Rosenblatt's Percepceptron Theorem guaranties us that a linearly
separable function (R^n --> {0,1}) can be learned in finite time.
Question: Is it possible to guarantee learning of a continuous-valued
function (R^n --> (0,1)) which can be represented on a
perceptron in finite time?
This paper answers this question (and other ones too) in the
affirmative:
The Multivalued and Continuous Perceptrons
by
George M. Georgiou
Rosenblatt's perceptron is extended to (1) a multivalued
perceptron and (2) to a continuous-valued perceptron. It shown that
any function that can be represented by the multivalued perceptron
can be learned in a finite number of steps, and any function that
can be represented by the continuous perceptron can be learned with
arbitrary accuracy in a finite number of steps. The whole apparatus
is defined in the complex domain. With these perceptrons
learnability is extended to more complicated functions than the
usual linearly separable ones. The complex domain promises to
be a fertile ground for neural networks research.
A postscript version of the paper (compressed, uuencoded, ~75k) can be
obtained by e-mailing me.
Comments and questions on the proofs are welcome.
--George
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Dr. George M. Georgiou E-mail: georgiou at wiley.csusb.edu
Computer Science Department TEL: (909) 880-5332
California State University FAX: (909) 880-7004
5500 University Pkwy
San Bernardino, CA 92407, USA
