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 http://publications.idiap.ch/attachments/reports/1997/rr97-10.pdf PUBLIC http://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 http://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.