Nested Barycentric Coordinate System as an Explicit Feature Map

Lee-Ad Gottlieb, Eran Kaufman, Aryeh Kontorovich, Gabriel Nivasch, Ofir Pele
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:766-774, 2021.

Abstract

We introduce a new embedding technique based on barycentric coordinate system. We show that our embedding can be used to transforms the problem of polytope approximation into that of finding a linear classifier in a higher (but nevertheless quite sparse) dimensional representation. This embedding in effect maps a piecewise linear function into a single linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former. We demonstrate that our embedding has applications to the problems of approximating separating polytopes – in fact, it can approximate any convex body and multiple convex bodies – as well as to classification by separating polytopes and piecewise linear regression.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-gottlieb21a, title = { Nested Barycentric Coordinate System as an Explicit Feature Map }, author = {Gottlieb, Lee-Ad and Kaufman, Eran and Kontorovich, Aryeh and Nivasch, Gabriel and Pele, Ofir}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {766--774}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/gottlieb21a/gottlieb21a.pdf}, url = {https://proceedings.mlr.press/v130/gottlieb21a.html}, abstract = { We introduce a new embedding technique based on barycentric coordinate system. We show that our embedding can be used to transforms the problem of polytope approximation into that of finding a linear classifier in a higher (but nevertheless quite sparse) dimensional representation. This embedding in effect maps a piecewise linear function into a single linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former. We demonstrate that our embedding has applications to the problems of approximating separating polytopes – in fact, it can approximate any convex body and multiple convex bodies – as well as to classification by separating polytopes and piecewise linear regression. } }
Endnote
%0 Conference Paper %T Nested Barycentric Coordinate System as an Explicit Feature Map %A Lee-Ad Gottlieb %A Eran Kaufman %A Aryeh Kontorovich %A Gabriel Nivasch %A Ofir Pele %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-gottlieb21a %I PMLR %P 766--774 %U https://proceedings.mlr.press/v130/gottlieb21a.html %V 130 %X We introduce a new embedding technique based on barycentric coordinate system. We show that our embedding can be used to transforms the problem of polytope approximation into that of finding a linear classifier in a higher (but nevertheless quite sparse) dimensional representation. This embedding in effect maps a piecewise linear function into a single linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former. We demonstrate that our embedding has applications to the problems of approximating separating polytopes – in fact, it can approximate any convex body and multiple convex bodies – as well as to classification by separating polytopes and piecewise linear regression.
APA
Gottlieb, L., Kaufman, E., Kontorovich, A., Nivasch, G. & Pele, O.. (2021). Nested Barycentric Coordinate System as an Explicit Feature Map . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:766-774 Available from https://proceedings.mlr.press/v130/gottlieb21a.html.

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