hyperplane

简明释义

英[ˈhaɪpəˌpleɪn]美[ˈhaɪpərˌpleɪn]

n. [数] 超平面

英英释义

单词用法

同义词

反义词

例句

1.Necessary and sufficient conditions are given on the symmetry of points and n-1 dimensional hyperplane of function of many variables in then-dimensional Euclidean space.

在n维欧氏空间内，给出了多元函数分别关于点、n-1维超平面对称的充要条件。

2.Distribution of first fitting points on hyperplane is derived in the diffusion process.

求出了一类扩散过程关于超平面的首中点分布。

3.Free arrangement is a very important kind of hyperplane arrangement.

自由构形是超平面构形中一类重要的构形。

4.The idea of this algorithm is to find the optimal solution in the feasible region by an iterative step from one basic standard hyperplane to another.

此算法的基本思想是在规划问题的可行域中由所建的一个切割面到另一个切割面的不断推进来求取最优的。

5.On the other hand, support vectors are the points that define the hyperplane.

另一方面，支持向量定义的平面的点。

6.Pole placement method is used in the selection of hyperplane to simplify the design. This method is more convenient and with less computation than the method based on the quadratic performance index.

在开关超平面的选择上，采用了极点配置法，而不是基于二次型性能指标，从而减少了计算量，简化了设计。

7.The former attempts to find an optimal hyperplane that maximize margin between two classes, and the later are designed to provide an explanation of the classification using logical rules.

前者寻求最大化两类间隔的最优分类超平面，后者用逻辑规则解释分类。

8.The fuzzy membership of each sample is defined by affinity among samples, and by the training determine a threshold, noises and outliers are removed, which influence optimal separating hyperplane.

应用基于样本之间的紧密度确定每个样本的模糊隶属度，通过训练确定阀值，去除影响得到最优分类超平面的噪声和野点。

9.In machine learning, the decision boundary can often be represented as a hyperplane (超平面) in a high-dimensional space.

在机器学习中，决策边界通常可以表示为高维空间中的一个hyperplane (超平面)。

10.Support Vector Machines (SVM) aim to find the optimal hyperplane (超平面) that separates different classes in the dataset.

支持向量机(SVM)旨在找到最佳的hyperplane (超平面)，以分隔数据集中的不同类别。

11.The hyperplane (超平面) divides the feature space into two distinct regions for classification tasks.

该hyperplane (超平面)将特征空间划分为两个不同的区域以进行分类任务。

12.In a three-dimensional space, a hyperplane (超平面) is simply a two-dimensional plane.

在三维空间中，一个hyperplane (超平面) 只是一个二维平面。

13.To optimize the model, we need to adjust the parameters of the hyperplane (超平面).

为了优化模型，我们需要调整hyperplane (超平面)的参数。

作文

In the realm of mathematics and data science, the concept of a hyperplane plays a crucial role in various applications, especially in machine learning and optimization problems. A hyperplane is defined as a subspace of one dimension less than its ambient space. For instance, in a three-dimensional space, a hyperplane would be a two-dimensional plane that divides the space into two half-spaces. This geometric interpretation is not only fascinating but also essential for understanding how we can separate different classes of data points in multidimensional spaces. Consider the example of a binary classification problem where we want to distinguish between two types of flowers based on their features, such as petal length and width. In this case, each flower can be represented as a point in a two-dimensional space. The goal is to find a hyperplane that separates these two classes effectively. The hyperplane acts as a decision boundary; if a new flower's features fall on one side of the hyperplane, it is classified as one type, and if it falls on the other side, it is classified as the other type. Mathematically, a hyperplane can be expressed with a linear equation of the form: Ax + By + C = 0 in two dimensions, or more generally, in n dimensions as: w1x1 + w2x2 + ... + wnxn + b = 0, where w represents the weights (or coefficients) and b is the bias term. The coefficients determine the orientation of the hyperplane, while the bias shifts it away from the origin. This equation helps us visualize how the hyperplane functions as a separator in higher dimensions. The importance of hyperplanes extends beyond just classification tasks. They are also fundamental in optimization problems, such as support vector machines (SVMs), which aim to find the optimal hyperplane that maximizes the margin between different classes. The optimal hyperplane is the one that not only separates the classes but does so with the largest possible distance to the nearest data points from either class. This property enhances the generalization capability of the model, making it robust against overfitting. Furthermore, hyperplanes are utilized in various fields, including economics, physics, and computer graphics. For example, in economics, a hyperplane can represent constraints in a resource allocation problem, helping to visualize feasible regions in a multi-dimensional budget constraint scenario. In physics, hyperplanes may describe phase transitions in state spaces, providing insights into the behavior of complex systems. To sum up, the concept of a hyperplane is not just a mathematical abstraction; it is a powerful tool that helps us understand and solve real-world problems across multiple disciplines. By grasping the idea of a hyperplane, we can enhance our analytical skills and improve our ability to make informed decisions based on data. The versatility and applicability of hyperplanes make them an indispensable part of modern mathematics and its applications in technology and science.

在数学和数据科学领域，超平面的概念在各种应用中发挥着至关重要的作用，尤其是在机器学习和优化问题中。超平面被定义为维度比其环境空间低一个维度的子空间。例如，在三维空间中，超平面将是一个二维平面，将空间分成两个半空间。这种几何解释不仅令人着迷，而且对于理解我们如何在多维空间中分离不同类别的数据点至关重要。 考虑一个二元分类问题的例子，我们想根据花瓣长度和宽度等特征区分两种类型的花。在这种情况下，每朵花可以表示为二维空间中的一个点。目标是找到一个有效地分隔这两个类别的超平面。超平面充当决策边界；如果一朵新花的特征落在超平面的一侧，则将其分类为一种类型；如果落在另一侧，则将其分类为另一种类型。 从数学上讲，超平面可以用线性方程表示，形式为：在二维中，Ax + By + C = 0，或更一般地，在n维中为：w1x1 + w2x2 + ... + wnxn + b = 0，其中w代表权重（或系数），b是偏置项。系数决定了超平面的方向，而偏置则将其从原点移开。该方程帮助我们可视化超平面如何在更高维度中作为分隔符的功能。 超平面的重要性不仅仅局限于分类任务。它们在优化问题中也至关重要，例如支持向量机（SVM），其目标是找到最佳的超平面，以最大化不同类别之间的间距。最佳的超平面不仅能分隔类别，而且能以最大的可能距离分隔最近的数据点。这一特性增强了模型的泛化能力，使其在防止过拟合方面更加稳健。 此外，超平面在经济学、物理学和计算机图形学等多个领域都有应用。例如，在经济学中，超平面可以表示资源分配问题中的约束，帮助可视化多维预算约束场景中的可行区域。在物理学中，超平面可能描述状态空间中的相变，为复杂系统的行为提供洞察。 总之，超平面的概念不仅仅是一个数学抽象；它是一个强大的工具，帮助我们理解和解决跨多个学科的现实问题。通过掌握超平面的概念，我们可以增强分析能力，提高基于数据做出明智决策的能力。超平面的多样性和适用性使其成为现代数学及其在技术和科学应用中不可或缺的一部分。