Yu. P. Laptin

Approaches to construction of linear classifiers in the case of many classes

Some approaches to the problem of constructing linear classifiers, including embedded ones, are studied for the case of many classes. Sufficient conditions for linear separability of classes are formulated, and specifics of the problem statement when sets are not linearly separable are considered. Different approaches to construction of optimal linear classifiers are studied, and the results of numerical experiments are presented. The properties of embedded (convex piecewise linear) classifiers are studied. It is shown that, for an arbitrary family of finite nonintersecting sets, there is an embedded linear classifier that correctly separates the points of these sets.

Determination of molecular glioblastoma subclasses on the basis of analysis of gene expression

Two glioblastoma groups, which are distinguished from each other by expression level of 416 genes (p ≤ 0.05), were determined using a mathematical model of linear Boolean programming on the basis of gene expression data, obtained by microarray analysis of the glioblastomas and available in Gene Expression Omnibus (GEO) data base. The expression level of 15 genes was more than two-fold higher in the first group of glioblastoma (80 samples) in comparison with the second group (144 samples) and 401 genes and more than two-fold lower as compared to the second group. Ten of 15 genes, which expression level prevailed in the first group, encode the proteins involved in cell cycle regulation and cell proliferation. A significant percentage of 401 genes are the genes that encode proteins involved in the functioning of neural cells and participating in the processes such as synaptic transmission, neurogenesis, the formation of myelin sheath, axon formation. Kohonen map, built on the basis of the data of 15 genes with prevailed expression in the first group and 60 (of 401) genes, whose expression level elevated in the second group, confirmed the existence of two glioblastoma groups with specific gene expression profiles. Distribution of the glioblastomas into two groups may reflect two pathways of astrocytic glioma development, one of which leads to the formation of tumors with higher levels of gene expression, which protein products are involved in cell cycle regulation and proliferation. On the other hand, the existence of two molecular variants may reflect different states of glioblastoma progression.

A geometric approach to the problem of reconstruction of the sample behavior in hidden dimensions

We investigate a direct geometric approach to the problem of reconstruction of the behavior of a sample of hidden dimensions. A method for an improved description of cluster sampling, based on the interpretation of nonlinearities in the empirical distribution of both local projections of a uniform distribution on a smooth manifold, defined in the hidden dimension, is given. This method can be used to resolve a number of critical features in the empirical distributions. The a priori assumptions under which many variants of reconstruction of sampling behavior in the hidden dimensions are limited are considered.