Datasets that answer difficult clinical questions are expensive in part due to the need for medical expertise and patient informed consent. We investigate the effect of small sample size on the performance of a text categorization algorithm. We show how to determine whether the dataset is large enough to train support vector machines. Since it is not possible to cover all aspects of sample size...

We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d: i) Given n points in Rd, compute their minimum enclosing cylinder. ii) Given two n-point sets in Rd, decide whether they can be separated by two hyperplanes. iii) Given a system of n linear inequalities with d variables, find a maximum-size feasible subsystem. We show that (the...

In this paper a special higher order neuron, the hypersphere neuron, is introduced. By embedding Euclidean space in a conformal space, hyperspheres can be expressed as vectors. The scalar product of points and spheres in conformal space, gives a measure for how far a point lies inside or outside a hypersphere. It will be shown that a hypersphere neuron may be implemented as a perceptron with tw...

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y ∈ K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w...

Finding nearest neighbor points in a large scale high dimensional data set is of wide interest in computer vision. One popular and efficient approach is to encode each data point as a binary code in Hamming space using separating hyperplanes. One condition which is often implicitly assumed is that the separating hyperplanes should be mutually orthogonal. With the aim of increasing the represent...

Given a Banach space we consider the $\sigma$-ideal of all its subsets which are covered by countably many hyperplanes and investigate standard cardinal characteristics as additivity, covering number, uniformity, cofinality. We determine their values for separable spaces, approximate them nonseparable spaces. The remaining questions reduce to deciding if following can be proved in ZFC (i.e. Zer...