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POVectorization: friend or foe? bsxfun/arrayfun to avoid loops, repmat, permute, squeeze, etc
 

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  1. POVectorization: friend or foe? bsxfun/arrayfun to avoid loops, repmat, permute, squeeze, etc
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    <p>This question is related to this <a href="https://stackoverflow.com/questions/20029235/simplifying-a-computation-so-it-could-be-done-using-matrix-operations">question</a> and probably to <a href="https://stackoverflow.com/questions/19896630/apply-bsxfun-or-arrayfun-to-every-row-of-a-matrix">this other</a> as well.</p> <p>Suppose you have two matrices A and B. A is M-by-N and B is N-by-K. I want to obtain an M-by-K matrix C such that <code>C(i, j) = 1 - prod(1 - A(i, :)' .* B(:, j))</code>. I have tried some solutions in Matlab - I am here comparing their computation performance.</p> <pre><code>% Size of matrices: M = 4e3; N = 5e2; K = 5e1; GG = 50; % GG instances rntm1 = zeros(GG, 1); % running time of first algorithm rntm2 = zeros(GG, 1); % running time of second algorithm rntm3 = zeros(GG, 1); % running time of third algorithm rntm4 = zeros(GG, 1); % running time of fourth algorithm rntm5 = zeros(GG, 1); % running time of fifth algorithm for gg = 1:GG A = rand(M, N); % M-by-N matrix of random numbers A = A ./ repmat(sum(A, 2), 1, N); % M-by-N matrix of probabilities (?) B = rand(N, K); % N-by-K matrix of random numbers B = B ./ repmat(sum(B), N, 1); % N-by-K matrix of probabilities (?) %% First solution % One-liner solution: tic C = squeeze(1 - prod(1 - repmat(A, [1 1 K]) .* permute(repmat(B, [1 1 M]), [3 1 2]), 2)); rntm1(gg) = toc; %% Second solution % Full vectorization, using meshgrid, arrayfun and reshape (from Luis Mendo, second link above) tic [ii jj] = meshgrid(1:size(A, 1), 1:size(B, 2)); D = arrayfun(@(n) 1 - prod(1 - A(ii(n), :)' .* B(:, jj(n))), 1:numel(ii)); D = reshape(D, size(B, 2), size(A, 1)).'; rntm2(gg) = toc; clear ii jj %% Third solution % Partial vectorization 1 tic E = zeros(M, K); for hh = 1:M tmp = repmat(A(hh, :)', 1, K); E(hh, :) = 1 - prod((1 - tmp .* B), 1); end rntm3(gg) = toc; clear tmp hh %% Fourth solution % Partial vectorization 2 tic F = zeros(M, K); for hh = 1:M for ii = 1:K F(hh, ii) = 1 - prod(1 - A(hh, :)' .* B(:, ii)); end end rntm4(gg) = toc; clear hh ii %% Fifth solution % No vectorization at all tic G = ones(M, K); for hh = 1:M for ii = 1:K for jj = 1:N G(hh, ii) = G(hh, ii) * prod(1 - A(hh, jj) .* B(jj, ii)); end G(hh, ii) = 1 - G(hh, ii); end end rntm5(gg) = toc; clear hh ii jj C D E F G end prctile([rntm1 rntm2 rntm3 rntm4 rntm5], [2.5 25 50 75 97.5]) % 3.6519 3.5261 0.5912 1.9508 2.7576 % 5.3449 6.8688 1.1973 3.3744 3.9940 % 8.1094 8.7016 1.4116 4.9678 7.0312 % 8.8124 10.5170 1.9874 6.1656 8.8227 % 9.5881 12.0150 2.1529 6.6445 9.5115 mean([rntm1 rntm2 rntm3 rntm4 rntm5]) % 7.2420 8.3068 1.4522 4.5865 6.4423 std([rntm1 rntm2 rntm3 rntm4 rntm5]) % 2.1070 2.5868 0.5261 1.6122 2.4900 </code></pre> <p>The solutions are equivalent but the algorithms with partial vectorization are way more efficient in terms of memory and execution time. Even the triple loop seems to perform better than arrayfun! Is there any approach that is actually better than the third, only partially vectorized solution?</p> <p><strong>EDIT</strong>: Dan's solutions are the best so far. Let rntm6, rntm7 and rntm8 be the runtime of his first, second and third solution. Then:</p> <pre><code>prctile(rntm6, [2.5 25 50 75 97.5]) % 0.6337 0.6377 0.6480 0.7110 1.2932 mean(rntm6) % 0.7440 std(rntm6) % 0.1970 prctile(rntm7, [2.5 25 50 75 97.5]) % 0.6898 0.7130 0.9050 1.1505 1.4041 mean(rntm7) % 0.9313 std(rntm7) % 0.2276 prctile(rntm8, [2.5 25 50 75 97.5]) % 0.5949 0.6005 0.6036 0.6370 1.3529 mean(rntm8) % 0.6753 std(rntm8) % 0.1890 </code></pre>
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    1. COA note: I would be weary calling `arrayfun` a way to fully vectorize, internally it also just loops. And loops in Matlab these days are actually pretty efficient. So often arrayfun just adds overhead
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