• 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.

• IEIE Transactions on Smart Processing and Computing
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• v.4 no.6
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• pp.413-421
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• 2015

Regularized Blind Deconvolution is a problem applicable in degraded images in order to bring the original image out of blur. Multichannel blind Deconvolution considered as an optimization problem. Each step in the optimization is considered as variable splitting problem using an algorithm called Alternating Minimization Algorithm. Each Step in the Variable splitting undergoes Augmented Lagrangian method (ALM) / Bregman Iterative method. Regularization is used where an ill posed problem converted into a well posed problem. Two well known regularizers are Tikhonov class and Total Variation (TV) / L2 model. TV can be isotropic and anisotropic, where isotropic for L2 norm and anisotropic for L1 norm. Based on many probabilistic model and Fourier Transforms Image deblurring can be solved. Here in this paper to improve the performance, we have used an adaptive regularization filtering and isotropic TV model Lp norm. Image deblurring is applicable in the areas such as medical image sensing, astrophotography, traffic signal monitoring, remote sensors, case investigation and even images that are taken using a digital camera / mobile cameras.

• 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.

• 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.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.487-494
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• 2019

We classify all the extreme and exposed bilinear forms of the unit ball of ${\mathcal{L}}(^2l^3_{\infty})$ which leads to a complete formula of ${\parallel}f{\parallel}$ for every $f{\in}{\mathcal{L}}(^2l^3_{\infty})^*$. It follows from this formula that every extreme bilinear form of the unit ball of ${\mathcal{L}}(^2l^3_{\infty})$ is exposed.

• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.231-241
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• 2013

On the framework of the 2-adic group $\mathcal{Z}_2$, we study a Sobolev-like inequality where we estimate the $L^2$ norm by a geometric mean of the BV norm and the $\dot{B}_{\infty}^{-1,{\infty}}$ norm. We first show, using the special topological properties of the $p$-adic groups, that the set of functions of bounded variations BV can be identified to the Besov space ˙$\dot{B}_1^{1,{\infty}}$. This identification lead us to the construction of a counterexample to the improved Sobolev inequality.

• 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.

• 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.

• 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 $\{{\alpha}_{\kappa}\}{\in}{\Gamma}$.

• The Mathematical Education
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• v.22 no.2
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• pp.1-4
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• 1984