Maximal margin classifiers are a core technology in modern machine learning. They have strong theoretical justifications and have shown empirical successes. We provide an alternative justification for maximal margin hyperplane classifiers by relating them to Bayes optimal classifiers that use Parzen windows estimations with Gaussian kernels. For any value of the smoothing parameter (the width o...

Hyperplane codes are a class of convex codes that arise as the output of a one layer feed-forward neural network. Here we establish several natural properties of nondegenerate hyperplane codes, in terms of the polar complex of the code, a simplicial complex associated to any combinatorial code. We prove that the polar complex of a non-degenerate hyperplane code is shellable and show that all cu...

Given a set S of n ≥ d points in general position in Rd, a random hyperplane split is obtained by sampling d points uniformly at random without replacement from S and splitting based on their affine hull. A random hyperplane search tree is a binary space partition tree obtained by recursive application of random hyperplane splits. We investigate the structural distributions of such random trees...

The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a h...

The requirement of transformation invariance of a probability distribution is employed to derive prior probabilities for the coefficients of the equation describing a hyperplane. In two dimensions, this is a straight line, in three dimensions an ordinary plane etc. We treat the general case of n dimensions and propose a procedure to normalize the resulting distributions in order to make them pr...

one of the important issues in data envelopment analysis (dea) is sensitivity analysis. heretofore the existent studies have considered the data modification of inputs and outputs in one or multiple dmus. in this paper the number of dmus is increased and a stability region is obtained in  by applying defining hyperplanes in which if the added dmu (only one dmu) is in this region then all of the...