如何在Matlab中一次拍摄两个以上矩阵的外部积? [英] How to take outer product of more than two matrices in one shot, in matlab?

问题描述

我要计算y = a⊗a⊗a，其中a是n-by-1向量，而⊗是外部产品运营商.在这种情况下，y应该是一个n×n×n张量.

I want to compute y = a⊗a⊗a, where a is a n-by-1 vector, and ⊗ is the outer product operator. In this case y should be an n-by-n-by-n tensor.

如果是y = a⊗a，这很容易.我只是这样做:

If y = a⊗a, it is easy. I simply do:

y = a * a' 

但是在第一种情况下该怎么办?如果存在两个以上的向量，如何在MATLAB中有效地计算该外部乘积?

But what to do in the first case? How do I compute this outer product efficiently in MATLAB if there are more than two vectors?

推荐答案

在y = u⊗v的多维(张量)情况下，我认为您需要像这样移动第二个操作数的尺寸:

In a multi-dimensional (tensor) case of y = u⊗v, I believe that you need to shift the dimensions of the second operand like so:

v_t = permute(v, circshift(1:(ndims(u) + ndims(v)), [0, ndims(u)]));

，然后将它们乘以bsxfun:

y = bsxfun(@times, u, v_t);

常规矩阵乘法仅针对矢量和二维矩阵定义，因此在一般情况下我们不能使用它.

The regular matrix multiplication is defined only for vector and 2-D matrices, so we couldn't use it in the general case.

还要注意，如果第二个操作数是一维矢量，则此计算仍然会失败，因为ndims对于矢量返回2而不是1.为此，让我们定义自己的计算尺寸的函数:

Also note that this computation still fails if the second operand is a 1-D vector, because ndims returns 2 instead of 1 for vectors. For this purpose, lets define our own function that counts dimensions:

my_ndims = @(x)(isvector(x) + ~isvector(x) * ndims(x));

要完成答案，您可以定义一个新函数(例如例如一个匿名函数)，

To complete the answer, you can define a new function (e.g. an anonymous function), like so:

outprod = @(u, v)bsxfun(@times, u, permute(v, circshift(1:(my_ndims(u) + my_ndims(v)), [0, my_ndims(u)])));

，然后根据需要多次使用.例如，y = a×a×a的计算方式如下:

and then use it as many times as you want. For example, y = a×a×a would be computed like so:

y = outprod(outprod(a, a), a);

当然，您可以编写一个更好的函数，该函数接受可变数量的参数以节省输入时间.遵循以下原则:

Of course, you can write a better function that takes a variable number of arguments to save you some typing. Something along these lines:

function y = outprod(u, varargin)
    my_ndims = @(x)(isvector(x) + ~isvector(x) * ndims(x));
    y = u;
    for k = 1:numel(varargin)
        v = varargin{k};
        v_t = permute(v, circshift(1:(my_ndims(y) + my_ndims(v)),[0, my_ndims(y)]));
        y = bsxfun(@times, y, v_t);
    end

我希望我数学正确！

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