查询输入向量的SVM映射?和SVM优化方程 [英] Query about SVM mapping of input vector? And SVM optimization equation

问题描述

我已经阅读了许多论文，并在很高的水平上了解了支持向量机的基本概念.您给它一个训练输入向量，该向量具有一组功能，并基于优化函数"如何评估此输入向量，将其称为x，(假设我们正在谈论文本分类)，即与该输入向量相关联的文本x被分类为两个预定义类之一，仅在二进制分类的情况下.

I have read through a lot of papers and understand the basic concept of a support vector machine at a very high level. You give it a training input vector which has a set of features and bases on how the "optimization function" evaluates this input vector lets call it x, (lets say we're talking about text classification), the text associated with the input vector x is classified into one of two pre-defined classes, this is only in the case of binary classification.

因此，我的第一个问题是通过上述过程完成的，所有论文首先说，该训练输入向量x映射到了更高(也许是无限大)的维空间.那么，这种映射可以实现什么?为什么需要这样做呢?假设输入向量x具有5个特征，那么谁来决定将x映射到哪个更高维度"?

So my first question is through this procedure described above, all the papers say first that this training input vector x is mapped to a higher (maybe infinite) dimensional space. So what does this mapping achieve or why is this required? Lets say the input vector x has 5 features so who decides which "higher dimension" x is going to be mapped to?

第二个问题是关于以下优化方程的:

Second question is about the following optimization equation:

min 1/2 wi(转置)* wi + CΣi= 1..nξi

min 1/2 wi(transpose)*wi + C Σi = 1..n ξi

所以我知道w与图中支持向量的超平面裕度有关，我知道C是一种惩罚，但是我不知道这是什么惩罚.在这种情况下，ξi也代表什么.

so I understand that w has something to do with the margins of the hyperplane from the support vectors in the graph and I know that C is some sort of a penalty but I dont' know what it is a penalty for. And also what is ξi representing in this case.

对第二个问题的简单解释将不胜感激，因为我对阅读技术论文了解得还不够运气.

A simple explanation of the second question would be much appreciated as I have not had much luck understanding it by reading technical papers.

推荐答案

1)更高维度的空间是通过内核机制实现的.但是，在评估测试样本时，不需要显式计算较高维的空间. (显然是必须这样的，因为我们不能在计算机上表示无限维.)例如，径向基函数内核隐含无限维空间，但是我们不需要显式映射到该无限维空间.我们只需要计算K(x_sv，x_test)，其中x_sv是支持向量之一，x_test是测试样本.

1) The higher dimension space happens through the kernel mechanism. However, when evaluating the test sample, the higher dimension space need not be explicitly computed. (Clearly this must be the case because we cannot represent infinite dimensions on a computer.) For instance, radial basis function kernels imply infinite dimensional spaces, yet we don't need to map into this infinite dimension space explicitly. We only need to compute, K(x_sv,x_test), where x_sv is one of the support vectors and x_test is the test sample.

通过训练过程和参数选择特定的高维空间，这些过程和参数选择了一组支持向量及其相应的权重.

The specific higher dimensional space is chosen by the training procedure and parameters, which choose a set of support vectors and their corresponding weights.

2)C是与无法完美分类训练集的成本相关的权重.优化方程表示要在两种不理想的情况下进行折衷:不良分类和低利润. ξi变量表示无法对训练集的实例i进行分类的程度，即实例i的训练误差.

2) C is the weight associated with the cost of not being able to classify the training set perfectly. The optimization equation says to trade-off between the two undesirable cases of non-perfect classification and low margin. The ξi variables represent by how much we're unable to classify instance i of the training set, i.e., the training error of instance i.

有关在任何地方(IMO)都会得到的最直观的解释，请参见Chris Burges在SVM上的教程.

See Chris Burges' tutorial on SVM's for about the most intuitive explanation you're going to get of this stuff anywhere (IMO).

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