vector - Matlab: need some help for a seemingly simple vectorization of an operation

Question

I would like to optimize this piece of Matlab code but so far I have failed. I have tried different combinations of repmat and sums and cumsums, but all my attempts seem to not give the correct result. I would appreciate some expert guidance on this tough problem.

S=1000; T=10;
X=rand(T,S),
X=sort(X,1,'ascend');
Result=zeros(S,1);
for c=1:T-1
    for cc=c+1:T
        d=(X(cc,:)-X(c,:))-(cc-c)/T;
        Result=Result+abs(d');
    end
end

Basically I create 1000 vectors of 10 random numbers, and for each vector I calculate for each pair of values (say the mth and the nth) the difference between them, minus the difference (n-m). I sum over of possible pairs and I return the result for every vector.

I hope this explanation is clear,

Thanks a lot in advance.

Answer 1

It is at least easy to vectorize your inner loop:

Result=zeros(S,1);
for c=1:T-1
   d=(X(c+1:T,:)-X(c,:))-((c+1:T)'-c)./T;
   Result=Result+sum(abs(d),1)';
end

Here, I'm using the new automatic singleton expansion. If you have an older version of MATLAB you'll need to use bsxfun for two of the subtraction operations. For example, X(c+1:T,:)-X(c,:) is the same as bsxfun(@minus,X(c+1:T,:),X(c,:)).

What is happening in the bit of code is that instead of looping cc=c+1:T, we take all of those indices at once. So I simply replaced cc for c+1:T. d is then a matrix with multiple rows (9 in the first iteration, and one fewer in each subsequent iteration).

Surprisingly, this is slower than the double loop, and similar in speed to Jodag's answer.

Next, we can try to improve indexing. Note that the code above extracts data row-wise from the matrix. MATLAB stores data column-wise. So it's more efficient to extract a column than a row from a matrix. Let's transpose X:

X=X';
Result=zeros(S,1);
for c=1:T-1
   d=(X(:,c+1:T)-X(:,c))-((c+1:T)-c)./T;
   Result=Result+sum(abs(d),2);
end

This is more than twice as fast as the code that indexes row-wise.

But of course the same trick can be applied to the code in the question, speeding it up by about 50%:

X=X';
Result=zeros(S,1);
for c=1:T-1
   for cc=c+1:T
      d=(X(:,cc)-X(:,c))-(cc-c)/T;
      Result=Result+abs(d);
   end
end

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My takeaway message from this exercise is that MATLAB's JIT compiler has improved things a lot. Back in the day any sort of loop would halt code to a grind. Today it's not necessarily the worst approach, especially if all you do is use built-in functions.