%Aigaion2 BibTeX export from Idiap Publications %Saturday 03 August 2024 04:31:01 PM @INPROCEEDINGS{Mayo97a1, author = {Mayoraz, Eddy}, editor = {Gerstner, W. and Germond, A. and Hasler, M. and Nicoud, J. -D.}, projects = {Idiap}, title = {On the Complexity of Recognizing Iterated Differences of Polyhedra}, booktitle = {Proceedings of the International Conference on Artificial Neural Networks (ICANN'97)}, series = {Lecture Notes in Computer Science}, number = {1327}, year = {1997}, publisher = {Springer-Verlag}, note = {IDIAP-RR 97-10}, crossref = {mayo97a}, abstract = {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.}, pdf = {https://publications.idiap.ch/attachments/reports/1997/rr97-10.pdf}, postscript = {ftp://ftp.idiap.ch/pub/reports/1997/rr97-10.ps.gz}, ipdmembership={learning}, } crossreferenced publications: @TECHREPORT{Mayo97a, author = {Mayoraz, Eddy}, projects = {Idiap}, title = {On the Complexity of Recognizing Iterated Differences of Polyhedra}, type = {Idiap-RR}, number = {Idiap-RR-10-1997}, year = {1997}, institution = {IDIAP}, note = {Published in the Proceedings of ICANN'97}, abstract = {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.}, pdf = {https://publications.idiap.ch/attachments/reports/1997/rr97-10.pdf}, postscript = {ftp://ftp.idiap.ch/pub/reports/1997/rr97-10.ps.gz}, ipdmembership={learning}, }