I have two sparse matrices in Matlab, A and B,
and I want to compute a three-dimensional matrix C such that
C(i,j,k) = A(i,j) * B(j,k)
can I do this without a loop?
(Side question: Is there a name for this operation?)
Edit:
Seems my question has already been asked (just for full matrices):
Create a 3-dim matrix from two 2-dim matrices
For full matrices:
You can do it using bsxfun and shiftdim:
C = bsxfun(#times, A, shiftdim(B,-1))
Explanation: Let A be of size M x N and B of size N x P. Applying shiftdim(B,-1) gives a 1 x N x P array. bsxfun implicitly replicates A along the third dimension and shiftdim(B,-1) along the first to compute the desired element-wise product.
Another possibility, usually less efficient than bsxfun, is to repeat the arrays explicity along the desired dimensions, using repmat:
C = repmat(A, [1 1 size(B,2)]) .* repmat(shiftdim(B,-1), [size(A,1) 1 1])
For sparse matrices:
The result cannot be sparse, as sparse ND-arrays are not supported.. But you can do the computations with sparse A and B using linear indexing:
ind1 = repmat(1:numel(A),1,size(B,2));
ind2 = repmat(1:numel(B),size(A,1),1);
ind2 = ind2(:).';
C = NaN([size(A,1),size(A,2),size(B,2)]); %// preallocate with appropriate shape
C(:) = full(A(ind1).*B(ind2)); %// need to use full if C is to be 3D
Answer to your side question: the name for this operation is a hash join.
Related
[1 2 1]'\[1 2 3]' This is a numerical example. This example gives an answer of 1.333
From the documentation:
x = A\B
If A is a rectangular m-by-n matrix with m ~= n, and B is a matrix with m rows, then A\B returns a least-squares solution to the system of equations A*x= B.
Furthermore the ' compute the conjugate transposed of a matrix. In your case you have two real matrices so you just get the transposed each time.
For a vector a = [1 2 3]', I want to extend it to matrix [1:1+n;2:2+n;3:3+n]. Just like extend a number x to a sequential vector by x:x+a.
You can use bsxfun to add 0:n to each element in a
b = bsxfun(#plus, a, 0:n)
Or if you're using MATLAB R2016b, you can replace the bsxfun approach with the following thanks to automatic broadcasting of dimensions.
b = a + 0:n
For a 3N by 3N by 3N matrix A, I would like to derive a N by N by N matrix B whose entries come from summation over blocks in A.
For example, B(1,1,1) = sum of all elements of A(1:3,1:3,1:3).
Basically, A is kind of a high resolution matrix and B is a low resolution matrix from summing over entries in A.
If memory is not a concern, you can use a "labelling" approach: build a 3-component label to group the elements of A, and use that label as the first input argument to accumarray to do the sum. The label uses integers from 1 to N, so the result of accumarray already has the desired shape (NxNxN).
N = 5;
F = 3; %// block size per dimension
A = rand(15,15,15); %// example data. Size FN x FN x FN
[ii jj kk] = ind2sub(size(A), 1:numel(A));
label = ceil([ii.' jj.' kk.']/F);
result = accumarray(label, A(:));
reshape + sum based approach and as such has to be pretty efficient -
sumrows = sum(reshape(A,3,[]),1); %// Sum along rows
sumcols = sum(reshape(sumrows,N,3,[]),2); %// Sum along cols
B = reshape(sum(reshape(sumcols,N*N,3,[]),2),N,N,N); %// Sum along 3rd dim
If you are crazy about one-liners, here's that combining all steps into one -
B = reshape(sum(reshape(sum(reshape(sum(reshape(A,3,[]),1),N,3,[]),2),N*N,3,[]),2),N,N,N);
For a 2D matrix, this would work:
B = reshape(sum(im2col(A, [3 3], 'distinct')), [N N]);
NB: You need the image processing toolbox.
But for 3D matrices, I don't know of any built-in function equivalent to im2col. You might have to use a loop. Left as an exercise to the reader ;)
I feel there should be an easy solution but I can't find it:
I have the sparse matrices A B with the same dimension n*n. I want to create matrix C which copies values in A where B is non-zero.
This is my approach:
[r,c,v] = find(B);
% now I'd like to create an array of values using indices r and c,
% but this doesn't work (wrong syntax)
v2 = A(r,c);
% This won't work either
idx = find(B); % linear indexing, too high-dimensional
v2 = A(idx);
% and create C
C = sparse(r,c,v2,n,n);
Here are some more details:
My matrices are very large, so the solution needs to be efficient. C(B~=0) = B(B~=0); won't do it, unfortunately.
Linear indexing won't work either as the matrices are too large (Matrix is too large to return linear indices.).
Is there really no way to use 2-dimensional indices?
Thanks for your help!
I think C = A .* (B~=0); should work. Only non-zeros will be accessed in the entrywise multiplication of two sparse matrices so it will be fast.
Is there a way to combine 2 vectors in MATLAB such that:
mat = zeros(length(C),length(S));
for j=1:length(C)
mat(j,:)=C(j)*S;
end
Using normal MATLAB syntax similar to:
mat = C * S(1:length(S))
This gives a "Inner matrix dimensions must agree error" because it's trying to do normal matrix operations. This is not a standard Linear Algebra operation so I'm not sure how to correctly express it in MATLAB, but it seems like it should be possible without requiring a loop, which is excessively slow in MATLAB.
From your description, it sounds like a simple matrix operation. You just have to make sure you have the right dimensions for C and S. C should be a column vector (length(C)-by-1) and S should be a row vector (1-by-length(S)). Assuming they are the right dimensions, just do the following:
mat = C*S;
If you're not sure of their dimensions, this should work:
mat = (C(:))*(S(:)');
EDIT: Actually, I went a little crazy with the parentheses. Some of them are unnecessary, since there are no order-of-operation concerns. Here's a cleaner version:
mat = C(:)*S(:)';
EXPLANATION:
The matrix multiplication operator in MATLAB will produce either an inner product (resulting in a scalar value) or an outer product (resulting in a matrix) depending on the dimensions of the vectors it is applied to.
The last equation above produces an outer product because of the use of the colon operator to reshape the dimensions of the vector arguments. The syntax C(:) reshapes the contents of C into a single column vector. The syntax S(:)' reshapes the contents of S into a column vector, then transposes it into a row vector. When multiplied, this results in a matrix of size (length(C)-by-length(S)).
Note: This use of the colon operator is applicable to vectors and matrices of any dimension, allowing you to reshape their contents into a single column vector (which makes some operations easier, as shown by this other SO question).
Try executing this in MATLAB:
mat = C*S'
As In:
C = [1; 2; 3];
S = [2; 2; 9; 1];
mat = zeros(length(C),length(S));
for j=1:length(C)
mat(j,:)=C(j)*S;
end
% Equivalent code:
mat2 = C*S';
myDiff = mat - mat2
Do you mean the following?
mat = zeros(length(C),length(S));
for j=1:length(C)
mat(j,:)=C(j)*S;
end
If so, it's simply matrix multiplication:
C' * S % if C and S are row vectors
C * S' % if C and S are column vectors
If you don't know whether C and S are row vectors or column vectors, you can use a trick to turn them into column vectors, then transpose S before multiplying them:
C(:) * S(:)'
I'm not entirely clear on what you're doing - it looks like your resulting matrix will consist of length(C) rows, where the ith row is the vector S scaled by the ith entry of C (since subscripting a vector gives a scalar). In this case, you can do something like
mat = repmat(C,[1 length(S)]) .* repmat(S, [length(C) 1])
where you tile C across columns, and S down rows.
Try this:
C = 1:3
S = 1:5
mat1 = C'*S
mat2 = bsxfun(#times, C',S)
(esp. good when the function you need isn't simpler MATLAB notation)
--Loren
Try using meshgrid:
[Cm, Sm] = meshgrid(C, S);
mat = Cm .* Sm;
edit: nevermind, matrix multiplication will do too. You just need one column vector C and one row vector S. Then do C * S.