# An Optimized Discrete Data Classification Method inN-Dimensional

Authors: Dongyung Kim
Journal: Computational and Mathematical Methods
 (2022)

---

## Abstract

We propose a discrete data classification method of scattered data inN-dimensional by solving the minimax problem for a set of points. The current research is extended from 2-dimensional and 3-dimensional to N-dimensional. The problem can be applied to artificial intelligence classification problems (machine learning, deep learning), point data analysis problems (data science problem), the optimized design of nanoscale circuits, and the location of facility problems, circle detection on 2D image, or sphere detection on depth image. We generalized the discrete data classification methodology in N-dimensional. Finally, we resolved to find an exact solution of the location of a manifold for our suggested problem in N-dimensional.


---

## Body

## 1. Introduction

Data classification has been emphasized to developing artificial intelligence (machine learning, deep learning) performance from the big data. And from the image big data, image classification is an important part of the data classification [1, 2]. Some classification methods are support vector machine (SVM) [3], the decision tree [4], artificial neural network [5], and the naive Bayes classifier [6]. The measurement of data classification is key role of the performance of the methodology. In the measurement of data classification, we describe the minimax problem. Our constrained optimization problem is then formulated as an unconstrained minimax problem of finding X∗ such that
(1)J∞X∗=minX∈ℝNJ∞X.The organization of the paper is as follows. Section1 is the introduction. 1.1. Literature review, Section 2 states the problem statement 2.1 and Innovative Research Contribution 2.2. The algorithm and the data classification methodology to solve our problem are described in Section 3. We prove that the algorithm searches an exact minimax solution. The data classification methodology is verified. Then, in Section 4, numerical results are presented. Our methodology is validated. Conclusions are given in Section 5.


### 1.1. Literature Review

A class of similar problems to (1) has a long history of development in operation research. It goes back to Pierre de Fermat who considered the case of djX=Pj−X, n=2 and p=1 with equal weights whose solution is called Fermat point. Consider the case of djX=Pj−X, for a general n and p=1 with equal weights. Then, the problem is to find the geometric median of the set of points Pj,j=1,⋯,n, which is a standard problem in facility location to minimize the cost of transportation. The minimizer X∗ of J1X is known as the Fermat-Weber [7] point or 1-median. The Fermat-Weber problem has drawn much attention from mathematicians and facility location scientists and engineers; see, for instance [8–20] and the references therein. For Euclidean metric dx,y=x1−y12+x2−y22+x3−y32 only, see the references [12–16, 18–21], and for rectilinear (Manhattan) metric dx,y=x1−y1+x2−y2 and Euclidean metric, see the references [10, 17]. For a survey paper on the Fermat-Weber problem, see Wesolowsky [22–25].The case ofdjX=Pj−X−Q1−X, with equal weights without the constraint on the circle passing through any point was dealt by Drezner et al. [21] for p=1,2, and ∞ and that with equal and unequal weights by Brimberg et al. [26] for p=∞.For nonlinear minimax problems with successive approximation methods for finding a stationary point is also studied by Demjanov [27]. This literature is helpful for us to suggest a new minimax model with a measurement for the general N-dimensional.


## 1.1. Literature Review

A class of similar problems to (1) has a long history of development in operation research. It goes back to Pierre de Fermat who considered the case of djX=Pj−X, n=2 and p=1 with equal weights whose solution is called Fermat point. Consider the case of djX=Pj−X, for a general n and p=1 with equal weights. Then, the problem is to find the geometric median of the set of points Pj,j=1,⋯,n, which is a standard problem in facility location to minimize the cost of transportation. The minimizer X∗ of J1X is known as the Fermat-Weber [7] point or 1-median. The Fermat-Weber problem has drawn much attention from mathematicians and facility location scientists and engineers; see, for instance [8–20] and the references therein. For Euclidean metric dx,y=x1−y12+x2−y22+x3−y32 only, see the references [12–16, 18–21], and for rectilinear (Manhattan) metric dx,y=x1−y1+x2−y2 and Euclidean metric, see the references [10, 17]. For a survey paper on the Fermat-Weber problem, see Wesolowsky [22–25].The case ofdjX=Pj−X−Q1−X, with equal weights without the constraint on the circle passing through any point was dealt by Drezner et al. [21] for p=1,2, and ∞ and that with equal and unequal weights by Brimberg et al. [26] for p=∞.For nonlinear minimax problems with successive approximation methods for finding a stationary point is also studied by Demjanov [27]. This literature is helpful for us to suggest a new minimax model with a measurement for the general N-dimensional.


## 2. Problem Statement and Research Contribution

### 2.1. Problem Statement

We generalize the discrete data classification method from 2-dimensional and 3-dimensional toN-dimensional. In the N-dimensional case, let Pjx1j,x2j,⋯,xNj,j=1,⋯,n, be a given set of discrete points on the space. Additionally, suppose that two additional points Q1 and Q2 are given which are distinct from Pj,j=1,⋯,n. We are interested in the constrained optimization problem of finding a N-dimensional manifold that is closest to all discrete points Pj,j=1,⋯,n, among the manifolds that are constrained to pass through Q1 and Q2, see Figure 1 for the previous suggested problem setting in the 2-dimensional case. In 2 dimensions, the closeness of a circle to a set of discrete points is given by the weighted maximum distance from the circle to the points. In 3 dimensions, the closeness of a sphere to a set of discrete points is given by the weighted maximum distance from the sphere to the points, see Figure 2. Generally, the closeness of a manifold to a set of discrete points is given by the weighted maximum distance from the manifold to the points in the N-dimensional. Denote by X the center of a manifold which passes through the two points Q1 and Q2 and by DwX the n-dimensional vector
(2)DwX=w1d1X,⋯,wndnX,where djX=Pj−X−Q1−X and the weight wj∈0,1,j=1,⋯,n.Figure 1
The constrained optimization problem of finding a circle in 2D that is closest to all points among all the circle that are constrained to pass throughQ1 and Q2.Figure 2
The constrained optimization problem of finding a sphere in 3D that is closest to all points among all the sphere that are constrained to pass throughQ1 and Q2.Set the objective functionJpX=DwXp, where ·p denotes the ℓp-norm for 1≤p≤∞.Then, we resolve the minimax problem of (1).Figure2 is in the 3-dimensional case, and we introduce the 2-dimensional case in Figure 1. Since the spheres are constrained to pass Q1 and Q2, the centers should lie on the straight line which bisects the line segment Q1Q2¯ perpendicularly. Those three points are on the same plane. We extended these problems to N-dimensional space. To simplify the problem, let us translate and rotate Q1, Q2, and Pj,j=1,⋯,n so that the locations of Q1 and Q2 are a1,a2,⋯,aN−1,0 and −a1,−a2,⋯,−aN−1,0, where 2a12+a22+⋯+aN−12 is the distance between Q1 and Q2. Also denote the coordinates of Pj by x1j,x2j,⋯,xNj,j=1,⋯,n. Since the center of the N-dimensional manifold lies on the N-axis, problem (1) is then reduced to a one-dimensional problem. Denoting by 0,0,⋯,t the coordinate of N-dimensional, the radius of the manifold which passes through Q1 and Q2 is t2+a12+a22+⋯+aN−12.Forj=1,⋯,n, let ψjt denote the weighted distance ϕjt=ψjt, where
(3)ψjt=wjdjX,djX=x1j2+x2j2+⋯+xN−1j2+t−xNj2−t2+a12+a22+⋯+aN−12.SinceDwX=ψ1t,⋯,ψnt, problem (1) can be rewritten as finding t∗∈ℝ such that
(4)ϕt∗=mintϕtwhereϕt≔maxjϕjt=J∞X.


### 2.2. Innovative Research Contribution

Our previous suggested optimized discrete data classification method (Kim method) in 2-dimensional case can be applied to the detection of a circle on the image [28, 29]. The sphere detection method (Kim method in 3-dimensional) can be applied to the 3-dimensional image with the depth intensity. And our suggested method can be applied to the optimized design of nanoscale circuits to reduce the defect rate in 2-dimensional and 3-dimensional.Our method is very effective for big discrete data classification problems of machine learning models and algorithms [30].As described in the beginning of the section, in this paper, we generalized to the optimization problem of finding a manifold which passes through two given points. We propose the discrete data classification methodology (Kim method) inN-dimensional to resolve this minimax problem, obtaining an exact solution algorithm which is very fast. We generalized this methodology to N-dimensional and obtained the mathematical verification of N-dimensional and physical validation with simple cases.


## 2.1. Problem Statement

We generalize the discrete data classification method from 2-dimensional and 3-dimensional toN-dimensional. In the N-dimensional case, let Pjx1j,x2j,⋯,xNj,j=1,⋯,n, be a given set of discrete points on the space. Additionally, suppose that two additional points Q1 and Q2 are given which are distinct from Pj,j=1,⋯,n. We are interested in the constrained optimization problem of finding a N-dimensional manifold that is closest to all discrete points Pj,j=1,⋯,n, among the manifolds that are constrained to pass through Q1 and Q2, see Figure 1 for the previous suggested problem setting in the 2-dimensional case. In 2 dimensions, the closeness of a circle to a set of discrete points is given by the weighted maximum distance from the circle to the points. In 3 dimensions, the closeness of a sphere to a set of discrete points is given by the weighted maximum distance from the sphere to the points, see Figure 2. Generally, the closeness of a manifold to a set of discrete points is given by the weighted maximum distance from the manifold to the points in the N-dimensional. Denote by X the center of a manifold which passes through the two points Q1 and Q2 and by DwX the n-dimensional vector
(2)DwX=w1d1X,⋯,wndnX,where djX=Pj−X−Q1−X and the weight wj∈0,1,j=1,⋯,n.Figure 1
The constrained optimization problem of finding a circle in 2D that is closest to all points among all the circle that are constrained to pass throughQ1 and Q2.Figure 2
The constrained optimization problem of finding a sphere in 3D that is closest to all points among all the sphere that are constrained to pass throughQ1 and Q2.Set the objective functionJpX=DwXp, where ·p denotes the ℓp-norm for 1≤p≤∞.Then, we resolve the minimax problem of (1).Figure2 is in the 3-dimensional case, and we introduce the 2-dimensional case in Figure 1. Since the spheres are constrained to pass Q1 and Q2, the centers should lie on the straight line which bisects the line segment Q1Q2¯ perpendicularly. Those three points are on the same plane. We extended these problems to N-dimensional space. To simplify the problem, let us translate and rotate Q1, Q2, and Pj,j=1,⋯,n so that the locations of Q1 and Q2 are a1,a2,⋯,aN−1,0 and −a1,−a2,⋯,−aN−1,0, where 2a12+a22+⋯+aN−12 is the distance between Q1 and Q2. Also denote the coordinates of Pj by x1j,x2j,⋯,xNj,j=1,⋯,n. Since the center of the N-dimensional manifold lies on the N-axis, problem (1) is then reduced to a one-dimensional problem. Denoting by 0,0,⋯,t the coordinate of N-dimensional, the radius of the manifold which passes through Q1 and Q2 is t2+a12+a22+⋯+aN−12.Forj=1,⋯,n, let ψjt denote the weighted distance ϕjt=ψjt, where
(3)ψjt=wjdjX,djX=x1j2+x2j2+⋯+xN−1j2+t−xNj2−t2+a12+a22+⋯+aN−12.SinceDwX=ψ1t,⋯,ψnt, problem (1) can be rewritten as finding t∗∈ℝ such that
(4)ϕt∗=mintϕtwhereϕt≔maxjϕjt=J∞X.


## 2.2. Innovative Research Contribution

Our previous suggested optimized discrete data classification method (Kim method) in 2-dimensional case can be applied to the detection of a circle on the image [28, 29]. The sphere detection method (Kim method in 3-dimensional) can be applied to the 3-dimensional image with the depth intensity. And our suggested method can be applied to the optimized design of nanoscale circuits to reduce the defect rate in 2-dimensional and 3-dimensional.Our method is very effective for big discrete data classification problems of machine learning models and algorithms [30].As described in the beginning of the section, in this paper, we generalized to the optimization problem of finding a manifold which passes through two given points. We propose the discrete data classification methodology (Kim method) inN-dimensional to resolve this minimax problem, obtaining an exact solution algorithm which is very fast. We generalized this methodology to N-dimensional and obtained the mathematical verification of N-dimensional and physical validation with simple cases.


## 3. Proposed Methodology of Data Classification Method

In this section, we proposed the data classification methodology. Verification and validation are critical. We propose the methodology with mathematical verification. Validation will be shown in Section 4.Lemma 1.
Supposes that there does not exist a manifold which passes throughQ1, Q2, and the data points Pj,j=1,⋯,n. A local optimum to the minimax problem is then taken at the intersection point of the graph of y=ϕk1t and y=ϕk2t for some k1 and k2.Assume that the functionϕt has a local minimum at a point t′ which is not an intersection point of any two graphs of y=ϕk1t and y=ϕk2t for any k1 and k2. Since the graph of y=ϕt is a finite number of piecewise smooth curves, there should exist some j and a sufficiently small positive ε such that ϕt=ϕjt on t′−ε,t′+ε, where t′ is a critical point of ϕ, and thus, it is a critical point of ϕj thereon. Such critical points should be either the zeros of the first derivative of ψjt or the zero of ϕjt. Thus, the exact critical point t′ of the function ϕjt should be either t±=a12+a22+⋯+aN−12xNj±xNja12+a22+⋯+aN−12R/a12+a22+⋯+aN−12−x1j2+x2j2+⋯+xN−1j2, with R=x1j2+x2j2+⋯+xN−1j2or(5)t0=x1j2+x2j2+⋯+xN−1j2+xNj2−a12+a22+⋯+aN−122xNj,which are zeros of ψj′t=wjt−xNj/x1j2+x2j2+⋯+xN−1j2+t−xNj2−t/t2+a12+a22+⋯+aN−12 or ϕjt, respectively. To derive a contradiction from the existence of such critical points, notice that the second derivative of ψjt is given as follows:
(6)ψj′′t=wjx1j2+x2j2+⋯+xN−1j2+t−xNj2−t−xNj2S−1/2x1j2+x2j2+⋯+xN−1j2+t−xNj2−t2+a12+a22+⋯+aN−12−t2t2+a12+a22+⋯+aN−12−1/2t2+a12+a22+⋯+aN−12,withS=x1j2+x2j2+⋯+xN−1j2+t−xNj2.We treat the two cases separately.(i)
First, let us consider the pointst′=t±. Observe that ψjt±ψj′′t±<0.This means that ifψjt± is greater (or less) than zero, then ψj′′t± is less (or greater) than zero, and thus, the function ψjt has a local maximum (or local minimum) at t±. Since ϕjt=ψjt, it follows that the function ϕjt has a local maximum at t±(ii)
Next, consider the pointt′=t0. Since the function ϕt has a local minimum at t0 and ϕt=ϕjt on t′−ε,t′+ε, it is trivial that ϕt0=ϕjt0=0Both cases (i) and (ii) lead to a contradiction to our assumption. Therefore, the local minimum ofϕt must be taken at a point t′ which is an intersection point of two graphs of y=ϕk1t and y=ϕk2t for some k1 and k2. This completes the proof.In Figure3, the graphs of y=ϕjt, y=ϕkt, and y=ϕjkt=maxϕjt,ϕkt are depicted. The local minima of y=ϕjkt are taken at the intersection points of y=ϕjt and y=ϕkt. Figure 3 is in 2-dimensional case.Theorem 2.
Lettjk’s be all the intersection points of the graphs of y=ϕjt and y=ϕkt for all j,k=1,⋯,n. Let t∗ be such that
(7)ϕt∗=minj,kϕtjk.
Ifϕt∗≤maxjwjyj, then ϕt∗ is a global minimum for problem (2); otherwise, problem (2) does not have a global minimum solution.Figure 3
The 2-dimensional case graphs ofy=ϕjt, y=ϕkt, and y=ϕjkt=maxϕjt,ϕkt. Two local minima of y=ϕjkt are taken at the intersection points of y=ϕjt and y=ϕkt.Let us begin with finding the candidates of global minimum ofϕt using the above Lemma 1. Since the equation ϕjt=ϕkt is equivalent to the equation ϕj2t=ϕk2t, we can find the intersection points of ϕj and ϕk by solving
(8)0=ϕj2t−ϕk2t=wjx1j2+x2j2+⋯+xN−1j2+t−xNj2−wkx1k2+x2k2+⋯+xN−1k2+t−xNk2−wj−wkt2+a12+a22+⋯+aN−12×wjx1j2+x2j2+⋯+xN−1j2+t−xNj2+wkx1k2+x2k2+⋯+xN−1k2+t−xNk2−wj+wkt2+a12+a22+⋯+aN−12≕I1wt×I2wt.The above equation is divided intoI1wt=0 or I2wt=0.To solve the equationI1wt=0, move the radical term wj−wkt2+a12+a22+⋯+aN−12 to the right side of the equation and square both sides. Then, we get
(9)2wjwkx1j2+x2j2+⋯+xN−1j2+t−xNj2T=wj2x1j2+x2j2+⋯+xN−1j2+t−xNj2+wk2x1k2+x2k2+⋯+xN−1k2+t−xNk2−wj−wk2t2+a12+a22+⋯+aN−12,withT=x1k2+x2k2+⋯+xN−1k2+t−xNk2.To isolate the radical expression on the left side, move all the other terms to the right and square both sides of the equation again. Then, one gets the following cubic polynomial:(10)pjkt=a0t3+a1t2+a2t+a3,where
(11)a0=4wj2wk2xNk+xNj−2wj4xNj+wk4xNk+2wj−wk2wj2xNj+wk2xNk,a1=wj4xNj2+2x1j2+x2j2+⋯+xN−1j2+xNj2+wk4xNk2+2x1k2+x2k2+⋯+xN−1k2+xNk2+2wj−wk4a12+a22+⋯+aN−12+2wj2wk2x1j2+x2j2+⋯+xN−1j2+xNj2+x1k2+x2k2+⋯+xN−1k2+xNk2−4xNjxNk−wk2wj−wk2x1k2+x2k2+⋯+xN−1k2+xNk2+a12+a22+⋯+aN−12−wj2wj−wk2x1j2+x2j2+⋯+xN−1j2+xNj2+a12+a22+⋯+aN−12−4wj2wk2x1k2+x2k2+⋯+xN−1k2+xNk2+x1j2+x2j2+⋯+xN−1j2+xNj2+4xNjxNk,a2=wj4−2xNjx1j2+x2j2+⋯+xN−1j2+xNj2+wk4−2xNkx1k2+x2k2+⋯+xN−1k2+xNk2+2wj2wk2−2xNjx1j2+x2j2+⋯+xN−1j2+xNj2−2xNkx1k2+x2k2+⋯+xN−1k2+xNk2−wk2wj−wk2−2xNka12+a22+⋯+aN−12−wj2wj−wk2−2xNja12+a22+⋯+aN−12+8wj2wk2xNjx1k2+x2k2+⋯+xN−1k2+xNk2+xNkx1j2+x2j2+⋯+xN−1j2+xNj2,a3=wj4x1j2+x2j2+⋯+xN−1j2+xNj22+wk4x1k2+x2k2+⋯+xN−1k2+xNk22+wj−wk4a12+a22+⋯+aN−122+2wj2wk2x1j2+x2j2+⋯+xN−1j2+xNj2x1k2+x2k2+⋯+xN−1k2+xNk2−wk2wj−wk2x1k2+x2k2+⋯+xN−1k2+xNk2a12+a22+⋯+aN−12−wj2wj−wk2x1j2+x2j2+⋯+xN−1j2+xNj2a12+a22+⋯+aN−12−4wj2wk2x1j2+x2j2+⋯+xN−1j2+xNj2x1k2+x2k2+⋯+xN−1k2+xNk2.Similarly, solving the equationI2wt=0 is equivalent to solving the following cubic polynomial:
(12)qjkt=b0t3+b1t2+b2t+b3.By using Cardano’s formula, one can find all the real roots ofpjkt=0 and qjkt=0. In this way, one can find the intersection points of y=ϕjt and y=ϕkt for j,k=1,⋯,n. Then, by comparing the values of the function ϕt at these points, one can find the candidate of global minimum at t∗. Of course, as the function ϕt is defined on ℝ, the global minimum may not exist. However, using the expression
(13)ϕjt=wjx1j2+x2j2+⋯+xN−1j2+xN2−a12−a22−⋯−aN−12−2xNjtx1j2+x2j2+⋯+xN−1j2+t−xNj2+t2+a12+a22+⋯+aN−12,we have ϕt⟶maxjwjyjas t⟶±∞. Thus, if ϕt∗≤maxjwjyj, then ϕt∗ is a global minimum. If ϕt∗>maxjwjyj, then the function ϕt does not have a global minimum. This completes the proof.Summarizing the above procedure in the proof of Theorem2, we propose the following algorithm for solving the minimax problem (1).Step 1.
Compute the distance2a12+a22+⋯+aN−12 between Q1 and Q2; choose the coordinate system such that Q1a1,a2,⋯,aN−1,0 and Q2−a1,−a2,⋯,−aN−1,0; by a rigid motion transform to Pj associate the coordinates x1j,x2j,⋯,xNj for j=1,⋯,n.Step 2.
IfQ1, Q2, and Pj,j=1,⋯,n are on one specific manifold, then it is done.Step 3.
Find all intersection pointstjk’s of the graphs z=ϕjt and z=ϕkt for all j,k.Step 4.
For all such intersection pointstjk’s, evaluate ϕitjk for all i=1,⋯,n; then, compute ϕtjk=maxiϕitjk, and find the minimum ϕt∗=minj,kϕtjk.Step 5.
Ifϕt∗≤maxjwjzj, then ϕt∗ is the global minimum. If ϕt∗>maxjwjzj, the global minimum of the function ϕt does not exist.This is our optimized suggested data classification method inN-dimensional (Kim method). Our N-dimensional discrete data classification methodology is mathematically verified in this section.


## 4. Numerical Results

In this section, we show validation with simple example and results, see Figure4. From Figure 4, we used 4 distinct points. The expected center is (0,0). Through our suggested methodology, we obtain the center 0.034375,0.034375 with the radius 1.7081. This is validated with an error <4.86135×10−2. With big data (200 points), we calculate our methodology, see Figure 5. In the data classification problem, we test two groups with 200 points, see Figure 6. We can classify the scattered data with our suggested methodology. Obviously, we can obtain the two distinct circles with two groups. Through the intersection area of the two circles, we can analyze the relation of the data.Figure 4
The validation figure of the constrained optimization problem of finding a circle in 2D that is closest to all 4 points, (3.58, 1.78), (-3.00, 2.61), (2.12, -3.45), and (-3.26, -2.34) among all the circle that are constrained to pass through (-4, -4) and (4, 4). The solution is the center (0.034375, 0.034375) with radius 1.7081.Figure 5
The constrained optimization problem of finding a circle in 2D that is closest to all 200 points among all the circle that are constrained to pass through (4, 6) and (-4, 1). The solution is the center (-0.93750, 5.0000) with radius 27.270.Figure 6
The validation figure of the constrained optimization problem of finding a circle in 2D that is closest to the different two groups; each group has 100 points. Group A (close dots) has the center (-1.7661, -8.8790) with minimax radius 9.2144, and group B (open dots) has the center (13.433, 18.536) with minimax radius 12.416.We show the comparison study with our suggested methodology and other methodology of the data classification [31], see Table 1 with comparison study of several methodology with the random data set from [31]. AUC is area under curve. Our method result of AUC (KIM) has 1.0000 in Table 1 because the two distinct circles are not overlapped. DT is a decision tree method. k−NN is k-nearest neighbor method. LogR is logistic regression method. NB is Naive Bayes method. C4.5 is a decision tree with divide-and-conquer. SVM is support vector machine. LC is linear classifier method [31]. Our methodology is to find the mathematical exact solution. We find the two distinct circles with centers and radii.Table 1
AUC values for dataset with sample size 200 with other methodology [31].
KIMDTk-NNLogR1.00000.79410.76830.6328NBC4.5SVMLC0.71260.74520.74480.6408We have developed a minimax circle and sphere code to find the solution of the suggested minimax problem. We show a simple test case with 20 input data including the weight, see Table2. The table shows the detailed input data with the weights (the weights are 1). X∗ is the numerical solution from the exact solution procedure of the suggested problems. We tested many times for various random input data, and the results are very satisfactory with the robustness. The computation is executed using the Fortran compiler in a window system with the architecture Intel 11th Gen Core i7-1165G7 of 2.80 GHz. These plots show that the algorithm is very efficiently.Table 2
Q1=4.00,6.00 and Q2=−4.00,1.00 are given with the equal weighted 20 points (value is 1). The minimax value is 15.025.
PjxjyjP10.3000.500P2-0.2001.53P31.971.90P4-2.308.63P55.7012.9P6-2.203.53P77.976.90P8-12.38.63P923.725.9P10-24.322.6P116.74.90P12-6.23.53P132.71.92P14-22.312.6P158.6716.9P16-4.6212.5P175.976.90P18-12.34.63P1914.71.90P20-12.315.6Solution-9.851519.262


## 5. Conclusions

We have investigated the suggested minimax problem by our suggested data classification methodology (Kim method) inN-dimensional space, and this methodology is verified and validated (VV). This problem is applied to artificial intelligence classification problems (machine learning, deep learning), point data analysis problems (data science problem), the optimized design of nanoscale circuits, and the location of facility problems and circle or sphere detection problem on the image. Here, we also consider the general weighted case. And we obtain the exact solution of the minimizing manifold to maximize the distance between the manifold through two fixed points and multiple points. By finding the local optima, we find the global optimum. After we proved the exact solution of the suggested problem, we proposed a very efficient algorithm. This is the optimized data classification methodology of N-dimensional scattered data (Kim Method). As future work, the proposed methodology can be applied to the image processing. We investigate the detection algorithm of the image processing with our methodology. In the different images, we will analyze the mutual information. And our methodology can be applied to analyze supply chain problems with data.



---

*Source: 8199872-2022-05-28.xml*