Abstract

The term “mathematical diagnostics” was introduced by V. F. Demyanov in the early 2000s. The simplest problem of mathematical diagnostics is to determine the relative position of some point p and the convex hull C of a finite number of given points in n-dimensional Euclidean space. Of interest is the answer to the following questions: does the point p belong to the set C or not? If p does not belong to C, then what is the distance from p to C? In the general problem of mathematical diagnostics, two convex hulls are considered. The question is whether they have common points. If there are no common points, then it is required to find the distance between these hulls. From an algorithmic point of view, the problems of mathematical diagnostics reduce to special linear- or quadratic-programming problems, which can be solved by finite methods. However, the implementation of this approach in the case of large data arrays runs into serious computational difficulties. Such situations can be dealt with by infinite but easily implemented methods, which allow one to obtain an approximate solution with the required accuracy in a finite number of iterations. These methods include the MDM method. It was developed by Mitchell, Demyanov, and Malozemov in 1971 for other purposes, but later found application in machine learning. From a modern point of view, the original version of the MDM method can be used to solve only the simplest problems of mathematical diagnostics. This article gives a natural generalization of the MDM method, oriented towards solving general problems of mathematical diagnostics. In addition, it is shown how, using the generalized MDM method, a solution to the problem of the linear separation of two finite sets, in which the separating strip has the largest width, is found.

中文翻译：

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求解数学诊断一般二次问题的MDM方法

摘要

“数学诊断”一词由 VF Demyanov 在 2000 年代初期提出。数学诊断最简单的问题是确定n维欧几里得空间中某个点p与有限个给定点的凸包C的相对位置。有趣的是以下问题的答案：点p是否属于集合C ？如果p不属于C ，那么从p到C 的距离是多少？在数学诊断的一般问题中，考虑两个凸包。问题是他们是否有共同点。如果没有公共点，则需要找到这些船体之间的距离。从算法的角度来看，数学诊断问题可以简化为特殊的线性或二次规划问题，可以通过有限方法求解。然而，在大数据数组的情况下实施这种方法会遇到严重的计算困难。这种情况可以通过无限但易于实现的方法来处理，这些方法允许人们在有限次数的迭代中获得具有所需精度的近似解。这些方法包括 MDM 方法。它是由 Mitchell、Demyanov 和 Malozemov 于 1971 年出于其他目的而开发的，但后来发现它在机器学习中的应用。从现代的角度来看，原始版本的 MDM 方法只能用于解决最简单的数学诊断问题。本文给出了 MDM 方法的自然概括，旨在解决数学诊断的一般问题。此外，还展示了如何使用广义 MDM 方法找到两个有限集的线性分离问题的解，其中分离带具有最大宽度。