Sparsifying the Fisher Linear Discriminant by Rotation

Many high dimensional classification techniques have been proposed in the literature based on sparse linear discriminant analysis (LDA). To efficiently use them, sparsity of linear classifiers is a prerequisite. However, this might not be readily available in many applications, and rotations of data are required to create the needed sparsity. In this paper, we propose a family of rotations to create the required sparsity. The basic idea is to use the principal components of the sample covariance matrix of the pooled samples and its variants to rotate the data first and to then apply an existing high dimensional classifier. This rotate-and-solve procedure can be combined with any existing classifiers, and is robust against the sparsity level of the true model. We show that these rotations do create the sparsity needed for high dimensional classifications and provide theoretical understanding why such a rotation works empirically. The effectiveness of the proposed method is demonstrated by a number of simulated and real data examples, and the improvements of our method over some popular high dimensional classification rules are clearly shown.Comment: 30 pages and 9 figures. This paper has been accepted by Journal of the Royal Statistical Society: Series B (Statistical Methodology). The first two versions of this paper were uploaded to Bin Dong's web site under the title "A Rotate-and-Solve Procedure for Classification" in 2013 May and 2014 January. This version may be slightly different from the published versio

Variance Estimation Using Refitted Cross-validation in Ultrahigh Dimensional Regression

Variance estimation is a fundamental problem in statistical modeling. In ultrahigh dimensional linear regressions where the dimensionality is much larger than sample size, traditional variance estimation techniques are not applicable. Recent advances on variable selection in ultrahigh dimensional linear regressions make this problem accessible. One of the major problems in ultrahigh dimensional regression is the high spurious correlation between the unobserved realized noise and some of the predictors. As a result, the realized noises are actually predicted when extra irrelevant variables are selected, leading to serious underestimate of the noise level. In this paper, we propose a two-stage refitted procedure via a data splitting technique, called refitted cross-validation (RCV), to attenuate the influence of irrelevant variables with high spurious correlations. Our asymptotic results show that the resulting procedure performs as well as the oracle estimator, which knows in advance the mean regression function. The simulation studies lend further support to our theoretical claims. The naive two-stage estimator which fits the selected variables in the first stage and the plug-in one stage estimators using LASSO and SCAD are also studied and compared. Their performances can be improved by the proposed RCV method

Fabrication and room temperature operation of semiconductor nano-ring lasers using a general applicable membrane transfer method

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Appl. Phys. Lett. 110, 171105 (2017) and may be found at https://doi.org/10.1063/1.4982621.Semiconductor nanolasers are potentially important for many applications. Their design and fabrication are still in the early stage of research and face many challenges. In this paper, we demonstrate a generally applicable membrane transfer method to release and transfer a strain-balanced InGaAs quantum-well nanomembrane of 260 nm in thickness onto various substrates with a high yield. As an initial device demonstration, nano-ring lasers of 1.5 μm in outer diameter and 500 nm in radial thickness are fabricated on MgF2 substrates. Room temperature single mode operation is achieved under optical pumping with a cavity volume of only 0.43λ03 (λ0 in vacuum). Our nano-membrane based approach represents an advantageous alternative to other design and fabrication approaches and could lead to integration of nanolasers on silicon substrates or with metallic cavity