## DIMACS TR: 95-39

## On the Power of Democratic Networks

### Author: Eddy Mayoraz

**
ABSTRACT
**

Linear Threshold Boolean units (\LTunits) are the basic processing
components of artificial neural networks of Boolean
activations. Quantization of their parameters is a central question in
hardware implementation, when numerical technologies are used to store
the configuration of the circuit. In the previous studies on the circuit
complexity of feedforward neural networks, no differences had been made
between a network with "small" integer weights and one composed of
majority units (\LTunits\ with weights in $\{-1,0,+1\}$), since any
connection of weight $w$ ($w$ integer) can be simulated by $|w|$
connections of value $\Sgn(w)$. This paper will focus on the circuit
complexity of democratic networks, \IdEst circuits of majority units with
at most one connection between each pair of units.
The main results presented are the following: any Boolean function can be
computed by a depth-3 non-degenerate democratic network and can be
expressed as a linear threshold function of majorities; \FN{AT-LEAST-k}
and \FN{AT-MOST-k} are computable by a depth-2, polynomial size
democratic network; the smallest sizes of depth-2 circuits computing
\FN{PARITY} are identical for a democratic network and for a usual
network; the \VC\ of the class of the majority functions is $n+1$, \IdEst
equal to that of the class of any linear threshold functions.

Paper available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1995/95-39.ps.gz

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