Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

9 Dynamical Systems

Morris W. Hirsch University of California at Berkeley


1. INTRODUCTION

In this chapter we look at neural nets as dynamical systems in the mathematical sense.

The following notations will be used. Rn denotes n-dimensional Euclidean space comprising all n-tuples of real numbers. We denote the inner or dot product of two vectors in Rn by x · y = 〈x, y〉 = Σ xiyi. The Euclidean norm of a vector x is

.

A map H : RnRm is smooth, or C1, provided it has continuous partial derivatives. In this case we denote the Jacobian matrix at p ∈ Rn by DH(p) = [(∂Hi/∂xj)(p)].We say H is C2 if the partial derivatives are C1. For integers k > 1 we recursively define H to be Ck if the partial derivatives are Ck-1. If f is a smooth real-valued function on Rn, the gradient vector grad f (p) of f at p ∈ Rn is the column vector of partial derivatives of f at p.

Before giving a formal mathematical treatment of dynamical systems, I indicate briefly how they arise in networks. Consider a net with n units labeled 1,. . ., n. For the moment assume that weights, thresholds, external inputs, and all other parameters are fixed. The activation of unit i at a given time is a real number, denoted here by ui. The state of the net is the activation vector u = (u1,. . . , un) ∈ Rn.

By starting at an initial activation state u and running the net, for each future time t > 0 we can determine the state x(t) to which state u will evolve at time t. Since we want to make a mathematical model of the net, we assume that, in principle, we can calculate any future state x(t) as a function of the time t and the initial state x(0) = u. The functions x(t), for all possible initial states u, constitute the activation dynamics of the net; they are called trajectories. To make explicit the fact that x(t) depends on u we use a notation such as x(t) = φt(u).

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