Parametric estimation and tests through divergences and duality technique

We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composite hypotheses, extending maximum likelihood technique. An other view at the maximum likelihood approach, for estimation and test, is given. We prove existence and consistency of the proposed estimates. The limit laws of the estimates and test statistics (including the generalized likelihood ratio one) are given both under the null and the alternative hypotheses, and approximation of the power functions is deduced. A new procedure of construction of confidence regions, when the parameter may be a boundary value of the parameter space, is proposed. Also, a solution to the irregularity problem of the generalized likelihood ratio test pertaining to the number of components in a mixture is given, and a new test is proposed, based on $\chi ^{2}$-divergence on signed finite measures and duality technique

Functional kernel estimators of conditional extreme quantiles

We address the estimation of "extreme" conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed kernel estimators. A Weissman-type estimator and kernel estimators of the conditional tail-index are derived, permitting to estimate extreme conditional quantiles of arbitrary order.Comment: arXiv admin note: text overlap with arXiv:1107.226

Self-consistent method for density estimation

The estimation of a density profile from experimental data points is a challenging problem, usually tackled by plotting a histogram. Prior assumptions on the nature of the density, from its smoothness to the specification of its form, allow the design of more accurate estimation procedures, such as Maximum Likelihood. Our aim is to construct a procedure that makes no explicit assumptions, but still providing an accurate estimate of the density. We introduce the self-consistent estimate: the power spectrum of a candidate density is given, and an estimation procedure is constructed on the assumption, to be released \emph{a posteriori}, that the candidate is correct. The self-consistent estimate is defined as a prior candidate density that precisely reproduces itself. Our main result is to derive the exact expression of the self-consistent estimate for any given dataset, and to study its properties. Applications of the method require neither priors on the form of the density nor the subjective choice of parameters. A cutoff frequency, akin to a bin size or a kernel bandwidth, emerges naturally from the derivation. We apply the self-consistent estimate to artificial data generated from various distributions and show that it reaches the theoretical limit for the scaling of the square error with the dataset size.Comment: 21 pages, 5 figure

Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functions

In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-inner-products, we can obtain the explicit representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can introduce the reproduction property in a generalized native space by Fourier transform techniques such that it becomes a reproducing kernel Banach space, which can be even embedded into Sobolev spaces, and its reproducing kernel is set up by the related positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly in terms of positive definite functions, and their finite numbers of coefficients can be computed by fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Mat\'ern functions (Sobolev splines) so that their support vector machine solutions are well computable as the classical algorithms. Moreover, each of their reproducing bases includes information from multiple training data points. The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.Comment: 26 page

Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ can be isometrically embedded into or even be isometrically equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. We provide several examples, such as Mat\'ern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are isometrically equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator $\mathbf{P}$. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D. thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}

Reproducing Kernel Banach Spaces with the l1 Norm

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.Comment: 28 pages, an extra section was adde

Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces

Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributions in order to implement the kernel Bayes rule and build sequential data models. It was recently shown that transfer operators such as the Perron-Frobenius or Koopman operator can also be approximated in a similar fashion using covariance and cross-covariance operators and that eigenfunctions of these operators can be obtained by solving associated matrix eigenvalue problems. The goal of this paper is to provide a solid functional analytic foundation for the eigenvalue decomposition of RKHS operators and to extend the approach to the singular value decomposition. The results are illustrated with simple guiding examples