as principal component analysis (PCA) and independent component analysis (ICA) make few assumptions about the structure in the data and have good scaling properties, but they are limited to representing linear statistical regularities and assume that the distribution of the data is stationary. For many natural, complex signals, the latent variables often exhibit residual dependencies as well as nonstationary statistics. Here we present a hierarchical Bayesian model that is able to capture higher-order nonlinear structure and represent nonstationary data distributions. The model is a generalization of ICA in which the basis function coefficients are no longer assumed to be independent; instead, the dependencies in their magnitudes are captured by a set of density components. Each density component describes a common pattern of deviation from the marginal density of the pattern ensemble; in different combinations, they can describe nonstationary distributions. Adapting the model to image or audio data yields a nonlinear, distributed code for higher-order statistical regularities that reflect more abstract, invariant properties of the signal.

A Hierarchical Bayesian Model for Learning Nonlinear Statistical Regularities in Nonstationary Natural Signals

The theoretical principles that underlie the representation and computation of higher-order structure in natural images are poorly understood. Recently, there has been considerable interest in using information theoretic techniques, such as independent component analysis, to derive representations for natural images that are optimal in the sense of coding efficiency. Although these approaches have been successful in explaining properties of neural representations in the early visual pathway and visual cortex, because they are based on a linear model, the types of image structure that can be represented are very limited. Here, we present a hierarchical probabilistic model for learning higher-order statistical regularities in natural images. This non-linear model learns an efficient code that describes variations in the underlying probabilistic density. When applied to natural images the algorithm yields coarse-coded, sparse-distributed representations of abstract image properties such as object location, scale and texture. This model offers a novel description of higher-order image structure and could provide theoretical insight into the response properties and computational functions of lower level cortical visual areas.

Learning higher-order structures in natural images.

Efficient coding provides a powerful principle for explaining early sensory coding. Most attempts to test this principle have been limited to linear, noiseless models, and when applied to natural images, have yielded oriented filters consistent with responses in primary visual cortex. Here we show that an efficient coding model that incorporates biologically realistic ingredients – input and output noise, nonlinear response functions, and a metabolic cost on the firing rate – predicts receptive fields and response nonlinearities similar to those observed in the retina. Specifically, we develop numerical methods for simultaneously learning the linear filters and response nonlinearities of a population of model neurons, so as to maximize information transmission subject to metabolic costs. When applied to an ensemble of natural images, the method yields filters that are center-surround and nonlinearities that are rectifying. The filters are organized into two populations, with On- and Off-centers, which independently tile the visual space. As observed in the primate retina, the Off-center neurons are more numerous and have filters with smaller spatial extent. In the absence of noise, our method reduces to a generalized version of independent components analysis, with an adapted nonlinear "contrast" function; in this case, the optimal filters are localized and oriented.

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Efficient coding of natural images with a population of noisy Linear-Nonlinear neurons

September 22, 2004 Proceedings Trim Size: 9in x 6in psb2004 CLASSIFICATION OF NON-CODING RNA USING GRAPH REPRESENTATIONS OF SECONDARY STRUCTURE YAN KARKLIN Department of Computer Science Carnegie Melon University Pittsburgh, PA USA E-mail: yan+@cs.cmu.edu RICHARD F. MERAZ ∗ AND STEPHEN R. HOLBROOK Physical Biosciences Division Lawrence Berkeley National Laboratory Berkeley, CA USA E-mail: rfmeraz@gmail.com, srholbrook@lbl.gov Some genes produce transcripts that function directly in regulatory, catalytic, or structural roles in the cell. These non-coding RNAs are prevalent in all living organisms, and methods that aid the understanding of their functional roles are essential. RNA secondary structure, the pattern of base-pairing, contains the crit- ical information for determining the three dimensional structure and function of the molecule. In this work we examine whether the basic geometric and topolog- ical properties of secondary structure are suﬃcient to distinguish between RNA families in a learning framework. First, we develop a labeled dual graph represen- tation of RNA secondary structure by adding biologically meaningful labels to the dual graphs proposed by Gan et al [1]. Next, we deﬁne a similarity measure di- rectly on the labeled dual graphs using the recently developed marginalized kernels [2]. Using this similarity measure, we were able to train Support Vector Machine classiﬁers to distinguish RNAs of known families from random RNAs with similar statistics. For 22 of the 25 families tested, the classiﬁer achieved better than 70% accuracy, with much higher accuracy rates for some families. Training a set of clas- siﬁers to automatically assign family labels to RNAs using a one vs. all multi-class scheme also yielded encouraging results. From these initial learning experiments, we suggest that the labeled dual graph representation, together with kernel machine methods, has potential for use in automated analysis and classiﬁcation of unchar- acterized RNA molecules or eﬃcient genome-wide screens for RNA molecules from existing families. ∗ to whom correspondence should be addressed.

Yan Karklin

Papers

Emergence of complex cell properties by learning to generalize in natural scenes

A Hierarchical Bayesian Model for Learning Nonlinear Statistical Regularities in Nonstationary Natural Signals

Learning higher-order structures in natural images.

Efficient coding of natural images with a population of noisy Linear-Nonlinear neurons

Classification of non-coding RNA using graph representations of secondary structure