Many learning situations involve separation of labeled training instances by hyperplanes. Consistent separation is of theoretical interest, but the real goal is rather to minimize the number of errors using a bounded number of hyperplanes. Exact minimization of empirical error in a high-dimensional grid induced into the feature space by axis-parallel hyperplanes is NP-hard. We develop two appro...

data envelopment analysis (dea) is a non-parametric method for evaluating the relative technical efficiency for each member of a set of peer decision making units (dmus) with multiple inputs and multiple outputs. the original dea models use positive input and output variables that are measured on a ratio scale, but these models do not apply to the variables in which interval scale data can appe...

The Production Possibility Set (PPS) is defined as the set of all inputs and outputs of a system in which inputs can produce outputs‎. ‎In Data Envelopment Analysis (DEA)‎, ‎it is highly important to identify the defining hyperplanes and especially the strong defining hyperplanes of the empirical PPS‎. ‎Although DEA models can determine the efficiency of a Decision Making Unit (DMU)‎, ‎but they...

The paper presents a recursive algorithm for the investigation of a strict, linear separation in the Euclidean space. In the case when sets are linearly separable, it allows us to determine the coefficients of the hyperplanes. An example of using this algorithm as well as its drawbacks are shown. Then the algorithm of determining an optimal separation (in the sense of maximizing the distance be...

This paper will introduce and prove several theorems involving the separation of convex sets by hyperplanes, along with other interesting related results. It will begin with some basic separation results in Rn, such as the Hyperplane Separation Theorem of Hermann Minkowski, and then it will focus on and prove the extension of this theorem into normed vector spaces, known as the Hahn-Banach Sepa...

We give general identifiability conditions on the source matrix in Blind Signal Separation problem. They refine some previously known ones. We develop a subspace clustering algorithm, which is a generalization of the k-plane clustering algorithm, and is suitable for separation of sparse mixtures with bigger sparsity (i.e. when the number of the sensors is bigger at least by 2 than the number of...

Separation is a famous principle and separation properties are important for optimization theory and various applications. In practice, input data are rarely known exactly and it is advisable to deal with parameters. In this article, we are concerned with the basic characteristics (existence, description, stability etc.) of separating hyperplanes of two convex polyhedral sets depending on param...

This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on semispaces. There are some new results. In particular, we give new “colorful” extensions of the max-min Carathéodory theorem. In the end of the paper, we list some c...

This is a survey on support and separation properties of convex sets in the n-dimensional Euclidean space. It contains detailed account existing results, given either chronologically or related groups, exhibits them uniform way, including terminology notation. We first discuss classical Minkowski’s theorems bodies, next describe various generalizations these results to case arbitrary sets, whic...

Suppose we choose a group of data points, which could reasonably separate information regions. These data points that lie close to separation regions, selected among all the input data, are commonly called “support vectors”. Assume that we have group of data {xi, yi}that could be separated by a hyperplane. Thus we can write the following statements about the separating hyperplanes, { β.xi + β0 ...