Most Efficient Way to Construct the Matrices to Extract the Lower and Upper Triangle from a Vectorized Matrix
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Given a matrix X and its vector form I am after the most efficient way to build the matrices L and U which extracts the lower and upper triangle from X.
So in MATLAB code it would be something like that:
clear();
numRows = 3;
numCols = numRows;
mX = randn(numRows, numCols);
vX = mX(:);
% Lower Triangle are indices 2, 3, 6
mL = [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ; ...
0, 0, 1, 0, 0, 0, 0, 0, 0 ; ...
0, 0, 0, 0, 0, 1, 0, 0, 0 ];
% Upper Triangle are indices 4, 7, 8
mU = [ 0, 0, 0, 1, 0, 0, 0, 0, 0 ; ...
0, 0, 0, 0, 0, 0, 1, 0, 0 ; ...
0, 0, 0, 0, 0, 0, 0, 1, 0 ];
assert(isequal(mL * vX, mX(logical(tril(mX, -1)))));
assert(isequal(mU * vX, mX(logical(triu(mX, 1)))));
I am after sparse represenation of mU and mL in the most efficient way.
My current implementation is given by:
function [ mLU ] = GenerateTriangleExtractorMatrix( numRows, triangleFlag, diagFlag )
EXTRACT_LOWER_TRIANGLE = 1;
EXTRACT_UPPER_TRIANGLE = 2;
INCLUDE_DIAGONAL = 1;
EXCLUDE_DIAGONAL = 2;
switch(diagFlag)
case(INCLUDE_DIAGONAL)
numElements = 0.5 * numRows * (numRows + 1);
diagIdx = 0;
case(EXCLUDE_DIAGONAL)
numElements = 0.5 * (numRows - 1) * numRows;
diagIdx = 1;
end
vJ = zeros(numElements, 1);
if(triangleFlag == EXTRACT_LOWER_TRIANGLE)
elmntIdx = 0;
for jj = 1:numRows
for ii = (jj + diagIdx):numRows
elmntIdx = elmntIdx + 1;
vJ(elmntIdx) = ((jj - 1) * numRows) + ii;
end
end
elseif(triangleFlag == EXTRACT_UPPER_TRIANGLE)
elmntIdx = numElements + 1;
for jj = numRows:-1:1
for ii = (jj - diagIdx):-1:1
elmntIdx = elmntIdx - 1;
vJ(elmntIdx) = ((jj - 1) * numRows) + ii;
end
end
end
mLU = sparse(1:numElements, vJ, 1, numElements, numRows * numRows, numElements);
end
Is there a more efficient way to generate vJ without extensive allocation of memory (In order to allow generating really large matrices)?
Thank You.
24 Comments
Matt J
on 23 Apr 2020
But that would mean your constraints are of the form mL*X(:)<=b. But since each row of mL contains only a single non-zero element, this means the constraint is equivalent to a simple bound X(j)<=b. In Matlab, you would never have to construct a matrix to represent such a constraint. You would use the vector input arguments lb and ub to specify those. I assume Gurobi has something similar.
Accepted Answer
James Tursa
on 22 Apr 2020
Edited: James Tursa
on 22 Apr 2020
Here is a mex routine that generates the sparse double matrices mL and mU directly, so no wasted memory in creating them. Seems to run about 3x-5x faster than m-code for somewhat large sizes.
/* S = GenerateTriangleExtractorMatrixMex(numRows,triangleFlag,diagFlag)
*
* S = double sparse matrix
* numRows = integer > 0
* triangleFlag = 1 , extract lower triangle
* 2 , extract upper triangle
* diagFlag = 1 , include diagonal
* 2 , exclude diagonal
* where
*
* M = an numRows X numRows matrix of non-zero terms
* assert(isequal(S * M(:), mX(logical(tril(M, -1))))); % for lower
* assert(isequal(S * M(:), mX(logical(triu(M, 1))))); % for upper
*
* Programmer: James Tursa
* Date: 2020-April-22
*/
#include "mex.h"
void mexFunction( int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[])
{
mwSize numRows, triangleFlag, diagFlag, numElements;
mwIndex *Ir, *Jc;
mwIndex i, j, k, m;
double *pr;
if( nrhs != 3 || !mxIsNumeric(prhs[0]) || !mxIsNumeric(prhs[1]) || !mxIsNumeric(prhs[2]) ||
mxGetNumberOfElements(prhs[0]) != 1 || mxGetNumberOfElements(prhs[1]) != 1 ||
mxGetNumberOfElements(prhs[2]) != 1 ) {
mexErrMsgTxt("Need three numeric scalar inputs");
}
if( nlhs > 1 ) {
mexErrMsgTxt("Too many outputs");
}
numRows = mxGetScalar(prhs[0]);
triangleFlag = mxGetScalar(prhs[1]);
diagFlag = mxGetScalar(prhs[2]);
if( numRows < 1 ) {
mexErrMsgTxt("Invalid numRows, should be > 0");
}
if( triangleFlag != 1 && triangleFlag != 2 ) {
mexErrMsgTxt("Invalid triangleFlag, should be 1 or 2");
}
if( diagFlag != 1 && diagFlag != 2 ) {
mexErrMsgTxt("Invalid diagFlag, should be 1 or 2");
}
if( diagFlag == 1 ) {
numElements = numRows * (numRows + 1) / 2; /* include diagonal */
} else {
numElements = (numRows - 1) * numRows / 2; /* exclude diagonal */
}
plhs[0] = mxCreateSparse(numElements, numRows*numRows, numElements, mxREAL);
pr = (double *) mxGetData(plhs[0]);
Ir = mxGetIr(plhs[0]);
Jc = mxGetJc(plhs[0]);
Jc[0] = 0;
diagFlag--;
k = 0;
m = 1;
if( triangleFlag == 1 ) { /* Lower */
for( j=0; j<numRows; j++ ) {
for( i=0; i<numRows; i++ ) {
if( i >= j+diagFlag ) {
*pr++ = 1.0;
*Ir++ = k++;
Jc[m] = Jc[m-1] + 1;
} else {
Jc[m] = Jc[m-1];
}
m++;
}
}
} else { /* Upper */
for( j=0; j<numRows; j++ ) {
for( i=0; i<numRows; i++ ) {
if( i+diagFlag <= j ) {
*pr++ = 1.0;
*Ir++ = k++;
Jc[m] = Jc[m-1] + 1;
} else {
Jc[m] = Jc[m-1];
}
m++;
}
}
}
}
You mex the routine as follows (you need a supported C compiler installed):
mex GenerateTriangleExtractorMatrixMex.c
And some test code:
% GenerateTriangleExtractorMatrix_test.m
n = 300;
disp('m-code timing')
tic
GenerateTriangleExtractorMatrix(10000,1,1);
toc
disp('mex code timing')
tic
GenerateTriangleExtractorMatrixMex(10000,1,1);
toc
for k=1:n
numRows = ceil(rand*5000+100);
numCols = numRows;
triangleFlag = (rand<0.5) + 1;
diagFlag = (rand<0.5) + 1;
Mm = GenerateTriangleExtractorMatrix(numRows,triangleFlag,diagFlag);
Mx = GenerateTriangleExtractorMatrixMex(numRows,triangleFlag,diagFlag);
if( ~isequal(Mm,Mx) )
error('Not equal');
end
end
disp('Random tests passed')
With a sample run:
>> GenerateTriangleExtractorMatrix_test
m-code timing
Elapsed time is 9.964882 seconds.
mex code timing
Elapsed time is 1.901741 seconds.
Random tests passed
4 Comments
More Answers (2)
Matt J
on 23 Apr 2020
Edited: Matt J
on 23 Apr 2020
Another approach to consider is to use my MatrixObj class
to construct an object that has the same effect as the operations mL*X and mL.'*Y, but doesn't require you to actually build the matrix,
N=5000;
tic;
mL0=GenerateTriangleExtractorMatrix( N, 1, 2);
toc
%Elapsed time is 0.678702 seconds.
tic;
B=tril(true(N),-1);
Bd=double(B(:));
mL=MatrixObj;
mL.Params.B=B;
mL.Params.Bd=Bd;
mL.Ops.mtimes=@(obj,z) z(obj.Params.B);
mL.Trans.mtimes=@mtimesT;
toc;
%Elapsed time is 0.086228 seconds.
function out=mtimesT(obj,z)
out=obj.Params.Bd;
out(obj.Params.B)=z;
end
In addition to requiring less time to construct, you can verify that it gives the same results as multiplications with mL and mL.',
>> X=rand(N^2,1); isequal(mL0.'*(mL0*X),mL.'*(mL*X))
ans =
logical
1
but with considerably less memory consumption:
>> whos mL mL0
Name Size Kilobytes Class Attributes
mL 1x1 219739 MatrixObj
mL0 12497500x25000000 390586 double sparse
Royi Avital
on 21 Apr 2020
2 Comments
Tommy
on 21 Apr 2020
The two methods are fairly similar - I also like that yours minimizes memory allocation. I ran a few simple fun tests:
I didn't dare try higher than 20,000 for numRows. It seems that your code may possibly perform better at higher values of numRows. In the second case (calculating both the upper and lower triangles) I had your code running both sets of for loops, one after the other (shown in red). In green is the result from your code if only the first set of loops runs, and you recognize that vJ for one triangle is easy to determine if you have vJ for the other triangle (N^2+1-flip(vJ)). So the only thing I'll conclude from this is, if you will eventually calculate both the lower and upper triangle matrices for a given size, it might be better to calculate them together and only find vJ once. I suppose it depends on how expensive N^2+1-flip(vJ) is.
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