Fernanda Botelho

A learning rule with generalized Hebbian synapses

Abstract We study the convergence behavior of a learning model with generalized Hebbian synapses.

Dynamical approximation by recurrent neural networks

Abstract We examine the approximating power of recurrent networks for dynamical systems through an unbounded number of iterations. It is shown that the natural family of recurrent neural networks with saturated linear transfer functions and synaptic weight matrices of rank 1 are essentially equivalent to feedforward neural networks with recurrent layers. Therefore, they inherit the universal approximation property of real-valued functions in one variable in a stronger sense, namely through an unbounded number of iterations and approximation guaranteed to be within O(1/n), with n neurons and possibly lateral synapses allowed in the hidden-layer. However, they are not as complex in their dynamical behavior as systems defined by Turing machines. It is further proved that every continuous dynamical system can be approximated through all iterations, by both finite analog and boolean networks, when one requires approximation of given arbitrary exact orbits of the (perhaps unknown) map. This result no longer holds when the orbits of the given map are only available as contaminated orbits of the approximant net due to the presence of random noise (e.g., due to digital truncations of analog activations). Neural nets can nonetheless approximate large families of continuous maps, including chaotic maps and maps sensitive to initial conditions. A precise characterization of what maps can be approximated fault-tolerantly by analog and discrete neural networks for unboundedly many iterations remains an open problem.

Absence of Cycles in Symmetric Neural Networks

For a given recurrent neural network, a discrete-time model may have asymptotic dynamics different from the one of a related continuous-time model. In this article, we consider a discrete-time model that discretizes the continuous-time leaky integrat or model and study its parallel, sequential, block-sequential, and distributed dynamics for symmetric networks. We provide sufficient (and in many cases necessary) conditions for the discretized model to have the same cycle-free dynamics of the corresponding continuous-time model in symmetric networks.

Homomorphisms on a class of commutative Banach algebras

We derive representations for homomorphisms and isomorphisms between Banach algebras of Lipschitz functions with values in a sequence space, including ∞. We show that such homomorphisms are automatically continuous and preserve the ∗ operation. We also give necessary conditions for the compactness of homomorphisms in these settings and give characterizations for the isometric isomorphisms.

Dynamical features simulated by recurrent neural networks

The evolution of two-dimensional neural network models with rank one connecting matrices and saturated linear transfer functions is dynamically equivalent to that of piecewise linear maps on an interval. It is shown that their iterative behavior ranges from being highly predictable, where almost every orbit accumulates to an attracting fixed point, to the existence of chaotic regions with cycles of arbitrarily large period.

Hermitian operators on Banach algebras of Lipschitz functions

For compact metric spaces (X, d), we show that the Lipschitz spaces Lip(X, d) and the little Lipschitz spaces lip(X, dα) with 0 < α < 1, equipped with the sum norm, support only trivial hermitian operators, that is, real multiples of the identity operator.

Projections in the Convex Hull of Surjective Isometries

We characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space

REAL COMPUTATION WITH CELLULAR AUTOMATA

Two definitions about computability of real-valued functions by cellular automata are proposed, each requiring exact computation (unlike Turing-based computability). Recursive functions, some polynomials, and even logistic and chaotic maps are shown to be exactly computable even under discrete space, time and states of the computer model, on representations more general than standard signed expansions. A number of consequences of these definitions are presented that point to computational primitives different from classical continuous objects based on addition and multiplication. Several open questions pertaining characterization of real-valued functions computable by cellular automata are briefly discussed, notably the encoding/representation problem and the halting criterion.

Boolean neural nets are observable

Abstract It is shown that arbitrary locally finite discrete neural networks are observable (have the shadowing property) in the sense that pseudo-orbits obtained by small perturbations of an orbit are approximated by actual orbits. The model includes discretizations of analog networks, arbitrary cellular automata, and a wide generalization of linear maps on a one dimensional grid. It follows that the true qualitative behavior of dynamical systems can be observed to infinite precision on computer simulations, despite unavoidable discretization and approximation errors.

Circular operators on minimal norm ideals of ℬ(ℋ)

In this article the notion of circular operator is extended to the Banach space setting. In particular, this property is considered for elementary operators of lengths one and two acting on minimal norm ideals of ℬ(ℋ). Necessary and sufficient conditions for the circularity of generalized derivations and Lüders operators are also obtained.