在实际中，为什么theta * X不是theta'* X? [英] Why theta*X not theta'*X in practical?

问题描述

Andrew Ng在ML上进行MOOC时，他从理论上解释了 theta'* X 给了我们假设，而在做功课时，我们使用了 theta * X .为什么会这样呢?

While doing MOOC on ML by Andrew Ng, he in theory explains theta'*X gives us hypothesis and while doing coursework we use theta*X. Why it's so?

推荐答案

在数学中，向量"始终定义为垂直堆叠的数组，例如 ，并表示3中的单点维空间.

In mathematics, a 'vector' is always defined as a vertically-stacked array, e.g. , and signifies a single point in a 3-dimensional space.

水平"向量通常表示一系列观察结果，例如 是3个标量观测值的元组.

A 'horizontal' vector, typically signifies an array of observations, e.g. is a tuple of 3 scalar observations.

同样，矩阵可以被认为是向量的集合.例如，以下是四个3维向量的集合:

Equally, a matrix can be thought of as a collection of vectors. E.g., the following is a collection of four 3-dimensional vectors:

可以将标量视为大小为1x1的矩阵，因此它的转置与原始标量相同.

A scalar can be thought of as a matrix of size 1x1, and therefore its transpose is the same as the original.

通常，n×m矩阵 W 也可以看作是从m维向量 x 到n维向量 y ，因为将该矩阵与m维向量相乘会得到一个新的n维向量.如果您的矩阵" W 是"1xn"，则表示从n维向量到标量的转换.

More generally, an n-by-m matrix W can also be thought of as a transformation from an m-dimensional vector x to an n-dimensional vector y, since multiplying that matrix with an m-dimensional vector will yield a new n-dimensional one. If your 'matrix' W is '1xn', then this denotes a transformation from an n-dimensional vector to a scalar.

因此，从概念上讲，习惯上是从数学表示法的角度介绍问题的，例如 y = Wx .

Therefore, notationally, it is customary to introduce the problem from the mathematical notation point of view, e.g. y = Wx.

但是，出于计算的原因，有时将计算作为向量乘以矩阵"而不是矩阵乘以向量"更有意义.由于(Wx)'=== x'W'，有时我们会解决类似的问题，并将 x'视为水平向量.另外，如果 W 不是矩阵，而是标量，则 Wx 表示标量乘法，因此在这种情况下 Wx === xW

However, for computational reasons, sometimes it makes more sense to perform the calculation as a "vector times a matrix" rather than "matrix times a vector". Since (Wx)' === x'W', sometimes we solve the problem like that, and treat x' as a horizontal vector. Also, if W is not a matrix, but a scalar, then Wx denotes scalar multiplication, and therefore in this case Wx === xW.

我不知道您所说的练习，但我的假设是他在课程中引入了 theta 作为适当的垂直向量，然后对其进行了转换以进行适当的计算，即从n维向量到标量的转换(这是您的预测).

I don't know the exercises you speak of, but my assumption would be that in the course he introduced theta as a proper, vertical vector, but then transposed it to perform proper calculations, i.e. a transformation from a vector of n-dimensions to a scalar (which is your prediction).

然后在练习中，大概是您或正在处理标量"theta"，因此没有转置它的点，为方便起见， 或都被保留为theta.em> ，现在将theta定义为水平(即转置)矢量，出于某种原因(例如打印方便)开始，然后在执行必要的转换时留在该状态.

Then in the exercises, presumably you were either dealing with a scalar 'theta' so there was no point transposing it, and was left as theta for convenience or, theta was now defined as a horizontal (i.e. transposed) vector to begin with for some reason (e.g. printing convenience), and then was left in that state when performing the necessary transformation.

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