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## Asymptotic Properties of Nonparametric Estimation on Manifold

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-dimensional 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 via nonparametric estimation, and ‘global part’ in which global low-dimensional coordinates on the DM are constructed by solving the certain convex optimization problem for specific cost function depending on the local characteristics. Both statistical properties of ‘local part’ and its average over manifold are considered in the paper. The article is an extension of the paper (Yanovich, 2016) for the case of nonparametric estimation.

Symmetric random walks in $R^d$ and $Z^d$ are considered. It is assumed that the jump distribution density has moderate tails, i.e., several density moments are finite, including the second one. The global (for all $x$ and $t$) asymptotic behavior at infinity of the transition probability (fundamental solution of the corresponding parabolic convolution operator) is found. Front propagation of ecological waves in the corresponding population dynamics models is described.

We use the characterization of distribution symmetry in terms of order statistics in order to obtain new tests of symmetry based on U-empirical distribution functions. We calculate their limiting distributions and large deviations and explore their local Bahadur efficiency against location alternatives which turns out to be rather high.

We propose new tests of exponentiality of integral and of Kolmogorov type based on a characterization of exponentiality proposed by Ahsanullah. Bahadur efficiency of new tests is computed, conditions of local asymptotic optimality are described.

We propose a novel multi-texture synthesis model based on generative adversarial networks (GANs) with a user-controllable mechanism. The user control ability allows to explicitly specify the texture which should be generated by the model. This property follows from using an encoder part which learns a latent representation for each texture from the dataset. To ensure a dataset coverage, we use an adversarial loss function that penalizes for incorrect reproductions of a given texture. In experiments, we show that our model can learn descriptive texture manifolds for large datasets and from raw data such as a collection of high-resolution photos. We show our unsupervised learning pipeline may help segmentation models. Moreover, we apply our method to produce 3D textures and show that it outperforms existing baselines.

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.

Let X be an unknown nonlinear smooth q-dimensional Data manifold (D-manifold) embedded in a p-dimensional space (p> q) covered by a single coordinate chart. It is assumed that the manifold's condition number is positive so X has no self-intersections. Let Xn={X1, X2,..., Xn}⊂ X⊂ Rp be a sample randomly selected from the D-manifold Xindependently of each other according to an unknown probability measure on X with strictly positive density.

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.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.