How to take outer product of more than two matrice

问题:

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.

If y = a⊗a, it is easy. I simply do:

y = a * a' 

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?

回答1:

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)]));

and then multiply them with bsxfun:

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

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

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));

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)])));

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);

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

I hope I got the math right!

回答2:

You can use as well the kron function:

kron(a * a', a)

or when four outer (kronecker tensor) products needed:

kron(kron(a * a', a), a)

and so on. The last one gives you a m x n matrix, where m = n * n * n.

If adding dimensions is desired as going on with the products, you may use the reshape function:

reshape(kron(a * a', a), [n, n, n])

or

reshape(kron(kron(a * a', a), a), [n, n, n, n])

and so on. The last one gives you a n x n x n x n tensor.

回答3:

The problem with using kron as in a previous solution is that it throws off canonical indexing of the outerproduct.

Instead, ndgrid is ideal for this scenario:

a = [1; 2; 3];
b = [4; 5];
c = [6; 7; 8; 9];

[xx, yy, zz] = ndgrid(1:length(a), 1:length(b), 1:length(c));

% desired outerproduct
M = a(xx) .* b(yy) .* c(zz);

On paper, we can check that the desired solution M is the datacube:

M(:,:,1) = | M(:,:,2) = | M(:,:,3) = | M(:,:,4) = 
           |            |            |
24    30   | 28    35   | 32    40   | 36    45
48    60   | 56    70   | 64    80   | 72    90
72    90   | 84   105   | 96   120   | 108   135

Using the Kronecker product approach

M2 = reshape(kron(a * b', c), [length(a), length(b), length(c)]);

we would get:

M2(:,:,1) = | M2(:,:,2) = | M2(:,:,3) = | M2(:,:,4) = 
            |             |             |
24    36    | 64    84    | 30    45    | 80   105
28    48    | 72    96    | 35    60    | 90   120
32    56    | 72   108    | 40    70    | 90   135

Datacube M2 has the same elements as M, but these elements are rearranged. This is because kron(a * b', c) does not contain the slices of M in contiguous blocks to facilitate direct application of the reshape function. To compute outerproduct this way, we would need to apply a rearrangement operation/function (determinable, but labrious and time consuming) to the elements of kron(a * b', c).

A further advantage of using ndgrid is that it generalizes easily to higher orders.