Information theory provides principled ways to analyze different inference and learning problems, such as hypothesis testing, clustering, dimensionality reduction, classification, and so forth. However, the use of information theoretic quantities as test statistics, that is, as quantities obtained from empirical data, poses a challenging estimation problem that often leads to strong simplifications, such as Gaussian models, or the use of plug in density estimators that are restricted to certain representation of the data. In this paper, a framework to nonparametrically obtain measures of entropy directly from data using operators in reproducing kernel Hilbert spaces defined by infinitely divisible kernels is presented. The entropy functionals, which bear resemblance with quantum entropies, are defined on positive definite matrices and satisfy similar axioms to those of Renyi's definition of entropy. Convergence of the proposed estimators follows from concentration results on the difference between the ordered spectrum of the Gram matrices and the integral operators associated to the population quantities. In this way, capitalizing on both the axiomatic definition of entropy and on the representation power of positive definite kernels, the proposed measure of entropy avoids the estimation of the probability distribution underlying the data. Moreover, estimators of kernel-based conditional entropy and mutual information are also defined. Numerical experiments on independence tests compare favorably with state-of-the-art.

The matrix-based Renyi's a-order entropy functional was recently introduced using the normalized eigenspectrum of a Hermitian matrix of the projected data in a reproducing kernel Hilbert space (RKHS). However, the current theory in the matrix-based Renyi's a-order entropy functional only defines the entropy of a single variable or mutual information between two random variables. In information theory and machine learning communities, one is also frequently interested in multivariate information quantities, such as the multivariate joint entropy and different interactive quantities among multiple variables. In this paper, we first define the matrix-based Renyi's a-order joint entropy among multiple variables. We then show how this definition can ease the estimation of various information quantities that measure the interactions among multiple variables, such as interactive information and total correlation. We finally present an application to feature selection to show how our definition provides a simple yet powerful way to estimate a widely-acknowledged intractable quantity from data. A real example on hyperspectral image (HSI) band selection is also provided.

Deep convolutional neural networks (CNNs) trained on visual objects have shown intriguing ability to predict some response properties of visual cortical neurons. However, the factors (e.g., if the model is trained or not, receptive field size) and computations (e.g., convolution, rectification, pooling, normalization) that give rise to such ability, at what level, and the role of intermediate processing stages in explaining changes that develop across areas of the cortical hierarchy are poorly understood. We focused on the sensitivity to textures as a paradigmatic example, since recent neurophysiology experiments provide rich data pointing to texture sensitivity in secondary (but not primary) visual cortex (V2). We initially explored the CNN without any fitting to the neural data and found that the first two layers of the CNN showed qualitative correspondence to the first two cortical areas in terms of texture sensitivity. We therefore developed a quantitative approach to select a population of CNN model neurons that best fits the brain neural recordings. We found that the CNN could develop compatibility to secondary cortex in the second layer following rectification and that this was improved following pooling but only mildly influenced by the local normalization operation. Higher layers of the CNN could further, though modestly, improve the compatibility with the V2 data. The compatibility was reduced when incorporating random rather than learned weights. Our results show that the CNN class of model is effective for capturing changes that develop across early areas of cortex, and has the potential to help identify the computations that give rise to hierarchical processing in the brain (code is available in GitHub).

Deep convolutional neural networks (CNNs) are becoming increasingly popular models to predict neural responses in visual cortex. However, contextual effects, which are prevalent in neural processing and in perception, are not explicitly handled by current CNNs, including those used for neural prediction. In primary visual cortex, neural responses are modulated by stimuli spatially surrounding the classical receptive field in rich ways. These effects have been modeled with divisive normalization approaches, including flexible models, where spatial normalization is recruited only to the degree that responses from center and surround locations are deemed statistically dependent. We propose a flexible normalization model applied to midlevel representations of deep CNNs as a tractable way to study contextual normalization mechanisms in midlevel cortical areas. This approach captures nontrivial spatial dependencies among midlevel features in CNNs, such as those present in textures and other visual stimuli, that arise from tiling high-order features geometrically. We expect that the proposed approach can make predictions about when spatial normalization might be recruited in midlevel cortical areas. We also expect this approach to be useful as part of the CNN tool kit, therefore going beyond more restrictive fixed forms of normalization.