• Communications for Statistical Applications and Methods
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• v.31 no.2
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• pp.203-212
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• 2024

The area under the ROC curve (AUC) is one of the most common criteria used to measure the overall performance of binary classifiers for a wide range of machine learning problems. In this article, we propose a L₁-penalized AUC-optimization classifier that directly maximizes the AUC for high-dimensional data. Toward this, we employ the AUC-consistent surrogate loss function and combine the L₁-norm penalty which enables us to estimate coefficients and select informative variables simultaneously. In addition, we develop an efficient optimization algorithm by adopting k-means clustering and proximal gradient descent which enjoys computational advantages to obtain solutions for the proposed method. Numerical simulation studies demonstrate that the proposed method shows promising performance in terms of prediction accuracy, variable selectivity, and computational costs.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.781-791
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• 2016

Let $d_*(1,w)^2 ={\mathbb{R}}^2$ with the octagonal norm of weight w. It is the two dimensional real predual of Lorentz sequence space. In this paper we classify the smooth points of the unit ball of the space of symmetric bilinear forms on $d_*(1,w)^2$. We also show that the unit sphere of the space of symmetric bilinear forms on $d_*(1,w)^2$ is the disjoint union of the sets of smooth points, extreme points and the set A as follows: $$S_{{\mathcal{L}}_s(^2d_*(1,w)^2)}=smB_{{\mathcal{L}}_s(^2d_*(1,w)^2)}{\bigcup}extB_{{\mathcal{L}}_s(^2d_*(1,w)^2)}{\bigcup}A$$, where the set A consists of $ax_1x_2+by_1y_2+c(x_1y_2+x_2y_1)$ with (a = b = 0, $c={\pm}{\frac{1}{1+w^2}}$), ($a{\neq}b$, $ab{\geq}0$, c = 0), (a = b, 0 < ac, 0 < ${\mid}c{\mid}$ < ${\mid}a{\mid}$), ($a{\neq}{\mid}c{\mid}$, a = -b, 0 < ac, 0 < ${\mid}c{\mid}$), ($a={\frac{1-w}{1+w}}$, b = 0, $c={\frac{1}{1+w}}$), ($a={\frac{1+w+w(w^2-3)c}{1+w^2}}$, $b={\frac{w-1+(1-3w^2)c}{w(1+w^2)}}$, ${\frac{1}{2+2w}}$ < c < ${\frac{1}{(1+w)^2(1-w)}}$, $c{\neq}{\frac{1}{1+2w-w^2}}$), ($a={\frac{1+w(1+w)c}{1+w}}$, $b={\frac{-1+(1+w)c}{w(1+w)}}$, 0 < c < $\frac{1}{2+2w}$) or ($a={\frac{1=w(1+w)c}{1+w}}$, $b={\frac{1-(1+w)c}{1+w}}$, $\frac{1}{1+w}$ < c < $\frac{1}{(1+w)^2(1-w)}$).

• Journal of Integrative Natural Science
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• v.1 no.3
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• pp.216-220
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• 2008

In general the Banach space $L^1(T^1)$ doesn't admit convergence in norm. Thus the convergence in norm of the partial sums can not be characterized in terms of Fourier coefficients without additional assumptions about the sequence$\{^{\^}f(\xi)\}$. The problem of $L^1(T^1)$-convergence consists of finding the properties of Fourier coefficients such that the necessary and sufficient condition for (1.2) and (1.3). This paper showed that let $\{{\alpha}_{\kappa}\}{\in}BV$ and ${\xi}{\Delta}a_{\xi}=o(1),\;{\xi}{\rightarrow}{\infty}$. Then (1.1) is a Fourier series if and only if $\{{\alpha}_{\kappa}\}{\in}{\Gamma}$.

• Korean Journal of Remote Sensing
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• v.33 no.5_1
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• pp.455-467
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• 2017

When performing target detection using hyperspectral imagery, a feature extraction process is necessary to solve the problem of redundancy of adjacent spectral bands and the problem of a large amount of calculation due to high dimensional data. This study proposes a new band selection method using the $L_{2,1}$-norm regression model to apply the feature selection technique in the machine learning field to the hyperspectral band selection. In order to analyze the performance of the proposed band selection technique, we collected the hyperspectral imagery and these were used to analyze the performance of target detection with band selection. The Adaptive Cosine Estimator (ACE) detection performance is maintained or improved when the number of bands is reduced from 164 to about 30 to 40 bands in the 350 nm to 2500 nm wavelength band. Experimental results show that the proposed band selection technique extracts bands that are effective for detection in hyperspectral images and can reduce the size of the data without reducing the performance, which can help improve the processing speed of real-time target detection system in the future.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.2
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• pp.25-37
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• 2001

We consider the finite element method applied to nonlinear Sobolev equation with smooth data and demonstrate for arbitrary order ($k{\geq}2$) finite element spaces the optimal rate of convergence in $L_{\infty}\;W^{1,{\infty}}({\Omega})$ and $L_{\infty}(L_{\infty}({\Omega}))$ (quasi-optimal for k = 1). In other words, the nonlinear Sobolev equation can be approximated equally well as its linear counterpart. Furthermore, we also obtain superconvergence results in $L_{\infty}(W^{1,{\infty}}({\Omega}))$ for the difference between the approximate solution and the generalized elliptic projection of the exact solution.

• Geophysics and Geophysical Exploration
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• v.19 no.4
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• pp.220-227
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• 2016

Marine seismic data have not only primary signals from subsurface but also ghost signals reflected from the sea surface. The ghost decreases temporal resolution of seismic data because it attenuates specific frequency components. For eliminating the ghost signals effectively, the exact ghost delaytimes and reflection coefficients are required. Because of undulation of the sea surface and vertical movements of airguns and streamers, the ghost delaytime varies spatially and randomly while acquiring seismic data. The reflection coefficient is a function of frequency, incidence angle of plane-wave and the sea state. In order to estimate the proper ghost delaytimes considering these characteristics, we compared the ghost delaytimes estimated with L-1 norm, L-2 norm and kurtosis of the deghosted trace and its autocorrelation on synthetic data. L-1 norm of autocorrelation showed a minimal error and the reflection coefficient was calculated using Kirchhoff approximation equation which can handle the effect of wave height. We applied the estimated ghost delaytimes and the calculated reflection coefficients to remove the source and receiver ghost effects. By removing ghost signals, we reconstructed the frequency components attenuated near the notch frequency and produced the migrated stack section with enhanced temporal resolution.

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.731-751
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• 2023

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into L^r-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.15 no.8
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• pp.3086-3101
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• 2021

To supply precise marketing and differentiated service for the electric power service department, it is very important to predict the customers with high sensitivity of electric power failure. To solve this problem, we propose a novel grouped 𝑙_1/2 sparsity constrained logistic regression method for sensitivity assessment of electric power failure. Different from the 𝑙₁ norm and k-support norm, the proposed grouped 𝑙_1/2 sparsity constrained logistic regression method simultaneously imposes the inter-class information and tighter approximation to the nonconvex 𝑙₀ sparsity to exploit multiple correlated attributions for prediction. Firstly, the attributes or factors for predicting the customer sensitivity of power failure are selected from customer sheets, such as customer information, electric consuming information, electrical bill, 95598 work sheet, power failure events, etc. Secondly, all these samples with attributes are clustered into several categories, and samples in the same category are assumed to be sharing similar properties. Then, 𝑙_1/2 norm constrained logistic regression model is built to predict the customer's sensitivity of power failure. Alternating direction of multipliers (ADMM) algorithm is finally employed to solve the problem by splitting it into several sub-problems effectively. Experimental results on power electrical dataset with about one million customer data from a province validate that the proposed method has a good prediction accuracy.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.139-156
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• 2014

In this paper, we investigate the error estimates of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the order k = 1 Raviart-Thomas mixed finite element and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of order $h^{\frac{3}{2}}$ in the $L^2$-norm and order h in the $L^{\infty}$-norm for the control variable are proved.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1209-1219
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• 2018

We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky [7]: let H be the Hilbert transform and let a, b be real constants. Then for 1 < p < ${\infty}$ the norm of the operator aI + bH from $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$ is equal to $$\({\max_{x{\in}{\mathbb{R}}}}{\frac{{\mid}ax-b+(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}ax-b-(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p}{{\mid}x+{\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}x-{\tan}{\frac{\pi}{2p}}{\mid}^p}}\)^{\frac{1}{p}}$$. Our proof avoids passing through the analogous result for the conjugate function on the circle, as in [7], and is given directly on the line. We also provide new approximate extremals for aI + bH in the case p > 2.