On the effect of analog noise in discrete-time analog computations
We introduce a model for analog computation with discrete time in the presence of analog noise that is flexible enough to cover the most important concrete cases, such as noisy analog neural nets and networks of spiking neurons. This model subsumes the classical model for digital computation in the presence of noise. We show that the presence of arbitrarily small amounts of analog noise reduces the power of analog computational models to that of finite automata, and we also prove a new type of upper bound for the VC-dimension of computational models with analog noise.
The computational power of continuous time neural networks
We investigate the computational power of continuous-time neural networks with Hopfield-type units. We prove that polynomial-size networks with saturated-linear response functions are at least as powerful as polynomially space-bounded Turing machines.
Exponential Transients in Continuous-Time Symmetric Hopfield Nets
We establish a fundamental result in the theory of continuous-time neural computation, by showing that so called continuous-time symmetric Hopfield nets, whose asymptotic convergence is always guaranteed by the existence of a Liapunov function may, in the worst case, possess a transient period that is exponential in the network size. The result stands in contrast to e.g. the use of such network models in combinatorial optimization applications. peerReviewed
On the Computational Complexity of Binary and Analog Symmetric Hopfield Nets
We investigate the computational properties of finite binary- and analog-state discrete-time symmetric Hopfield nets. For binary networks, we obtain a simulation of convergent asymmetric networks by symmetric networks with only a linear increase in network size and computation time. Then we analyze the convergence time of Hopfield nets in terms of the length of their bit representations. Here we construct an analog symmetric network whose convergence time exceeds the convergence time of any binary Hopfield net with the same representation length. Further, we prove that the MIN ENERGY problem for analog Hopfield nets is NP-hard and provide a polynomial time approximation algorithm for this p…
Computation of the Multivariate Oja Median
The multivariate Oja median (Oja, 1983) is an affine equivariant multivariate location estimate with high efficiency. This estimate has a bounded influence function but zero breakdown. The computation of the estimate appears to be highly intensive. We consider different, exact and stochastic, algorithms for the calculation of the value of the estimate. In the stochastic algorithms, the gradient of the objective function, the rank function, is estimated by sampling observation. hyperplanes. The estimated rank function with its estimated accuracy then yields a confidence region for the true sample Oja median, and the confidence region shrinks to the sample median with the increasing number of…
Some Afterthoughts on Hopfield Networks
In the present paper we investigate four relatively independent issues, which complete our knowledge regarding the computational aspects of popular Hopfield nets. In Section 2 of the paper, the computational equivalence of convergent asymmetric and Hopfield nets is shown with respect to network size. In Section 3, the convergence time of Hopfield nets is analyzed in terms of bit representations. In Section 4, a polynomial time approximate algorithm for the minimum energy problem is shown. In Section 5, the Turing universality of analog Hopfield nets is studied. peerReviewed
A Survey of Continuous-Time Computation Theory
Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuous-time computation. However, while special-case algorithms and devices are being developed, relatively little work exists on the general theory of continuous- time models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. Final Draft peerReviewed
Universal computation by finite two-dimensional coupled map lattices
Extended abstract presented at the 4th Workshop on Physics and Computation (PhysComp 96), 22-24 Nov 1996. Boston, Massachusetts acceptedVersion peerReviewed