用熵函数优化超平面，实现高效分类。
Classification by Separating Hypersurfaces: An Entropic Approach
- 通过最小化熵函数寻找分离超平面参数
- 可处理线性与非线性分类任务，效果优于传统方法
- 适合需要鲁棒决策边界的机器学习场景
我们研究一个分类问题：给定由向量表示的个体属性（在 ${\mathbb R}^N$ 中），目标是找到一个超平面将两类点分隔开。该问题源于感知机模型，至今仍是机器学习的核心。本文提出一种新方法：在以原点为中心的 $N$ 维有界超立方体内搜索参数向量，并通过最小化定义在未知变量空间上的熵函数，获得一个正向量（在 ${\mathbb R}^M$ 中）。该方法可推广至多项式曲面，允许使用更复杂的决策边界进行数据分离。相比支持向量机和梯度下降等传统线性或二次优化技术，本方法更具鲁棒性。数值实验表明，该方法在多种分类任务中均表现出高效性和通用性，涵盖线性和非线性可分情形。
原文摘要 · Abstract (English)
We consider the following classification problem: Given a population of individuals characterized by a set of attributes represented as a vector in ${\mathbb R}^N$, the goal is to find a hyperplane in ${\mathbb R}^N$ that separates two sets of points corresponding to two distinct classes. This problem, with a history dating back to the perceptron model, remains central to machine learning. In this paper we propose a novel approach by searching for a vector of parameters in a bounded $N$-dimensional hypercube centered at the origin and a positive vector in ${\mathbb R}^M$, obtained through the minimization of an entropy-based function defined over the space of unknown variables. The method extends to polynomial surfaces, allowing the separation of data points by more complex decision boundaries. This provides a robust alternative to traditional linear or quadratic optimization techniques, such as support vector machines and gradient descent. Numerical experiments demonstrate the efficiency and versatility of the method in handling diverse classification tasks, including linear and non-linear separability.
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