Statistical models of natural stimuli provide an important tool for researchers in the fields of machine learning and computational neuroscience. A canonical way to quantitatively assess and compare the performance of statistical models is given by the likelihood. One class of statistical models which has recently gained increasing popularity and has been applied to a variety of complex data are deep belief networks. Analyses of these models, however, have been typically limited to qualitative analyses based on samples due to the computationally intractable nature of the model likelihood. Motivated by these circumstances, the present article provides a consistent estimator for the likelihood that is both computationally tractable and simple to apply in practice. Using this estimator, a deep belief network which has been suggested for the modeling of natural image patches is quantitatively investigated and compared to other models of natural image patches. Contrary to earlier claims based on qualitative results, the results presented in this article provide evidence that the model under investigation is not a particularly good model for natural images
Tractable generalizations of the Gaussian distribution play an important role for the analysis of high-dimensional data. One very general super-class of Normal distributions is the class of $ u$-spherical distributions whose random variables can be represented as the product $x = rcdot u$ of a uniformly distribution random variable $u$ on the $1$-level set of a positively homogeneous function $ u$ and arbitrary positive radial random variable $r$. Prominent subclasses of $ u$-spherical distributions are spherically symmetric distributions ($ u(x)=|x|_2$) which have been further generalized to the class of $L_p$-spherically symmetric distributions ($ u(x)=|x|_p$). Both of these classes contain the Gaussian as a special case. In general, however, $ u$-spherical distributions are computationally intractable since, for instance, the normalization constant or fast sampling algorithms are unknown for an arbitrary $ u$. In this paper we introduce a new subclass of $ u$-spherical distributions by choosing $ u$ to be a nested cascade of $L_p$-norms. This class is still computationally tractable, but includes all the aforementioned subclasses as a special case. We derive a general expression for $L_p$-nested symmetric distributions as well as the uniform distribution on the $L_p$-nested unit sphere, including an explicit expression for the normalization constant. We state several general properties of $L_p$-nested symmetric distributions, investigate its marginals, maximum likelihood fitting and discuss its tight links to well known machine learning methods such as Independent Component Analysis (ICA), Independent Subspace Analysis (ISA) and mixed norm regularizers. Finally, we derive a fast and exact sampling algorithm for arbitrary $L_p$-nested symmetric distributions, and introduce the Nested Radial Factorization algorithm (NRF), which is a form of non-linear ICA.
Orientation selectivity is the most striking feature of simple cell coding in V1 which has been shown to emerge from the reduction of higher-order correlations in natural images in a large variety of statistical image models. The most parsimonious one among these models is linear Independent Component Analysis (ICA), whereas second-order decorrelation transformations such as Principal Component Analysis (PCA) do not yield oriented filters. Because of this finding it has been suggested that the emergence of orientation selectivity may be explained by higher-order redundancy reduction. In order to assess the tenability of this hypothesis, it is an important empirical question how much more redundancies can be removed with ICA in comparison to PCA, or other second-order decorrelation methods. This question has not yet been settled, as over the last ten years contradicting results have been reported ranging from less than five to more than hundred percent extra gain for ICA. Here, we aim at resolving this conflict by presenting a very careful and comprehensive analysis using three evaluation criteria related to redundancy reduction: In addition to the multi-information and the average log-loss we compute, for the first time, complete rate-distortion curves for ICA in comparison with PCA. Without exception, we find that the advantage of the ICA filters is surprisingly small. Furthermore, we show that a simple spherically symmetric distribution with only two parameters can fit the data even better than the probabilistic model underlying ICA. Since spherically symmetric models are agnostic with respect to the specific filter shapes, we conlude that orientation selectivity is unlikely to play a critical role for redundancy reduction.
