Continuous Generalized Gradient Descent

This article derives characterizations and computational algorithms for continuous general gradient descent trajectories in high-dimensional parameter spaces for statistical model selection, prediction, and classification. Examples include proportional gradient shrinkage as an extension of LASSO and LARS, threshold gradient descent with right-continuous variable selectors, threshold ridge regression, and many more with proper combinations of variable selectors and functional forms of a kernel. In all these problems, general gradient descent trajectories are continuous piecewise analytic vector-valued curves as solutions to matrix differential equations. We show the monotonicity and convergence of the proposed algorithms in the loss or negative likelihood functions. We prove that approximations of continuous solutions via infinite series expansions are computationally more efficient and accurate compared with discretization methods. We demonstrate the applicability of our algorithms through numerical experiments with real and simulated datasets.     

Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values

We present some extreme continuous selector theorems, synthesizing the author's results; namely, we study existence and properties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the space of Bochner integrable functions.     

Application of Michael's theorem and its converse to sublinear operators

A theorem of Michael on continuous selectors and its converse are used in this article to study subdifferentials of continuous sublinear operators with values in a cone of lower semicontinuous functions. It is proved that such operators are subdifferentiable (i.e., have nonempty subdifferentials) if their domains are separable Banach spaces. Sublinear operators that are not subdifferentiable are found.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 67–75, July, 1992.     

Some properties of set-valued stochastic integrals

The present paper is devoted to properties of set-valued stochastic integrals defined as some special type of set-valued random variables. In particular, it is shown that if the probability base is separable or probability measure is nonatomic then defined set-valued stochastic integrals can be represented by a sequence of Itô?s integrals of nonanticipative selectors of integrated set-valued processes. Immediately from Michael?s continuous selection theorem it follows that the indefinite set-valued stochastic integrals possess some continuous selections. The problem of integrably boundedness of set-valued stochastic integrals is considered. Some remarks dealing with stochastic differential inclusions are also given.     

Selectors in non-Archimedean spaces

Continuous selectors on the hyperspace F(X) are studied, when X is a non-Archimedean space. It is shown that a non-Archimedean space has a continuous selector if and only if it is topologically well orderable. Another characterization is given in terms of density and complete metrizability.     

A note on the universal consistency of the kernel distribution function estimator

The problem of universal consistency of data driven bandwidth selectors for the kernel distribution estimator is analyzed. We provide a uniform in bandwidth result for the kernel estimate of a continuous distribution function. Our smoothness assumption is minimal in the sense that if the true distribution function has some discontinuity then the kernel estimate is no longer consistent.     

Nuclear Typing of Frames vs Spatial Selectors

Nuclei which are defined over a class of frames are called nuclear typings. There is the dual notion of a spatial selector, and the relationship between nuclear typings and spatial selectors emanates from the duality between spatial frames and sober spaces. Especially interesting is the interplay between typings that are well-behaved with respect to certain frame quotients and selectors which similarly behave well in passage to closed sets.     

The cumulation of incompetence in organizations

An absorbing Markov chain model with a continuous time parameter is used to describe the effects on organizational efficiency of selection for vacancies by selectors with incomplete information on the candidates. Analytical results indicate that the efficiency decreases to zero, eventually at an exponential rate. Simulation studies for large hierarchical organizations of moderate “steepness” (number of employees per boss) and with moderate accuracy of the selection procedure suggest that the amount of time before a substantial decrease in efficiency occurs is generally large compared to a lifetime.     

Categories of partial frames

This article discusses the basic categorical algebra for categories of partial frames. Categories of partial frames are labelled by subset selectors that indicate which joins exist. Constructions for limits, colimits, and free functors connecting various categories of partial frames are given. Examples of partial frame categories are given. Subset selectors which preserve surjections are virtually the same as rules which select all subsets smaller than a given cardinal. Received December 1, 2004; accepted in final form March 28, 2005.     

Separately continuous selectors

The existence of separately continuous selectors of a separately lower and upper semicontinuous many-valued multivariate map is proved, provided its domain is a compact set and its range is a convex closed subset of a metrizable convex compact set. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 209–216, February, 1998. This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00264.     

A note on a consistency property for permutations

In cooperative game theory allocation of earnings to players may take place on the basis of selectors or—more restrictively—consistent selectors, or on the basis of a permutation representing the queueing of the players. This note gives a graph theoretic characterization of those situations in which the latter allocation method results in allocation with consistent selectors.     

Closed classes of k-valued logic,all of whose congruences are trivial

In this article a criterion for the existence of only trivial congruences on a closed class of k-valued logic containing selectors is formulated and proved. All homomorphic, but nonisomorphic, images of a given closed class of k-valued logic containing selectors are described.Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 499–509, October, 1977.The author thanks S. V. Yablonskii for guidance and assistance with the article.     

Optimal implementation of selectors in a data-base

The records of a data base can be accessed from other records or from a set of data items (inverted access, primary and secondary index of IMS, search keys of CODASYL etc.) which we call selectors. The implementation of this selectors can use different techniques as hash coding, inverted lists or hierarchical index (indexed sequential, B-trees etc…) We consider here the last one and we search for a given set of selectors an optimal index structure. We show how this problem can be put as the search of an optimal rooted tree among the partial subgraphs of a given graph G (this problem is known in graph theory as Steiner problem) and we give several properties which allow the graph G to be notabily reduced. Then we present a branch and bound algorithm to solve this problem.     

L p-Continuous extreme selectors of multifunctions with decomposable values: Relaxation theorems

Some density theorems of L _p-continuous selectors whose values are extreme points are proved for a class of multivalued maps. applications to the Darboux problem for a differential inclusion are presented.     

Plug-in method for nonparametric regression

The problem of bandwidth selection for non-parametric kernel regression is considered. We will follow the Nadaraya–Watson and local linear estimator especially. The circular design is assumed in this work to avoid the difficulties caused by boundary effects. Most of bandwidth selectors are based on the residual sum of squares (RSS). It is often observed in simulation studies that these selectors are biased toward undersmoothing. This leads to consideration of a procedure which stabilizes the RSS by modifying the periodogram of the observations. As a result of this procedure, we obtain an estimation of unknown parameters of average mean square error function (AMSE). This process is known as a plug-in method. Simulation studies suggest that the plug-in method could have preferable properties to the classical one. Supported by the MSMT: LC 06024.     

Convergence rates for unconstrained bandwidth matrix selectors in multivariate kernel density estimation

Progress in selection of smoothing parameters for kernel density estimation has been much slower in the multivariate than univariate setting. Within the context of multivariate density estimation attention has focused on diagonal bandwidth matrices. However, there is evidence to suggest that the use of full (or unconstrained) bandwidth matrices can be beneficial. This paper presents some results in the asymptotic analysis of data-driven selectors of full bandwidth matrices. In particular, we give relative rates of convergence for plug-in selectors and a biased cross-validation selector.     

L p -continuous extreme selectors of multifunctions with decomposable values: Existence theorems

The existence theorems of L _p -continuous selectors that values are extreme points are proved for a class of multivalued maps. Applications to multivalued maps appearing in multivalued differential equations are presented.     

Set-valved Markov Processes and Their Representation Theorems

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Abstract Characteristic of the Class of Unitary Positional Algebras of Operations

We find an abstract characteristic of the class of unitary positional algebras of operations, i.e., algebras that contain a complete collection of selectors.     

Lp-Continuous extreme selectors of multifunctions with decomposable values: Relaxation theorems

Some density theorems of L _p-continuous selectors whose values are extreme points are proved for a class of multivalued maps. applications to the Darboux problem for a differential inclusion are presented.Supported in part by RFFI Grant 93-011-264.