The influence of oppositely classified examples on the generalization complexity of Boolean functions

A deep connection between the Vapnik-Chervonenkis entropy and the Rademacher complexity

In this paper, we derive a deep connection between the Vapnik-Chervonenkis (VC) entropy and the Rademacher complexity. For this purpose, we first refine some previously known relationships between the two notions of complexity and then derive new results, which allow computing an admissible range for the Rademacher complexity, given a value of the VC-entropy, and vice versa. The approach adopted in this paper is new and relies on the careful analysis of the combinatorial nature of the problem. The obtained results improve the state of the art on this research topic.

The generalization complexity measure for continuous input data

We introduce in this work an extension for the generalization complexity measure to continuous input data. The measure, originally defined in Boolean space, quantifies the complexity of data in relationship to the prediction accuracy that can be expected when using a supervised classifier like a neural network, SVM, and so forth. We first extend the original measure for its use with continuous functions to later on, using an approach based on the use of the set of Walsh functions, consider the case of having a finite number of data points (inputs/outputs pairs), that is, usually the practical case. Using a set of trigonometric functions a model that gives a relationship between the size of the hidden layer of a neural network and the complexity is constructed. Finally, we demonstrate the application of the introduced complexity measure, by using the generated model, to the problem of estimating an adequate neural network architecture for real-world data sets.