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## Manifold Learning: Generalization Ability and Tangent Proximity

One of the ultimate goals of Manifold Learning (ML) is to reconstruct an unknown nonlinear low-dimensional Data Manifold (DM) embedded in a high-dimensional observation space from a given set of data points sampled from the manifold. We derive asymptotic expansion and local lower and upper bounds for the maximum reconstruction error in a small neighborhood of an arbitrary point. The expansion and bounds are defined in terms of the distance between tangent spaces to the original Data manifold and the Reconstructed Manifold (RM) at the selected point and its reconstructed value, respectively. We propose an amplification of the ML, called Tangent Bundle ML, in which proximity is required not only between the DM and RM but also between their tangent spaces. We present a new geometrically motivated Grassman&Stiefel Eigenmaps algorithm that solves this problem and gives a new solution for the ML also.

The paper presents a new geometrically motivated method for non-linear regression based on Manifold learning technique. The regression problem is to construct a predictive function which estimates an unknown smooth mapping f from q-dimensional inputs to m-dimensional outputs based on a training data set consisting of given ‘input-output’ pairs. The unknown mapping f determines q-dimensional manifold M(f) consisting of all the ‘input-output’ vectors which is embedded in (q+m)-dimensional space and covered by a single chart; the training data set determines a sample from this manifold. Modern Manifold Learning methods allow constructing the certain estimator M* from the manifold-valued sample which accurately approximates the manifold. The proposed method called Manifold Learning Regression (MLR) finds the predictive function fMLR to ensure an equality M(fMLR) = M*. The MLR simultaneously estimates the m×q Jacobian matrix of the mapping f.

In many Data Analysis tasks, one deals with data that are presented in high-dimensional spaces. In practice original high-dimensional data are transformed into lower-dimensional representations (features) preserving certain subject-driven data properties such as distances or geodesic distances, angles, etc. Preserving as much as possible available information contained in the original high-dimensional data is also an important and desirable property of the representation. The real-world high-dimensional data typically lie on or near a certain unknown low-dimensional manifold (Data manifold) embedded in an ambient high-dimensional `observation' space, so in this article we assume this Manifold assumption to be fulfilled. An exact isometric manifold embedding in a low-dimensional space is possible in certain special cases only, so we consider the problem of constructing a `locally isometric and conformal' embedding, which preserves distances and angles between close points. We propose a new geometrically motivated locally isometric and conformal representation method, which employs Tangent Manifold Learning technique consisting in sample-based estimation of tangent spaces to the unknown Data manifold. In numerical experiments, the proposed method compares favourably with popular Manifold Learning methods in terms of isometric and conformal embedding properties as well as of accuracy of Data manifold reconstruction from the sample.

In many applications, the real high-dimensional data occupy only a very small part in the high dimensional ‘observation space’ whose intrinsic dimension is small. The most popular model of such data is Manifold model which assumes that the data lie on or near an unknown manifold Data Manifold, (DM) of lower dimensionality embedded in an ambient high-dimensional input space (Manifold assumption about high-dimensional data). Manifold Learning is a Dimensionality Reduction problem under the Manifold assumption about the processed data and its goal is to construct a low-di-mensional parameterization of the DM (global low-dimensional coordinates on the DM) from a finite dataset sampled from the DM. Manifold assumption means that local neighborhood of each manifold point is equivalent to an area of low-dimensional Euclidean space. Because of this, most of Manifold Learning algorithms include two parts: ‘local part’ in which certain characteristics reflecting low-dimensional local structure of neighborhoods of all sample points are constructed and ‘global part’ in which global low-dimensional coordinates on the DM are constructed by solving certain convex optimization problem for specific cost function depending on the local characteristics. Statistical properties of ‘local part’ are closely connected with local sampling on the manifold, which is considered in the study.

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.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The geographic information system (GIS) is based on the first and only Russian Imperial Census of 1897 and the First All-Union Census of the Soviet Union of 1926. The GIS features vector data (shapefiles) of allprovinces of the two states. For the 1897 census, there is information about linguistic, religious, and social estate groups. The part based on the 1926 census features nationality. Both shapefiles include information on gender, rural and urban population. The GIS allows for producing any necessary maps for individual studies of the period which require the administrative boundaries and demographic information.

It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXPNP, or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are *selfreducible*? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that EXPNP does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of EXP is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for NEXP.