Luis G. Sanchez Giraldo

Benjamin Colburn, Jose C. Principe, Luis G. Sanchez Giraldo

Kernel Adaptive Filtering (KAF) are mathematically principled methods which search for a function in a Reproducing Kernel Hilbert Space. While they work well for tasks such as time series prediction and system identification they are plagued by a linear relationship between number of training samples and model size, hampering their use on the very large data sets common in today's data saturated world. Previous methods try to solve this issue by sparsification. We describe a novel view of optimal filtering which may provide a route towards solutions in a RKHS which do not necessarily have this linear growth in model size. We do this by defining a RKHS in which the time structure of a stochastic process is still present. Using correntropy [11], an extension of the idea of a covariance function, we create a time based functional which describes some potentially nonlinear desired mapping function. This form of a solution may provide a fruitful line of research for creating more efficient representations of functionals in a RKHS, while theoretically providing computational complexity in the test set similar to Wiener solution.

Hassen Dhrif, Luis G. Sanchez Giraldo, Miroslav Kubat et al.

Evolutionary computation (EC) algorithms, such as discrete and multi-objective versions of particle swarm optimization (PSO), have been applied to solve the Feature selection (FS) problem, tackling the combinatorial explosion of search spaces that are peppered with local minima. Furthermore, high-dimensional FS problems such as finding a small set of biomarkers to make a diagnostic call add an additional challenge as such methods ability to pick out the most important features must remain unchanged in decision spaces of increasing dimensions and presence of irrelevant features. We developed a combinatorial PSO algorithm, called COMB-PSO, that scales up to high-dimensional gene expression data while still selecting the smallest subsets of genes that allow reliable classification of samples. In particular, COMB-PSO enhances the encoding, speed of convergence, control of divergence and diversity of the conventional PSO algorithm, balancing exploration and exploitation of the search space. Applying our approach on real gene expression data of different cancers, COMB-PSO finds gene sets of smallest size that allow a reliable classification of the underlying disease classes.

Luis G. Sanchez Giraldo, Jose C. Principe

A rekindled the interest in auto-encoder algorithms has been spurred by recent work on deep learning. Current efforts have been directed towards effective training of auto-encoder architectures with a large number of coding units. Here, we propose a learning algorithm for auto-encoders based on a rate-distortion objective that minimizes the mutual information between the inputs and the outputs of the auto-encoder subject to a fidelity constraint. The goal is to learn a representation that is minimally committed to the input data, but that is rich enough to reconstruct the inputs up to certain level of distortion. Minimizing the mutual information acts as a regularization term whereas the fidelity constraint can be understood as a risk functional in the conventional statistical learning setting. The proposed algorithm uses a recently introduced measure of entropy based on infinitely divisible matrices that avoids the plug in estimation of densities. Experiments using over-complete bases show that the rate-distortion auto-encoders can learn a regularized input-output mapping in an implicit manner.

Luis G. Sanchez Giraldo, Jose C. Principe

In this paper, we develop a framework for information theoretic learning based on infinitely divisible matrices. We formulate an entropy-like functional on positive definite matrices based on Renyi's axiomatic definition of entropy and examine some key properties of this functional that lead to the concept of infinite divisibility. The proposed formulation avoids the plug in estimation of density and brings along the representation power of reproducing kernel Hilbert spaces. As an application example, we derive a supervised metric learning algorithm using a matrix based analogue to conditional entropy achieving results comparable with the state of the art.