I was wondering if someone could help me with my problem.
Let say that I have the coordinates of MxN vectors in a tensor r of dimensions [M,N,3]. I would like to save in a 3M-by-3N block matrix all dyadic products r_0'*r_0, where r_0 is the vector r_0 = r(m,n,:) for some m and n, and I would like to do this without using for loops.
If haven't explain myself correctly, here is an example code that shows what I would like to obtain (but using for loops, of course):
N=10;
M=5;
r=rand(M,N,3);
Dyadic=zeros(3*M,3*N);
for m=1:M
a1=3*m-2;
a2=3*m;
for n=1:N
b1=3*n-2;
b2=3*n;
aux(3)=r(m,n,3);
aux(2)=r(m,n,2);
aux(1)=r(m,n,1);
Dyadic(a1:a2,b1:b2)=transpose(aux)*aux
end
end
Thanks in advance!
You need to use bsxfun(#times and then re-arrange elements to have the desired output -
%// Get the multipliication result
mat_mult = bsxfun(#times,permute(r,[1 2 4 3]),r);
%// OR if you would like to keep mat_mult as 3D that could be potentially faster -
%// mat_mult = bsxfun(#times,reshape(r,[],3),permute(reshape(r,[],3),[1 3 2]));
%// Re-arrange elements to have them the way you are indexing in nested loops
Dyadic = reshape(permute(reshape(mat_mult,M,N,3,[]),[3 1 4 2]),M*3,N*3);
The major play about this solution is really the re-arrangement of elements after we have the multiplication result.
Quick runtime tests with the input r as 1000 x 1000 x 3 sized array, show that this bsxfun based approach gives over 20x speedup over the nested loop code listed in the question!
Related
I've been trying to vectorize a for loop in Matlab because it bottlenecks the whole program but I have no clue about how to do it when iterating on different rows or columns of a single matrix inside of the loop. Here's the code :
// X is an n*k matrix, W is n*n
// W(i,j) stores the norm of the vector resulting from subtracting
// the j-th line of X from its i-th line.
for i = 1:n
for j = 1:n
W(i,j) = norm(X(i,:) - X(j,:))
end
end
Edit :
I chose Luis Mendo's answer as it was the most convenient for me and the one that's closer to the mathematical concept behind my algorithm, yet all three answers were correct and I'd advise to use the one that's the most convenient depending on which Matlab Toolboxes you have or the way you want to code.
I also noticed that the common point of all answers was to use a different format, for example : using an array that would store the indices, reshaping current arrays, using one more dimension...
So I think that's the right thing to explore if you have a similar problem.
Using combinatory:
% sample data
X = randi(9,5,4);
n = size(X,1);
% row index combinations
combIdx = combvec(1:n,1:n);
% difference between row combinations
D = X(combIdx(1,:),:)-X(combIdx(2,:),:);
% norm of each row
W = diag(sqrt(D*D'));
% reshape
W = reshape(W,n,[]);
Here is another solution that may be more efficient for big matrices.
For Matlab >=R2016 you can simple use
W = sqrt(sum(abs(reshape(X,n,1,k) - reshape(X,1,n,k)).^2,3))
(if your array is real-valued you can also skip the abs). For earlier versions of Matlab you need to add some repmat magic, namely:
W = sqrt(sum(abs(repmat(reshape(X,n,1,k),[1,n,1]) - repmat(reshape(X,1,n,k),[n,1,1])).^2,3));
P.S.: R2017b might make it even more convenient, at least the release notes mention a function called vecnorm that could replace the somewhat ugly sqrt(sum(abs(.).^2))) thing. But the documentation is not up yet so I don't know what exactly it will do. "
If you have the Statistics Toolbox, use pdist:
W = squareform(pdist(X));
No toolbox: good old bsxfun:
W = sqrt(sum(bsxfun(#minus, permute(X, [1 3 2]), permute(X, [3 1 2])).^2, 3));
I'd love to know if there is a more efficient way to multiply specific elements of multi-dimensional matrices that doesn't require a 'for' loop.
I have a region * time matrix for an individual (say, 50 regions and 1000 timepoints) and I want to multiply each pair of regions at each timepoint to create a new matrix of the products of each region pair at each time point (50 x 50 x 1000). The way that I'm currently running it is:
for t = 1:1000
for i = 1:50
for j = 1:50
new(i,j,t) = old(i,t) .* old(j,t)
As I'm sure you can imagine, this is super slow. Any ideas on how i can fix it up so that it will run more quickly?
%some example data easy to trace
old=[1:5]'
old(:,2)=old*i
%multiplicatiion
a=permute(old,[1,3,2])
b=permute(old,[3,1,2])
bsxfun(#times,a,b)
permute is used to make 3d-matrices with dimensions n*1*m and 1*n*m out of the n*m input matrix. Changing the dimensions this way, new(i,j,k) can be calculated using new(i,j,k)=a(i,1,k)*b(1,j,k). Applying such operations element-by-element is what bsxfun was designed for.
Regarding bsxfun, try to understand simple 2d-examples like bsxfun(#times,[1:7],[1,10,100]') first
I am trying to improve the performance of my code by converting some iterations into matrix operations in Matlab. One of these is the following code and I need to figure out how can I avoid using loop in the operation.
Here gamma_ic & bow are two dimensional matrices.
c & z are variables set from outer iterations.
for z=1:maxNumber,
for c=1:K,
n = 0;
for y2=1:number_documents,
n = n+(gamma_ic(y2,c)*bow(y2,z));
end
mu(z,c) = n / 2.3;
end
end
Appreciate your assistance.
Edit. Added The loop for c and z. The iteration goes on till the maximum indices in gamma_ic & bow. Added mu which is another two dimensional matrix to show usage of n.
This should work for you to get mu, which seems to be the desired output -
mu = bow(1:number_documents,1:maxNumber).'*gamma_ic(1:number_documents,1:K)./2.3
Im am currently writing a code to implement a numerical approximation to the 3D steady state heat equation using finite difference matrix methods. This involves discritising the 2nd order PDE into the matrix A and solving Ax=b. where x is temperature at each of the specified grid points. Further information on this type of question can be found here:
http://people.nas.nasa.gov/~pulliam/Classes/New_notes/Matrix_ODE.pdf
To complete this problem, I have represented the 3D matrix A by a 2D array calling the values in the 1D array b using an indexing function of the form:
i+(j-1)*Nx+Nx*Ny*(k-1)
for the (i,j,k)th element of the 3D matrix where Nx, Ny, Nz are the number of points in the x,y,z coordinates. There ends up being a lot of loop computation in order to create the matrix A and b and I was wondering what is the most computationally efficient and less memory exhaustive way to run these loops, i.e. is it better to use something like
for j=1:Ny
for i=2:Nx-1
b(i+(j-1)*Nx)=D4;
end
end
for j=1:Ny
for i=2:Nx-1
b(i+(j-1)*Nx+Nx*Ny*(Nz-1))=D3;
end
end
or should I condense these into a single loop like:
for j=1:Ny
for i=2:Nx-1
b(i+(j-1)*Nx)=D4;
b(i+(j-1)*Nx+Nx*Ny*(Nz-1))=D3;
end
end
I have preallocated both the arrays A and b. Is there a vectorised way to do this also?
Assuming Nx, Ny, Nz, D3 and D4 to be scalars and that you are using pre-allocation for b with zeros, you may try this vectorized approach -
I = 2:Nx-1; %// Vectors to represent i
J = 1:Ny; %// Vectors to represent j
ind1 = bsxfun(#plus,I,[(J-1)*Nx]'); %//' Indices, 1st set of nested loops
ind2 = bsxfun(#plus,I,[(J-1)*Nx+Nx*Ny*(Nz-1)]'); %//' Indices, 2nd set of loops
b(ind1) = D4; %// Assign values for 1st set
b(ind2) = D3; %// Assign values for 2nd set
The second method should be slightly faster since it performs the same number of calculations with fewer increments of the loop variables. You can look into MATLAB's built-in stopwatch commands tic and toc to time your code. http://www.mathworks.com/help/matlab/ref/tic.html
Something more vectorized might be possible but I would need to know more about the format of the arrays that contain D3 and D4. The reshape() function might be able to help.
A question about matlab and randomisation of a 3d matrix respecting the rows and columns.
I have a n x n x s matrix M and I want to mess it up a bit, but with some control.
I can achieve my wish with a for loop
for j=1:size(M,3)
r=randperm(size(M,1));
random_M(:,:,j)=M(r,r,j);
end
Is there a way to perform this without having to loop over j? I need many randomisation iterations and could afford the benefits of indexing.
Cheers!
edit: Some more thoughts following Alexandrew's comments
I have created a function that randomises a squeezed version of M:
function randomMat=randomiseMat(Mat)
[rows,cols]=size(Mat);
r=randperm(rows);
randomMat=Mat(r,r);
then, using arrayfun I seem to get what I want:
randomM=arrayfun(#(x) randomiseMat(M(:,:,x)),1:size(M,3),'UniformOutput', false)
however, randomM is now a cell array of size (1,size(M,3)) with each cell containing randomised array.
Is there a way to make it in a 3d matrix just like the input M?
You can calculate all the values for r in one go, and then use arrayfun:
[nRows,nCols,nPages] = size(M);
[~,r]=sort(rand(nRows,nPages));
%# you should test on a realistic example whether a for-loop
%# isn't faster here
outCell = arrayfun(#(x) M(r(:,x),r(:,x),x), 1:nPages,'UniformOutput',false);
randomM = cat(3,outCell{:});