CONF
Mayo97a1/IDIAP
On the Complexity of Recognizing Iterated Differences of Polyhedra
Mayoraz, Eddy
Gerstner, W.
Ed.
Germond, A.
Ed.
Hasler, M.
Ed.
Nicoud, J. -D.
Ed.
EXTERNAL
https://publications.idiap.ch/attachments/reports/1997/rr97-10.pdf
PUBLIC
https://publications.idiap.ch/index.php/publications/showcite/mayo97a
Related documents
Proceedings of the International Conference on Artificial Neural Networks (ICANN'97)
Lecture Notes in Computer Science
1327
475-480
1997
Springer-Verlag
IDIAP-RR 97-10
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.
REPORT
Mayo97a/IDIAP
On the Complexity of Recognizing Iterated Differences of Polyhedra
Mayoraz, Eddy
EXTERNAL
https://publications.idiap.ch/attachments/reports/1997/rr97-10.pdf
PUBLIC
Idiap-RR-10-1997
1997
IDIAP
Published in the Proceedings of ICANN'97
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.