• Journal of the Korean Mathematical Society
• /
• v.59 no.2
• /
• pp.217-233
• /
• 2022

Although machine learning shows state-of-the-art performance in a variety of fields, it is short a theoretical understanding of how machine learning works. Recently, theoretical approaches are actively being studied, and there are results for one of them, margin and its distribution. In this paper, especially we focused on the role of margin in the perturbations of inputs and parameters. We show a generalization bound for two cases, a linear model for binary classification and neural networks for multi-classification, when the inputs have normal distributed random noises. The additional generalization term caused by random noises is related to margin and exponentially inversely proportional to the noise level for binary classification. And in neural networks, the additional generalization term depends on (input dimension) × (norms of input and weights). For these results, we used the PAC-Bayesian framework. This paper is considering random noises and margin together, and it will be helpful to a better understanding of model sensitivity and the construction of robust generalization.

• The Korean Journal of Applied Statistics
• /
• v.17 no.1
• /
• pp.75-88
• /
• 2004

The Support Vector Machine(SVM) algorithm has paid attention on maximizing the shortest distance between sample points and discrimination hyperplane. This paper suggests the total margin algorithm which considers the distance between all data points and the separating hyperplane. The method extends existing support vector machine algorithm. In addition, this newly proposed method improves the generalization error bound. Numerical experiments show that the total margin algorithm provides good performance, comparing with the previous methods.

• Journal of the Chungcheong Mathematical Society
• /
• v.21 no.2
• /
• pp.223-230
• /
• 2008

In this paper, we present an upper bound of the reciprocal sums of generalized subset-sum-distinct sequences with respect to the first terms of the sequences. And we show the suggested upper bound is best possible. This is a kind of generalization of [1] which contains similar result for classical subset-sum-distinct sequences.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.4
• /
• pp.823-826
• /
• 2010

Let p be a prime. We generalized Sturm's bound to a group ${\Gamma}_{0}^{+}(p)$ of which the genus is zero.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.1
• /
• pp.7-10
• /
• 2009

With the introduction of a new parameter $n{\geq}1$, Kim generalized an upper bound for the exponential function that implies the inequality between the arithmetic and geometric means. By a change of variable, this generalization is equivalent to exp $(\frac{n(x-1)}{n+x-1})\;\leq\;\frac{n-1+x^n}{n}$ for real ${n}\;{\geq}\;1$ and x > 0. In this paper, we show that this inequality is true for real x > 1 - n provided that n is an even integer.

• Korean Journal of Mathematics
• /
• v.12 no.2
• /
• pp.117-125
• /
• 2004

We investigate some properties of functions of ${\Phi}{\Lambda}$-bound variation on a closed bounded interval, which is the generalization of bounded variation.

• Journal of the military operations research society of Korea
• /
• v.9 no.1
• /
• pp.75-85
• /
• 1983

This study is concerned with the problem of routing vehicles stationed at a central depot to supply customers with known demands, in such a way as to minimize the total distance travelled. The problem is referred to as the vehicle routing problem and is a generalization of the multiple traveling salesmen problem that has many practical applications. A branch-and-bound algorithm for the exact solution of the vehicle routing problem is presented. The algorithm finds the optimal number of vehicles as well as the minimum distance routes. A numerical example is given.

• Journal of the Korean Data and Information Science Society
• /
• v.19 no.1
• /
• pp.25-36
• /
• 2008

Error Correcting Output Codes (ECOC) can improve generalization performance when applied to multicategory classification problem. In this study we propose a new criterion to select hyperparameters included in ECOC scheme. Instead of margins of a data we propose to use the probability of misclassification error since it makes the criterion simple. Using this we obtain an upper bound of leave-one-out error of OVA(one vs all) method. Our experiments from real and synthetic data indicate that the bound leads to good estimates of parameters.

• Journal of Korean Society of Industrial and Systems Engineering
• /
• v.13 no.22
• /
• pp.65-69
• /
• 1990

Two different algorithms for traveling salesman problem(TSP) will be discussed. One is the engineering approach to the TSP. The other one is Branch-and-Bound algorithm to take advantage of the special structure of combinational problems. Also a generalization of TSP will be presented.

• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 1997.10a
• /
• pp.52-55
• /
• 1997

In this paper, both the generalization of one artificial variable technique to the general bound problem and the efficient implementation of the technique are suggested. When the steepest-edge method is used as a pricing rule in the simplex method, it is easy to update the reduced cost and the simplex multiplier every iteration. Therefore, one artificial variable technique is more efficient than Wolfe's method in which the reduced cost and simplex multiplier must be recalculated in every iteration. When implementing the one artificial variable technique on the LP problems with the general bound restraints on the variables, an arbitrary basic solution which satisfies the bound restraints is sought first, and the artificial column which adjusts the infeasibility is introduced. The phase one of the simplex method minimizes the one artificial variable. The efficient implementation technique includes the splitting, scaling, storage of the artificial column, and the cure of infeasibility problem.