Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization'' by V. Koltchinskii

Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization'' by V. Koltchinskii [arXiv:0708.0083]Comment: Published at http://dx.doi.org/10.1214/009053606000001055 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Generalization error for multi-class margin classification

In this article, we study rates of convergence of the generalization error of multi-class margin classifiers. In particular, we develop an upper bound theory quantifying the generalization error of various large margin classifiers. The theory permits a treatment of general margin losses, convex or nonconvex, in presence or absence of a dominating class. Three main results are established. First, for any fixed margin loss, there may be a trade-off between the ideal and actual generalization performances with respect to the choice of the class of candidate decision functions, which is governed by the trade-off between the approximation and estimation errors. In fact, different margin losses lead to different ideal or actual performances in specific cases. Second, we demonstrate, in a problem of linear learning, that the convergence rate can be arbitrarily fast in the sample size $n$ depending on the joint distribution of the input/output pair. This goes beyond the anticipated rate $O(n^{-1})$. Third, we establish rates of convergence of several margin classifiers in feature selection with the number of candidate variables $p$ allowed to greatly exceed the sample size $n$ but no faster than $\exp(n)$.Comment: Published at http://dx.doi.org/10.1214/07-EJS069 in the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Conversions between barycentric, RKFUN, and Newton representations of rational interpolants

We derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm [Y. Nakatsukasa, O. S\`ete, L. N. Trefethen, arXiv preprint 1612.00337, 2016] within the Rational Krylov Toolbox and for the solution of nonlinear eigenvalue problems

Perturbation-Assisted Sample Synthesis: A Novel Approach for Uncertainty Quantification

This paper introduces a novel generator called Perturbation-Assisted Sample Synthesis (PASS), designed for drawing reliable conclusions from complex data, especially when using advanced modeling techniques like deep neural networks. PASS utilizes perturbation to generate synthetic data that closely mirrors the distribution of raw data, encompassing numerical and unstructured data types such as gene expression, images, and text. By estimating the data-generating distribution and leveraging large pre-trained generative models, PASS enhances estimation accuracy, providing an estimated distribution of any statistic through Monte Carlo experiments. Building on PASS, we propose a generative inference framework called Perturbation-Assisted Inference (PAI), which offers a statistical guarantee of validity. In pivotal inference, PAI enables accurate conclusions without knowing a pivotal's distribution as in simulations, even with limited data. In non-pivotal situations, we train PASS using an independent holdout sample, resulting in credible conclusions. To showcase PAI's capability in tackling complex problems, we highlight its applications in three domains: image synthesis inference, sentiment word inference, and multimodal inference via stable diffusion

Network-based support vector machine for classification of microarray samples

Background: The importance of network-based approach to identifying biological markers for diagnostic classification and prognostic assessment in the context of microarray data has been increasingly recognized. To our knowledge, there have been few, if any, statistical tools that explicitly incorporate the prior information of gene networks into classifier building. The main idea of this paper is to take full advantage of the biological observation that neighboring genes in a network tend to function together in biological processes and to embed this information into a formal statistical framework. Results: We propose a network-based support vecto