Koltchinskii, Vladimir, Sakhanenko, Lyudmila, Cai, Songhe
Let $v$ be a vector field in a bounded open set $G\subset {\mathbb {R}}^d$. Suppose that $v$ is observed with a random noise at random points $X_i, i=1,...,n,$ that are independent and uniformly...
Let $\mathcal{F}$ be a class of measurable functions $f:S\mapsto [0,1]$ defined on a probability space $(S,\mathcal{A},P)$. Given a sample (X_1,...,X_n) of i.i.d. random variables taking values in S...
Rejoinder: 2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization [arXiv:0708.0083]
High Dimensional Probability (2006)
Giné, Evarist, Koltchinskii, Vladimir, Li, Wenbo, Zinn, Joel
About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better...
Empirical graph Laplacian approximation of Laplace--Beltrami operators: Large sample results (2006)
Giné, Evarist, Koltchinskii, Vladimir
Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension $\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the...
Concentration inequalities and asymptotic results for ratio type empirical processes (2006)
Giné, Evarist, Koltchinskii, Vladimir
Let $\mathcal{F}$ be a class of measurable functions on a measurable space $(S,\mathcal{S})$ with values in $[0,1]$ and let \[P_n=n^{-1}\sum_{i=1}^n\delta_{X_i}\] be the empirical measure based on an...
Concentration inequalities and asymptotic results for ratio type empirical processes (2006)
Giné, Evarist, Koltchinskii, Vladimir
Let ℱ be a class of measurable functions on a measurable space $(S,\mathcal{S})$ with values in [0,1] and let Pn=n−1∑i=1nδXi be the empirical measure based on an i.i.d. sample (X1,…,Xn) from...
Complexities of convex combinations and bounding the generalization error in classification (2005)
Koltchinskii, Vladimir, Panchenko, Dmitry
We introduce and study several measures of complexity of functions from the convex hull of a given base class. These complexity measures take into account the sparsity of the weights of a convex...
Weighted uniform consistency of kernel density estimators (2004)
Gine, Evarist, Koltchinskii, Vladimir, Zinn, Joel
Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let \Psi(t) be a positive continuous function such that \|\Psi f^{\beta}\|_{\infty}
Weighted uniform consistency of kernel density estimators (2004)
Giné, Evarist, Koltchinskii, Vladimir, Zinn, Joel
Let fn denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let Ψ(t) be a positive continuous function such that ‖Ψfβ‖∞
Koltchinskii, Vladimir, Panchenko, Dmitry, Lozano, Fernando
A problem of bounding the generalization error of a classifier f in H, where H is a "base" class of functions (classifiers), is considered. This problem frequently occurs in computer learning, where...
Complexities of convex combinations and bounding the generalization error in classification (2004)
Koltchinskii, Vladimir, Panchenko, Dmitry
We introduce and study several measures of complexity of functions from the convex hull of a given base class. These complexity measures take into account the sparsity of the weights of a convex...
Rademacher processes and bounding the risk of function learning (2004)
Koltchinskii, Vladimir, Panchenko, Dmitry
We construct data dependent bounds on the risk in function learning problems. The bounds are based on the local norms of the Rademacher process indexed by the underlying function class and they do...
Some Local Measures of Complexity of Convex Hulls and Generalization Bounds (2004)
Bousquet, Olivier, Koltchinskii, Vladimir, Panchenko, Dmitry
We investigate measures of complexity of function classes based on continuity moduli of Gaussian and Rademacher processes. For Gaussian processes, we obtain bounds on the continuity modulus on the...
Empirical margin distributions and bounding the generalization error of combined classifiers (2004)
Koltchinskii, Vladimir, Panchenko, Dmitry
We prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers. Such combinations could be implemented by neural networks or by...
Three papers on boosting: an introduction (2004)
Koltchinskii, Vladimir, Yu, Bin
The notion of boosting originated in the Machine Learning literature in the 1980's [VALIANT, L.G. (1984). A theory of the learnable. In Proc. 16th Annual ACM Symposium on Theory of Computing 436-445....
Discussions of boosting papers, and rejoinders (2004)
Bartlett, Peter L., Bickel, Peter J., Bühlmann, Peter, Freund, Yoav, Friedman, Jerome, Hastie, Trevor, ...
Discussions of: "Process consistency for AdaBoost" [Ann. Statist. 32 (2004), no. 1, 13-29] by W. Jiang; "On the Bayes-risk consistency of regularized boosting methods" [ibid., 30-55] by G. Lugosi and...
A note on the richness of convex hulls of VC classes (2003)
Lugosi, Gàbor; Pompeu Fabra University, Spain; Lugosi@upf.es, Mendelson, Shahar; The Australian National University, Australia; Shahar.mendelson@anu.edu.au, Koltchinskii, Vladimir; The University Of New Mexico, USA; Vlad@math.unm.edu
We prove the existence of a class A of subsets of Rd of VC dimension 1 such that the symmetric convex hull F of the class of characteristic functions of sets in A is rich in the following sense. For...
A note on the asymptotic distribution of Berk-Jones type statistics under the null hypothesis (2003)
Jon A. Wellner, Vladimir Koltchinskii
Proofs are given of the limiting null distributions of the statistics of Berk and Jones (1979) and of Einmahl and McKeague (2002). 1.
Ratio Limit Theorems for Empirical Processes (2003)
Evarist Gine, Vladimir Koltchinskii, Jon A. Wellner
Concentration inequalities are used to derive some new inequalities for ratio-type suprema of empirical processes. These general inequalities are used to prove several new limit theorems for...
Koltchinskii, Vladimir, Panchenko, Dmitriy, Lozano, Fernando
A problem of bounding the generalization error of a classifier %\break $f\in \conv(\mathcal{H})$, where $\mathcal{H}$ is a "base" class of functions (classifiers), is considered. This problem...
Some Local Measures of Complexity of Convex Hulls and Generalization Bounds (2002)
Olivier Bousquet, Vladimir Koltchinskii
We investigate measures of complexity of function classes based on continuity moduli of Gaussian and Rademacher processes. For Gaussian processes, we obtain bounds on the continuity modulus on the...
Bounds on Margin Distributions in Learning Problems (2002)
Soient (S, 4, P) un espace probabilis et Pn la mesure empirique supporte par l'chantillon (X,... ,Xn) de n variables alatoires i.i.d. tires selon P. Soit r une classe de fonctions k valeurs relles,...
Some Local Measures of Complexity of Convex Hulls and Generalization Bounds (2002)
Olivier Bousquet, Vladimir Koltchinskii, Dmitriy Panchenko
We investigate measures of complexity of function classes based on continuity moduli of Gaussian and Rademacher processes. For Gaussian processes, we obtain bounds on the continuity modulus on the...
Vladimir Koltchinskii, Dmitriy Panchenko, O Lozano
In this paper we present new bounds on the generalization error of a classifier f constructed as a convex combination of base classifiers from the class H. The algorithms of combining simple...
Some New Bounds on the Generalization Error of (2001)
Vladimir Koltchinskii, Dmitriy Panchenko, Fernando Lozano
In this paper we develop the method of bounding the generalization error of a classifier in terms of its margin distribution which was introduced in the recent papers of Bartlett and Schapire,...
Some New Bounds on the Generalization Error of Combined Classifiers. (2000)
Vladimir Koltchinskii, Dmitriy Panchenko, Fernando Lozano
In this paper we develop the method of bounding the generalization error of a classifier in terms of its margin distribution which was introduced in the recent papers of Bartlett and Schapire,...
Vladimir Koltchinskii, Dmitriy Panchenko
A problem of bounding the generalization error of a classifier f 2 conv(H); where H is a "base" class of functions (classifiers), is considered. This problem frequently occurs in computer learning,...
Bounding The Generalization Error Of Neural Networks And Combined Classifiers (2000)
Vladimir Koltchinskii, Dmitry Panchenko
Recently, several authors developed a new approach to bounding the generalization error of complex classifiers (of large or even infinite VC-dimension) obtained by combining simpler classifiers. The...
Rademacher Penalties And Structural Risk Minimization (2000)
this paper, we suggest a data based penalty, de#ned by ##n; N#:=R n #C N #;where #1:8# Rn #C# := sup
Random matrix approximation of spectra of integral operators (2000)
Koltchinskii, Vladimir, Giné, Evarist
Let$H:L_2(S,{\cal S},P) \rightarrow L_2(S,{\cal S},P)$ be a compact integral operator with a symmetric kernel h. Let${X_i,\ i\in\N}$ , be independent S-valued random variables with common probability...