Localization vs. Identification of Semi-Algebraic Sets

Authors:
Shai Ben-David;Michael Lindenbaum
Affiliations:
Department of Computer Science, Technion, Haifa 32000, Israel. E-mail: Email: shai@cs.technion.ac.il;Department of Computer Science, Technion, Haifa 32000, Israel. E-mail: Email: mic@cs.technion.ac.il
Venue:
Machine Learning
Year:
1998

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Abstract

How difficult is it to find the position of a known object using random samples? We study this question, which is central to Computer Vision andRobotics, in a formal way. We compare the information complexity of two types of tasks: the task of identificationof an unknown object from labeled examples input,and the task of localization in which the identity of thetarget is known and its location in some background scene has tobe determined.We carry out the comparison of these tasks using two measuringrods for the complexity of classes of sets; TheVapnik-Chervonenkis dimension and the ε-entropy of relevant classes. The VC-dimension analysis yields bounds onthe sample complexity of performing these tasks in the PAC-learningscenario whereas the ε-entropy parameterreflects the complexity of the relevant learning tasks when the examples are generatedby the uniform distribution (over the background scene).Our analysis provides a mathematical ground to the intuitionthat localization is indeed much easier than identification.Our upper-bounds on the hardness of localization are established by applying a new, algebraic-geometry based, general tool for the calculation of the VC-dimension of classes ofalgebraically defined objects. This technique was independently discovered by Goldberg and Jerrum.We believe that our techniques will prove useful for furtherVC-dimension estimation problems.