new paper on the effect of analog noise in neural computation

[Wolfgang Maass]

The following paper is now available for copying from

 http://www.math.jyu.fi/~orponen/papers/noisyac.ps

The paper has 19 pages.


" On the Effect of Analog Noise in Discrete-Time Analog Computations "

                               by                            

 Wolfgang Maass                and       Pekka Orponen
 Inst. for Theor. Comp. Sci.             Department of Mathematics
 Technische Universitaet Graz            University of Jyvaskyla 
 Klosterwiesgasse 32/2                   P.O. Box 35
 A-8010 Graz, Austria                    Jyvaskyla, Finland
 maass at igi.tu-graz.ac.at                 orponen at math.jyu.fi


Abstract:

We introduce a model for analog noise in analog computations 
with discrete time that is flexible enough to cover the most 
important concrete cases, such as analog noise in sigmoidal neural 
nets and networks of spiking neurons. The noise model can also
be applied to cases where there are dependencies among the
noise-sources, and to hybrid analog/digital systems. 

In contrast to previous models for noise in analog computations
(which demand that the output of the computation has to be 
100% reliable), we assume that the output of a noisy analog
computation has to be correct only with a certain probability
(which may be chosen to be very high).
We believe that this convention is more adequate for the analysis
of "real world" analog computations. In addition this convention 
is consistent with the common models for noisy digital computations 
in computational complexity theory. 

We show that under very general conditions the presence of analog 
noise reduces the power of analog computational models to that of a
finite automaton, and we exhibit bounds for the number of states of
such finite automaton.

We also prove a new type of upper bound for the VC-dimension 
of computational models with analog noise. In the case of a 
feedforward sigmoidal neural net this bound does not depend on 
the the total number of units in the net. 

An extended abstract of this paper will appear in the Proceedings
of NIPS '96.