The Geometry of Nonlinear Embeddings in Kernel Discriminant Analysis

Fisher's linear discriminant analysis is a classical method for classification, yet it is limited to capturing linear features only. Kernel discriminant analysis as an extension is known to successfully alleviate the limitation through a nonlinear feature mapping. We study the geometry of nonlinear embeddings in discriminant analysis with polynomial kernels and Gaussian kernel by identifying the population-level discriminant function that depends on the data distribution and the kernel. In order to obtain the discriminant function, we solve a generalized eigenvalue problem with between-class and within-class covariance operators. The polynomial discriminants are shown to capture the class difference through the population moments explicitly. For approximation of the Gaussian discriminant, we use a particular representation of the Gaussian kernel by utilizing the exponential generating function for Hermite polynomials. We also show that the Gaussian discriminant can be approximated using randomized projections of the data. Our results illuminate how the data distribution and the kernel interact in determination of the nonlinear embedding for discrimination, and provide a guideline for choice of the kernel and its parameters

Efficient quantile regression for heteroscedastic models

Quantile regression (QR) provides estimates of a range of conditional quantiles. This stands in contrast to traditional regression techniques, which focus on a single conditional mean function. Lee et al. [Regularization of case-specific parameters for robustness and efficiency. Statist Sci. 2012;27(3):350–372] proposed efficient QR by rounding the sharp corner of the loss. The main modification generally involves an asymmetric ℓ₂ adjustment of the loss function around zero. We extend the idea of ℓ₂ adjusted QR to linear heterogeneous models. The ℓ₂ adjustment is constructed to diminish as sample size grows. Conditions to retain consistency properties are also provided

Bayesian Restricted Likelihood Methods: Conditioning on Insufficient Statistics in Bayesian Regression

Bayesian methods have proven themselves to be successful across a wide range of scientific problems and have many well-documented advantages over competing methods. However, these methods run into difficulties for two major and prevalent classes of problems: handling data sets with outliers and dealing with model misspecification. We outline the drawbacks of previous solutions to both of these problems and propose a new method as an alternative. When working with the new method, the data is summarized through a set of insufficient statistics, targeting inferential quantities of interest, and the prior distribution is updated with the summary statistics rather than the complete data. By careful choice of conditioning statistics, we retain the main benefits of Bayesian methods while reducing the sensitivity of the analysis to features of the data not captured by the conditioning statistics. For reducing sensitivity to outliers, classical robust estimators (e.g., M-estimators) are natural choices for conditioning statistics. A major contribution of this work is the development of a data augmented Markov chain Monte Carlo (MCMC) algorithm for the linear model and a large class of summary statistics. We demonstrate the method on simulated and real data sets containing outliers and subject to model misspecification. Success is manifested in better predictive performance for data points of interest as compared to competing methods

Optical Torque from Enhanced Scattering by Multipolar Plasmonic Resonance

We present a theoretical study of the optical angular momentum transfer from a circularly polarized plane wave to thin metal nanoparticles of different rotational symmetries. While absorption has been regarded as the predominant mechanism of torque generation on the nanoscale, we demonstrate numerically how the contribution from scattering can be enhanced by using multipolar plasmon resonance. The multipolar modes in non-circular particles can convert the angular momentum carried by the scattered field, thereby producing scattering-dominant optical torque, while a circularly symmetric particle cannot. Our results show that the optical torque induced by resonant scattering can contribute to 80% of the total optical torque in gold particles. This scattering-dominant torque generation is extremely mode-specific, and deserves to be distinguished from the absorption-dominant mechanism. Our findings might have applications in optical manipulation on the nanoscale as well as new designs in plasmonics and metamaterials.Comment: main article 20 pages, 4 figures; supplementary material 6 pages, 2 figure

Predictive Model Degrees of Freedom in Linear Regression

Overparametrized interpolating models have drawn increasing attention from machine learning. Some recent studies suggest that regularized interpolating models can generalize well. This phenomenon seemingly contradicts the conventional wisdom that interpolation tends to overfit the data and performs poorly on test data. Further, it appears to defy the bias-variance trade-off. As one of the shortcomings of the existing theory, the classical notion of model degrees of freedom fails to explain the intrinsic difference among the interpolating models since it focuses on estimation of in-sample prediction error. This motivates an alternative measure of model complexity which can differentiate those interpolating models and take different test points into account. In particular, we propose a measure with a proper adjustment based on the squared covariance between the predictions and observations. Our analysis with least squares method reveals some interesting properties of the measure, which can reconcile the "double descent" phenomenon with the classical theory. This opens doors to an extended definition of model degrees of freedom in modern predictive settings.Comment: 47 pages, 18 figure

A Dominant Complement Fixation Pathway for Pneumococcal Polysaccharides Initiated by SIGN-R1 Interacting with C1q

The intricate system of serum complement proteins provides resistance to infection. A pivotal step in the complement pathway is the assembly of a C3 convertase, which digests the C3 complement component to form microbial binding C3 fragments recognized by leukocytes. The spleen and C3 provide resistance against blood-borne S. pneumoniae infection. To better understand the mechanisms involved, we studied SIGN-R1, a lectin that captures microbial polysaccharides in spleen. Surprisingly, conditional SIGN-R1 knockout mice developed deficits in C3 catabolism when given S. pneumoniae or its capsular polysaccharide intravenously. There were marked reductions in proteolysis of serum C3, deposition of C3 on organisms within SIGN-R1+ spleen macrophages, and formation of C3 ligands. We found that SIGN-R1 directly bound the complement C1 subcomponent, C1q, and assembled a C3 convertase, but without the traditional requirement for either antibody or factor B. The transmembrane lectin SIGN-R1 therefore contributes to innate resistance by an unusual C3 activation pathway

Perspectives of Apparel Sustainability Among Design Students from Different Cultural Backgrounds

Designers need to understand the breadth of strategies for developing more sustainable solutions (Shedroff, 2009). Design students’ concepts of sustainability are different depending on their cultural contexts and approaches to sustainability issues that universities have taken that vary across cultures. In this research, we examine how American and Korean design students perceive the importance of sustainability regarding their apparel as a user and professional and how they strategize sustainable practices. This cross-cultural comparison of American and Korean students perceptions of sustainability can provide valuable insight regarding how cultural factors shape and influence a group’s sustainable behavior and can help the instructor who wants to educate students as broadly as possible