6,327 research outputs found
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
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