scholarly journals Universal consistency of the k-NN rule in metric spaces and Nagata dimension

2020 ◽  
Vol 24 ◽  
pp. 914-934
Author(s):  
Benoît Collins ◽  
Sushma Kumari ◽  
Vladimir G. Pestov
Keyword(s):  
Metric Spaces ◽  
Direct Proof ◽  
Learning Rule ◽  
Dimensional Euclidean Space ◽  
Real Analysis ◽  
Uniform Distance ◽  
Finite Dimensional ◽  
Universal Consistency ◽  
Euclidean Setting ◽  
Nagata Dimension

The k nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space X that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the k-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-Euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

scholarly journals Approximations of Variational Problems in Terms of Variational Convergence

2017 ◽  
Vol 20 (K2) ◽  
pp. 107-116
Author(s):  
Diem Thi Hong Huynh
Keyword(s):  
Multiobjective Optimization ◽  
Metric Space ◽  
Metric Spaces ◽  
Equilibrium Problems ◽  
Variational Problems ◽  
Dimensional Case ◽  
Variational Convergence ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Definition Of

We show first the definition of variational convergence of unifunctions and their basic variational properties. In the next section, we extend this variational convergence definition in case the functions which are defined on product two sets (bifunctions or bicomponent functions). We present the definition of variational convergence of bifunctions, icluding epi/hypo convergence, minsuplop convergnece and maxinf-lop convergence, defined on metric spaces. Its variational properties are also considered. In this paper, we concern on the properties of epi/hypo convergence to apply these results on optimization proplems in two last sections. Next we move on to the main results that are approximations of typical and important optimization related problems on metric space in terms of the types of variational convergence are equilibrium problems, and multiobjective optimization. When we applied to the finite dimensional case, some of our results improve known one.

scholarly journals On the construction of resolving control in the problem of getting close at a fixed time moment

2021 ◽  
Vol 58 ◽  
pp. 73-93
Author(s):  
V.N. Ushakov ◽  
A.V. Ushakov ◽  
O.A. Kuvshinov
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Time Moment ◽  
Fixed Time ◽  
Dimensional Euclidean Space ◽  
Controlled System ◽  
Finite Dimensional ◽  
Solvability Set

The problem of getting close of a controlled system with a compact space in a finite-dimensional Euclidean space at a fixed time is studied. A method of constructing a solution to the problem is proposed which is based on the ideology of the maximum shift of the motion of the controlled system by the solvability set of the getting close problem.

Equidistant Loci and the Minkowskian Geometries

1972 ◽  
Vol 24 (2) ◽  
pp. 312-327 ◽  
Author(s):  
B. B. Phadke
Keyword(s):  
Metric Spaces ◽  
Open Convex ◽  
Minkowskian Space ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Projective Metric ◽  
Affine Line ◽  
Finite Dimensional Banach Space ◽  
Homogeneous Norm ◽  
Symmetric Distance

The spaced of this paper is a metrization, with a not necessarily symmetric distance xy, of an open convex set D in the n-dimensional affine space An such that xy + yz = xz if and only if x, y, z lie on an affine line with y between x and z and such that all the balls px ≦ p are compact. These spaces are called straight desarguesian G-spaces or sometimes open projective metric spaces. The hyperbolic geometry is an example; a large variety of other examples is studied by contributors to Hilbert's problem IV. When D = An and all the affine translations are isometries for the metric xy, the space is called a Minkowskian space or sometimes a finite dimensional Banach space, the (not necessarily symmetric) distance of a Minkowskian space being a (positive homogeneous) norm. In this paper geometric conditions in terms of equidistant loci are given for the space R to be a Minkowskian space.

BIFURCATION OF EQUILIBRIA IN MICROMACHINED ELASTIC STRUCTURES WITH ELECTROSTATIC ACTUATION

2001 ◽  
Vol 11 (02) ◽  
pp. 469-482 ◽  
Author(s):  
P. K. C. WANG
Keyword(s):  
Microelectromechanical Systems ◽  
Nonlinear Partial Differential Equation ◽  
Dimensional Euclidean Space ◽  
Elastic Structures ◽  
Electrostatic Actuators ◽  
Finite Dimensional ◽  
Fold Bifurcation ◽  
Multiparameter Bifurcation ◽  
Limiting Case

The determination of critical actuator voltages for the onset of "pull-in" or buckling of micromachined elastic structures or microelectromechanical systems (MEMS) with a finite number of electrostatic actuators is posed as a multiparameter bifurcation problem associated with a nonlinear partial differential equation. By making suitable approximations, the problem is reduced to one involving a multiparameter family of mappings on a bounded subset of a finite-dimensional Euclidean space RP into itself. Then the problem is reposed as one involving the intersection of level sets of certain functions whose levels correspond to the variable parameters (squares of actuator voltages). A sufficient condition for fold bifurcation when the actuator voltages exceed certain critical values is deduced using simple geometric arguments. The paper concludes with a discussion on the bifurcation of equilibrium of such systems for the limiting case where the number of actuators tends to infinity.

scholarly journals Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 98-115 ◽  
Author(s):  
Eero Saksman ◽  
Tomás Soto
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Metric Space ◽  
Besov Spaces ◽  
Metric Spaces ◽  
Function Spaces ◽  
First Order ◽  
Trace Theorems ◽  
Euclidean Setting ◽  
Regular Subset

Abstract We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.

scholarly journals Actions of amenable equivalence relations on CAT(0) fields

2012 ◽  
Vol 34 (1) ◽  
pp. 21-54 ◽  
Author(s):  
MARTIN ANDEREGG ◽  
PHILIPPE HENRY
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Amenable Group ◽  
Equivalence Relations ◽  
General Notion ◽  
Rigidity Result ◽  
Finite Dimensional ◽  
Borel Field

AbstractWe present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

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