Localization vs. Identification of Semi-Algebraic Sets

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 and Robotics, 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 the target is known and its location in some background scene has to be determined. We carry out the comparison of these tasks using two measuring rods for the complexity of classes of sets; The Vapnik-Chervonenkis dimension and the†-entropy of relevant classes. The VC-dimension analysis yields bounds on the sample complexity of performing these tasks in the PAC-learning scenario whereas the†-entropy parameter reflects the complexity of the relevant learning tasks when the examples are generated by the uniform distribution (over the background scene). Our analysis provides a mathematical ground to the intuition that 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 of algebraically defined objects. This technique was independently discovered by Goldberg and Jerrum. We believe that our techniques will prove useful for further VC-dimension estimation problems.