I have a sparse matrix which only has elements in three diagonals. E.g.
I also have a column vector where I wish to multiply every element in each row of the sparse matrix by the corresponding element in each row of the column vector. Is there an efficient way to do this in MATLAB? If the sparse matrix is called A and the column vector B, I've only tried
A.*repmat(B,[1,9])
which is obviously inefficient.
Here's one way:
C = bsxfun(#times, A, B)
According to docs, the resulting matrix C is sparse:
Binary operators yield sparse results if both operands are sparse, and full results if both are full. For mixed operands, the result is full unless the operation preserves sparsity. If S is sparse and F is full, then S+F, S*F, and F\S are full, while S.*F and S&F are sparse. In some cases, the result might be sparse even though the matrix has few zero elements.
Related
Swift's library "Accelerate" has sparse matrix types and several classes of functions for sparse matrix multiplication with different argument types, and BLAS-like functions with sparse matrices and vectors.
Interestingly, there are no functions that produce a sparse vector from a sparse matrix dot product with a sparse vector. (Or at least I did not see any in
Accelerate's documentation
.)
It looks like the workflow using
SparseVector_double for d = S . v
could be:
Convert the sparse vector v into a dense vector (or matrix)
Use the function SparseMultiply
Make the dense result d sparse
Alternative workflows with the BLAS functions are possible, say, using the function
sparse_matrix_product_sparse_double, but, again, the result is dense and has to be converted into a sparse vector/matrix.
I have several questions:
Is my conjecture that there is no direct way of getting a sparse vector from a dot product correct?
What is the fastest and easiest way to convert the dense vector/matrix results from the dot product functions into a sparse vector/matrix?
I should just scan the 0's with a loop, or there are relevant library functions?
What are the reasons none of these functions produce sparse structures?
In Matlab, I have created a matrix A with size (244x2014723)
and a matrix B with size (244x1)
I was able to calculate the correlation matrix using corr(A,B) which yielded in a matrix of size 2014723x1. So, every column of matrix A correlates with matrix B and gives one row value in the matrix of size 2014723x1.
My question is when I ask for a covariance matrix using cov(A,B), I get an error saying A and B should be of same sizes. Why do I get this error? How is the method to find corr(A,B) any different from cov(A,B)?
The answer is pretty clear if you read the documentation:
cov:
If A and B are matrices of observations, cov(A,B) treats A and B as vectors and is equivalent to cov(A(:),B(:)). A and B must have equal size.
corr
corr(X,Y) returns a p1-by-p2 matrix containing the pairwise correlation coefficient between each pair of columns in the n-by-p1 and n-by-p2 matrices X and Y.
The difference between corr(X,Y) and the MATLAB® function corrcoef(X,Y) is that corrcoef(X,Y) returns a matrix of correlation coefficients for the two column vectors X and Y. If X and Y are not column vectors, corrcoef(X,Y) converts them to column vectors.
One way you could get the covariances of your vector with each column of you matrix is to use a loop. Another way (might be in-efficient depending on the size) is
C = cov([B,A])
and then look at the first row (or column) or C.
See link
In the more about section, the equation describing how cov is computed for cov(A,B) makes it clear why they need to be the same size. The summation is over only one variable which enumerates the elements of A,B.
I am generating a random binary matrix with a specific number of ones in each row. Now, I want to take each row in the matrix and multiply it by its transpose (i.e row1'*row1).
So, I am using row1=rnd_mat(1,:) to get the first row. However, in the multiplication step I get this error
"Both logical inputs must be scalar. To compute elementwise TIMES, use TIMES (.*) instead."
Knowing that I don't want to compute element-wise, I want to generate a matrix using the outer product. I tried to write row1 manually using [0 0 1 ...], and tried to find the outer product. I managed to get the matrix I wanted.
So, does anyone have some ideas on how I can do this?
Matrix multiplication of logical matrices or vectors is not supported in MATLAB. That is the reason why you are getting that error. You need to convert your matrix into double or another valid numeric input before attempting to do that operation. Therefore, do something like this:
rnd_mat = double(rnd_mat); %// Cast to double
row1 = rnd_mat(1,:);
result = row1.'*row1;
What you are essentially computing is the outer product of two vectors. If you want to avoid casting to double, consider using bsxfun to do the job for you instead:
result = bsxfun(#times, row1.', row1);
This way, you don't need to cast your matrix before doing the outer product. Remember, the outer product of two vectors is simply an element-wise multiplication of two matrices where one matrix is consists of a row vector where each row is a copy of the row vector while the other matrix is a column vector, where each column is a copy of the column vector.
bsxfun automatically broadcasts each row vector and column vector so that we produce two matrices of compatible dimensions, and performs an element by element multiplication, thus producing the outer product.
I have N kx1 sparse vectors and I need to multiply each of them by their transpose, creating N square matrices, which I then have to sum over. The desired output is a k by k matrix. I have tried doing this in a loop and using arrayfun, but both solutions are too slow. Perhaps one of you can come up with something faster. Below are specific details about the best solution I've come up with.
mdev_big is k by N sparse matrix, containing each of the N vectors.
fun_sigma_i = #(i) mdev_big(:,i)*mdev_big(:,i)';
sigma_i = arrayfun(fun_sigma_i,1:N,'UniformOutput',false);
sigma = sum(reshape(full([sigma_i{:}]),k,k,N),3);
The slow part of this process is making sigma_i full, but I cannot reshape it into a 3d array otherwise. I've also tried cat instead of reshape (slower), ndSparse instead of full (way slower), and making fun_sigma_i return a full matrix rather than a sparse one (slower).
Thanks for the help! ,
I have to perform this operation:
N = A'*P*A
The structure of the P matrix is block diagonal while the A matrix is largely sparse (also in a banded structure). The multiplication is performed in blocks. But the problem is storage.
The N matrix is too huge to store in full (out of memory when trying to allocate). So, I want to store in a sparse fashion. While the sparse command generates only the values in row,column format, can it be applied to store banded matrices with the row column as the index of the block?
I have tried spalloc given in the this question but it hasnt helped storing the row and index of the block.
Thank you.
Image for A P A' formation
The problem lies in the blocks. The blocks are themselves sparse. So is it possible to make blocks as sparse matrices themselves while saving.
So, if a block has a row = 1 and col = 1, then can this be done?
N(row,col) = sparse(A'*P*A)
There may be some additional tricks to play but the first thing to try is to make sure the full matrix N is never created in memory. The immediate problem is that if you call sparse(A'*P*A) then you multiple A'*P then (A'*P)*A and only then do you make it sparse and take out the zeros. Right before making it sparse, the entire non-sparse matrix representation of N is in memory. To force MATLAB to be smarter do the following:
SA = sparse(A);
N = SA'*sparse(P)*SA;
whos N
You should see that N is sparse but, more importantly, each multiplication result is sparse as well because you are multiplying a sparse matrix times a sparse matrix.