神经网络:什么是“线性可分离的"?意思是? [英] Neural Networks: What does "linearly separable" mean?

问题描述

我目前正在阅读Tom Mitchell撰写的《机器学习》一书.在谈论神经网络时，Mitchell指出:

I am currently reading the Machine Learning book by Tom Mitchell. When talking about neural networks, Mitchell states:

尽管感知器规则在以下情况下找到了成功的权重向量 训练示例是线性可分离的，可能无法收敛 如果示例不是线性可分离的. "

"Although the perceptron rule finds a successful weight vector when the training examples are linearly separable, it can fail to converge if the examples are not linearly separable. "

我在理解他的线性可分离"含义时遇到问题?维基百科告诉我:如果二维空间中的两套点可以完全用一条直线分开，则它们是线性可分离的."

I am having problems understanding what he means with "linearly separable"? Wikipedia tells me that "two sets of points in a two-dimensional space are linearly separable if they can be completely separated by a single line."

但是这如何应用于神经网络的训练集?输入(或动作单元)如何线性分离?

But how does this apply to the training set for neural networks? How can inputs (or action units) be linearly separable or not?

我不是最擅长的几何和数学-有人可以像我5岁时向我解释吗? ;)谢谢！

I'm not the best at geometry and maths - could anybody explain it to me as though I were 5? ;) Thanks!

推荐答案

假设您要编写一种算法，该算法基于大小和价格这两个参数来确定房屋是否会在同一年出售.或不.因此，您有2个输入(大小和价格)，而1个输出将出售或不出售.现在，当您收到训练集时，可能会发生输出未累积以使我们的预测变得容易的情况(您能否告诉我，根据第一张图，X是N还是S?第二张图怎么样):

Suppose you want to write an algorithm that decides, based on two parameters, size and price, if an house will sell in the same year it was put on sale or not. So you have 2 inputs, size and price, and one output, will sell or will not sell. Now, when you receive your training sets, it could happen that the output is not accumulated to make our prediction easy (Can you tell me, based on the first graph if X will be an N or S? How about the second graph):

        ^
        |  N S   N
       s|  S X    N
       i|  N     N S
       z|  S  N  S  N
       e|  N S  S N
        +----------->
          price


        ^
        |  S S   N
       s|  X S    N
       i|  S     N N
       z|  S  N  N  N
       e|    N N N
        +----------->
          price

位置:

S-sold,
N-not sold

如您在第一张图中所看到的，您无法真正用直线将两个可能的输出(已出售/未出售)分开，无论您如何尝试，总是会同时出现S和N在行的两边，这意味着您的算法将有很多possible行，但没有最终的正确行来拆分2个输出(当然也要预测新的输出，这是一开始的目标).这就是linearly separable(第二张图)数据集更容易预测的原因.

As you can see in the first graph, you can't really separate the two possible outputs (sold/not sold) by a straight line, no matter how you try there will always be both S and N on the both sides of the line, which means that your algorithm will have a lot of possible lines but no ultimate, correct line to split the 2 outputs (and of course to predict new ones, which is the goal from the very beginning). That's why linearly separable (the second graph) data sets are much easier to predict.

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