11月17日 Charles Micchelli教授学术报告

发布时间:2017-11-06

报告人：Professor Charles A. Micchelli
    Department of Mathematics & Statistics
    University at Albany, SUNY

时 间： 11月17日（周五） 上午10：00-12：00
    下午 3：00- 5：00

地 点：南校区336栋旧数学楼210

Lecture 1. On discrete least squares for multivariate kernels.

For a positive integer m we introduce an index set I of carnality m and an index set J of carnality n. Corresponding to these two sets we prescribe two distinct sets of vectors X of carnality n and Y of carnality m and a prescribed data sets of real numbers Z. Furthermore, we have a kernel K, defined on XxY which yields a linear combinations of the kernel evaluated at X which is defined on some open neighborhood of Y. We consider the multivariate discrete squares problem and provide some information about what happens when the components of the vectors X and Y are close to an arbitrary point s.

Lecture 2. Fixed point proximity algorithm for minimal norm interpolation.

Our goal in this paper is to address the following problem: from an unknown matrix X we are given inner products of this matrix with known prescribed matrices and wish to find X. This is the problem we shall consider in this paper by using the notion of minimal norm interpolation.

Lecture 3. On univariate functions with piecewise constant spectra.

In this paper, we consider univariate functions whose Fourier transform are piecewise constant functions. Such functions include' for example, the celebrated sinc function. Our goal is to provide detailed information about them when the breakpoints are periodic. In that case, we obtain various properties of these functions including stability, orthogonality, interpolation, refinability, biorthogonality and tight frames. This rather comprehensive list of desirable properties goes way beyond what was previously available for such functions. The success of our program relies upon material on periodic wavelet analysis which we readily make use of. The approach produces, among others, new refinable and wavelet functions with piecewise constant spectra that generate bi orthogonality wavelet bases and tight wavelet frames.

Lecture 4. Learning with optimal interpolation norms.

We analyze a class of norms defined via optimal interpolation problems involving the composition of norms and a linear operator. The construction, known as infimal post composition in convex analysis, is shown to encompass various norms which have been used as regularizers in machine learning, signal processing and statistics.

In particular, these include the latent group lasso, the overlapping group lasso, and certain norms used for learning tensors. We establish basic properties of this class of norms and we provide dual norms. The extension to more general classes of convex functions is also discussed. A stochastic block-coordinate version of the Douglas-Rachford algorithm is devised to solve the minimization problems involving these regularizers. A prominent feature of these algorithms is that it yields iterates that converge to a solution in the in the case of non smooth losses and a random block updates. Finally, we present numerical experiments with problems employing the latent group lasso penalty.

报告人简介：
Charles A. Micchelli教授是国际著名逼近论专家，他曾于1983年在国际数学家大会作45分钟的报告。Charles A. Micchelli曾在IBMT.J. Watson Research Center 工作，现任纽约州立大学奥尔巴尼分校数学与统计系教授。