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## Group Symmetric Robust Covariance Estimation

In this paper, we consider Tyler's robust covariance M-estimator under group symmetry constraints. We assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group, referred to as group symmetry. Examples of group symmetric structures include circulant, perHermitian, and proper quaternion matrices. We introduce a group symmetric version of Tyler's estimator (STyler) and provide an iterative fixed point algorithm to compute it. The classical results claim that at least n=p+1 sample points in general position are necessary to ensure the existence and uniqueness of Tyler's estimator, where p is the ambient dimension. We show that the STyler requires significantly less samples. In some groups, even two samples are enough to guarantee its existence and uniqueness. In addition, in the case of elliptical populations, we provide high probability bounds on the error of the STyler. These, too, quantify the advantage of exploiting the symmetry structure. Finally, these theoretical results are supported by numerical simulations.

We apply the suboptimal sequential nonparametric hypotheses testing approach for effectiveness of a statistical decision by sample space reducing. Numerical examples of the sample space reducing are given when an appropriate reducing makes it possible to construct robust sequential nonparametric hypotheses testing with a smaller mean duration time then one on the total sample space. © 2014 IEEE.

We study the problem of testing composite hypotheses versus composite alternatives when there is a slight deviation between the model and the real distribution. The used approach, which we called sub-optimal testing, implies an extension of the initial model and a modification of a sequential statistically significant test for the new model. The sub-optimal test is proposed and a non-asymptotic border for the loss function is obtained. Also we investigate correlation between the sub-optimal test and the sequential probability ratio test for the initial model.

In this paper we introduce a generalized learning algorithm for probabilistic topic models (PTM). Many known and new algorithms for PLSA, LDA, and SWB models can be obtained as its special cases by choosing a subset of the following “options”: regularization, sampling, update frequency, sparsing and robustness. We show that a robust topic model, which distinguishes specific, background and topic terms, doesn’t need Dirichlet regularization and provides controllably sparse solution.

Proceedings of the III International Conference in memory of V.I. Zubov "Stability and Control Processes (SCP 2015)".

We present robustness of the firm as an uninterrupted exchange of resources between the firm and owners of resources - stakeholders. We derive the model on the mutually accepted conditions of exchanges for the major resources and indicate the firm's limits to manipulate the exchange conditions. We also argue that temporary benevolent behavior of the firms towards one or several its stakeholders leads to accumulation of stakeholders' quasi-rent and contributes to the overall robustness of the firm.

In this paper we introduce a generalized learning algorithm for probabilistic topic models (PTM). Many known and new algorithms for PLSA, LDA, and SWB models can be obtained as its special cases by choosing a subset of the following “options”: regularization, sampling, update frequency, sparsing and robustness. We show that a robust topic model, which distinguishes specific, background and topic terms, doesn’t need Dirichlet regularization and provides controllably sparse solution.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.