Fast way to turn matrix columns into unit vectors - matlab

Question

I have a matrix:
S = [ -1.0400 4.9100 4.1000 -3.5450 -0.6600 -0.9300 4.3950 -1.0650 2.9850 -4.9800 0.2100;
-0.5200 -4.3150 -3.0950 0.5700 4.4700 1.1500 3.1350 0.6450 0.3750 -4.9150 -2.1150;
5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 ];
I want to convert the columns to unit vectors, so I use a for loop
for i=1:size(S,2)
S(:,i) = S(:,i) / norm( S(:,i) );
end
Is there a way to do this more efficiently in MATLAB?

Answer 1

TLDR
If you have MATLAB 2016b or newer, and no compatibility concerns, I would use
S = S ./ sqrt(sum(S.^2,1));
Edit: See benchmark at the bottom for performance benchmark of alternatives.
Context
We can just manually calculate the norm and divide column-wise.
By definition, norm(x) = sqrt( sum( x(:).^2 ) ). I've used (:) here to show that norm is calculated over the whole matrix. What's useful for us is that sum works column-wise by default, so the column-wise norm is defined like so:
nrm = sqrt( sum( x.^2 ) );
Note that if there's a possibility of your matrix S only having 1 row, you should ecplicitly enforce column-wise summation using nrm = sqrt(sum(x.^2,1)).
Now we have several options for division:
Implicit expansion (MATLAB R2016b or newer)
S = S ./ nrm;
Implicit expansion using bsxfun (all MATLAB versions)
S = bsxfun( #mrdivide, S, nrm );
Manual expansion using repmat (all MATLAB versions)
S = S ./ repmat(nrm, size(S,1), 1);
If you have MATLAB R2017b or newer, and again no compatability concerns, you can use vecnorm, which can be used in place of the manual norm calculation
S = S ./ vecnorm(S, 2, 1);
Benchmark:
Since you asked for performance, here is a simple benchmark for testing the speed of these different methods. Specifically the original loop in your question versus implicit expansion with either vecnorm or the manual calculation.
Results (run using R2017b)
size(S): 1e3*1e2 1e5*1e3 1e3*1e6
Looping: 0.0005 1.0186 12.7788
Implicit manual: 0.0001 1.1236 10.4031
Implicit vecnorm: 0.0002 0.5774 6.8058
Conclusions
For relatively small arrays, all of the methods are very fast and I would opt for code clarity over performance.
If you only want to use MATLAB versions which support it, vecnorm is approximately twice as quick as other methods for large matrices.
For matrices of the order 1e5*1e3, looping is comparable to implicit expansion.
Code
function benchie()
S = rand( 1e3, 1e2 )*5;
f1 = #() loopingNorm(S);
f2 = #() implicitManual(S);
f3 = #() implicitVecnorm(S);
fprintf( 'Looping: %.4f\nImplicit manual: %.4f\nImplicit vecnorm: %.4f\n', ...
timeit(f1), timeit(f2), timeit(f3) );
end
function S = loopingNorm(S)
for ii = 1:size(S,2)
S(:,ii) = S(:,ii) / norm( S(:,ii) );
end
end
function S = implicitManual(S)
S = S ./ sqrt(sum(S.^2,1));
end
function S = implicitVecnorm(S)
S = S ./ vecnorm( S, 2, 1 );
end