/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD).We present two distribution free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

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A kernel two-sample test

On robust estimation of the location parameter

1 Introduction to Wavelets 2 A Multiresolution Formulation of Wavelet Systems 3 Filter Banks and the Discrete Wavelet Transform 4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases 5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients 6 Regularity, Moments, and Wavelet System Design 7 Generalizations of the Basic Multiresolution Wavelet System 8 Filter Banks and Transmultiplexers 9 Calculation of the Discrete Wavelet Transform 10 Wavelet-Based Signal Processing and Applications 11 Summary Overview 12 References Bibliography Appendix A Derivations for Chapter 5 on Scaling Functions Appendix B Derivations for Section on Properties Appendix C Matlab Programs Index

Introduction to Wavelets and Wavelet Transforms: A Primer

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

We propose a collocation method for solving integral equations which model image restoration from out-of-focus images. Restoration of images from out-of-focus images can be formulated as an integral equation of the first kind, which is an ill-posed problem. We employ the Tikhonov regularization to treat the ill-posedness and obtain results of a well-posed second kind integral equation whose integral operator is the square of the original operator. The present of the square of the integral operator requires high computational cost to solve the equation. To overcome this difficulty, we convert the resulting second kind integral equation into an equivalent system of integral equations which do not involve the square of the integral operator. A multiscale collocation method is then applied to solve the system. A truncation strategy for the matrices appearing in the resulting discrete linear system is proposed to design a fast numerical solver for the system of integral equations. A quadrature method is used to compute the entries of the resulting matrices. We estimate the computational cost of the numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method for image restoration.

/pdf/a-collocation-method-solving-integral-equation-models-for-10eah6lhjs.pdf

A collocation method solving integral equation models for image restoration

Numerical solutions for weakly singular Fredholm integral equations of the second kind

An approximate sparsity model for inpainting

Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering This is often formulated as a minimum norm interpolation problem, regularized learning problem or, in general, a semi-discrete inverse problem, in certain functional spaces The choice of an appropriate space is crucial for solutions of these problems Motivated by sparse representations of the reconstructed functions such as compressed sensing and sparse learning, much of the recent research interest has been directed to considering these problems in certain Banach spaces in order to obtain their sparse solutions, which is a feasible approach to overcome challenges coming from the big data nature of most practical applications It is the goal of this paper to provide a systematic study of the representer theorems for these problems in Banach spaces There are a few existing results for these problems in a Banach space, with all of them regarding implicit representer theorems We aim at obtaining explicit representer theorems based on which convenient solution methods will then be developed For the minimum norm interpolation, the explicit representer theorems enable us to express the infimum in terms of the norm of the linear combination of the interpolation functionals For the purpose of developing efficient computational algorithms, we establish the fixed-point equation formulation of solutions of these problems We reveal that unlike in a Hilbert space, in general, solutions of these problems in a Banach space may not be able to be reduced to truly finite dimensional problems (with certain infinite dimensional components hidden) We demonstrate how this obstacle can be removed, reducing the original problem to a truly finite dimensional one, in the special case when the Banach space is $\ell_1(\mathbb{N})$

Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

It was observed in [4] that the Hilbert transform of the univariate B-spline preserves the B-spline recurrence. Motivated by this observation, we characterize translation invariant operators that preserve the multivariate B-spline recurrence and analogous results are also provided for the multivariate cube spline.

Yuesheng Xu

Papers

A collocation method solving integral equation models for image restoration

Numerical solutions for weakly singular Fredholm integral equations of the second kind

An approximate sparsity model for inpainting

Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

On translation invariant operators which preserve the B-spline recurrence