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VC‐dimension on manifolds: a first approach
Author(s) -
Ferri Massimo,
Frosini Patrizio
Publication year - 2007
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.927
Subject(s) - mathematics , dimension (graph theory) , simple (philosophy) , manifold (fluid mechanics) , euclidean geometry , context (archaeology) , product (mathematics) , euclidean space , euclidean distance , pure mathematics , index (typography) , algebra over a field , computer science , geometry , epistemology , mechanical engineering , paleontology , philosophy , world wide web , engineering , biology
Abstract The Vapnik–Chervonenkis‐dimension is an index of the capacity of a learning machine. It has been computed in several cases, but always in a Euclidean context. This paper extends the notion to classifiers acting in the more general environment of a manifold. General properties are proved, and some examples of simple classifiers on elementary manifolds are given. A large part of the research is directed toward a still open problem on product manifolds. Copyright © 2007 John Wiley & Sons, Ltd.