- Calculating the Vapnik-Chervonenkis dimension in a non-discrete setting (arXiv)
Author: Mohamed Neshba, Muhajir Mohamed, Sejari Yassin
Summary: 1984, Valiant [ 7 ] We introduced the Probably Approximately Correct (PAC) learning framework for the Boolean function class. Bloomer et al. [ 2] In 1989, we extended this model by introducing the VC dimension as a tool to characterize the learnability of PAC. The VC dimensions are based on the work of Vapnik and Chervonenkis in 1971. [8 ], he introduced a tool called growth function to characterize grinding properties. Since then, efforts have been made by researchers to develop algorithms that can determine the His VC dimension of a particular class and calculate the His VC dimension of any conceptual class. 1991, Linial, Mansour, Rivest [4] We presented an algorithm for computing the VC dimension in a discrete setting, assuming that both the concept class and the domain set are finite. However, no attempt has been made to design an algorithm that can compute the VC dimension in a general setting. Therefore, our research focuses on developing a method to approximately compute the VC dimension without imposing constraints on the conceptual class or its domain set. Our approach is based on the discovery that the experiential risk minimization (ERM) learning paradigm can be used as a new tool to characterize the disruptive properties of conceptual classes.
2. Inner product of F3q and Vapnik-Chervonenkis dimension (arXiv)
Author: A. Iosevic, B. McDonald, M. Sun
Summary: We are given a set E⊂F3q. Here Fq is a field with q elements. Consider a set of “classifiers” H3t(E)={hy:y∈E}. Here, if x⋅y=t, hy(x)=1, x∈E, otherwise 0. Prove that for |E|≥Cq114 and a sufficiently large constant C>0, the Vapnik-Chervonenkis dimension of H3t(E) is equal to 3. In particular, this means that for a sufficiently large subset of F3q: , the Vapnik-Chervonenkis dimension of H3t(E) is the same as the Vapnik-Chervonenkis dimension of H3t(F3q). In a sense, this proof leads us to consider the most complex possible configuration that can always be embedded in a subset of F3q of size ≥Cq114.
