Neural Networks for Complex Data

Abstract

Artificial neural networks are simple and efficient machine learning tools. Defined originally in the traditional setting of simple vector data, neural network models have evolved to address more and more difficulties of complex real world problems, ranging from time evolving data to sophisticated data structures such as graphs and functions. This paper summarizes advances on those themes from the last decade, with a focus on results obtained by members of the SAMM team of Université Paris 1.

Theoretical and Applied Aspects of the Self-Organizing Maps

“…When the data are not numerical, the SOM algorithm has to be adapted. See for example Kohonen [1985], Kohonen [1996], Kaski et al [1998a], Kohohen and Somervuo [1998], Kohonen [2001], Kohonen and Somervuo [2002], Kohonen [2013], Kohonen [2014], Cottrell et al [2012], where some of these adaptations are presented. Here we deal with categorical data collected in surveys and with abstract data which are known only by a dissimilarity matrix or a kernel matrix.…”

Dynamics of large cooperative pulsed-coupled networks

“…By hypothesis, all the cells of the network spike at instants t * 0 > 0 and t * p > t * 0 . Thus, due to the reset hypothesis in Definition 2.1, Formula (8), the state of the network at instant t * 0 , is…”

“…In Engineering, networks of coupled dynamical units are designed for control systems and communications [32]. Computational research on artificial intelligence, by means of artificial neuronal networks, is used to analyze, simulate, and investigate on data obtained from dynamical systems of interacting units with a large degree of complexity [8]. In Economics, networks of coupled units are used to investigate the equilibrium states in social systems of interacting agents [1].…”

An Extension of the Quadratic Error Function for Learning Imprecise Data in Multivariate Nonlinear Regression

“…Let X and Y be, respectively, R p and R q valued random vectors defined on the same probability space (Ω, F, P) and let (X t , Y t ) t∈N be independent identically distributed (i.i.d) replications of (X, Y). One seeks to build the best estimator 2 ] must be conjointly minimized and given the bias-variance balance to quantify the accuracy of the estimator. Let S � F θ (X), θ ∈ Θ, X ∈ R p } be not necessarily a nonlinear parametric regression model family with Y � F θ (X) + ε. F θ : R p ⟶ R q is a continuous function of θ for fixed X and measurable for all θ fixed and Θ is a compact (i.e., closed and bounded) subset of the set of possible parameters of the regression model family.…”