On the Complexity of Recognizing Iterated Differences of Polyhedra

The iterated difference of polyhedra $V = P_1 \backslash ( P_2 \backslash (... P_k ) ... )$ has been proposed independently in [Zwie-Aart-Wess92] and [Shon93] as a sufficient condition for $V$ to be exactly computable by a two-layered neural network. An algorithm checking whether $V$ included in $R^d$ is an iterated difference of polyhedra is proposed in [Zwie-Aart-Wess92]. However, this algorithm is not practically usable because it has a high computational complexity and it was only conjectured to stop with a negative answer when applied to a region which is not an iterated difference of polyhedra. This paper sheds some light on the nature of iterated difference of polyhedra. The outcomes are\,: (i) an algorithm which always stops after a small number of iterations, (ii) sufficient conditions for this algorithm to be polynomial and (iii) the proof that an iterated difference of polyhedra can be exactly computed by a two-layered neural network using only essential hyperplanes.


Editor(s):
Gerstner, W.
Germond, A.
Hasler, M.
Nicoud, J. -D.
Published in:
Proceedings of the International Conference on Artificial Neural Networks (ICANN'97), 1327, 475-480
Presented at:
Proceedings of the International Conference on Artificial Neural Networks (ICANN'97)
Year:
1997
Publisher:
Springer-Verlag
Keywords:
Note:
IDIAP-RR 97-10
Laboratories:




 Record created 2006-03-10, last modified 2018-03-17

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