On a model of visual cortex: learning invariance and selectivity

In this paper we present a class of algorithms for similarity learning on spaces of images. The general framework that we introduce is motivated by some well-known hierarchical pre-processing architectures for object recognition which have been developed during the last decade, and which have been in some cases inspired by functional models of the ventral stream of the visual cortex. These architectures are characterized by the construction of a hierarchy of âlocalâ feature representations of the visual stimulus. We show that our framework includes some well-known techniques, and that it is suitable for the analysis of dynamic visual stimuli, presenting a quantitative error analysis in this setting

A polynomial time algorithm for diophantine equations in one variable.

We show that the integer roots of of a univariate polynomial with integer coefficients can be computed in polynomial time. This result holds for the classical (i.e. Turing) model of computation and a sparse representation of polynomials (i.e. coefficients and exponents are written in binary, and only nonzero monomials are represented).On montre que les racines entières d'un polynôme en une variable à coefficients entiers peuvent être calculées en temps polynomial. Ce résultat est valable pour le modèle de calcul classique des machines de Turing et pour une représentation creuse des polynômes (coefficients et exposants sont écrits en binaire, et seuls les monômes non nul sont représentés)

Neurons That Confuse Mirror-Symmetric Object Views

Neurons in inferotemporal cortex that respond similarly to many pairs of mirror-symmetric images -- for example, 45 degree and -45 degree views of the same face -- have often been reported. The phenomenon seemed to be an interesting oddity. However, the same phenomenon has also emerged in simple hierarchical models of the ventral stream. Here we state a theorem characterizing sufficient conditions for this curious invariance to occur in a rather large class of hierarchical networks and demonstrate it with simulations

Notes on the Shannon Entropy of the Neural Response

In these notes we focus on the concept of Shannon entropy in an attempt to provide a systematic way of assessing the discrimination properties of the neural response, and quantifying the role played by the number of layers and the number of templates

Generalization and Properties of the Neural Response

Hierarchical learning algorithms have enjoyed tremendous growth in recent years, with many new algorithms being proposed and applied to a wide range of applications. However, despite the apparent success of hierarchical algorithms in practice, the theory of hierarchical architectures remains at an early stage. In this paper we study the theoretical properties of hierarchical algorithms from a mathematical perspective. Our work is based on the framework of hierarchical architectures introduced by Smale et al. in the paper "Mathematics of the Neural Response", Foundations of Computational Mathematics, 2010. We propose a generalized definition of the neural response and derived kernel that allows us to integrate some of the existing hierarchical algorithms in practice into our framework. We then use this generalized definition to analyze the theoretical properties of hierarchical architectures. Our analysis focuses on three particular aspects of the hierarchy. First, we show that a wide class of architectures suffers from range compression; essentially, the derived kernel becomes increasingly saturated at each layer. Second, we show that the complexity of a linear architecture is constrained by the complexity of the first layer, and in some cases the architecture collapses into a single-layer linear computation. Finally, we characterize the discrimination and invariance properties of the derived kernel in the case when the input data are one-dimensional strings. We believe that these theoretical results will provide a useful foundation for guiding future developments within the theory of hierarchical algorithms