# Speeding up summation inside comprehension

I currently do something similar to the code below (though I have axis arrays instead of regular arrays):

``````A = 1:100
B = 1:200
C = 1:300

x = rand(length(A),length(B),length(C))
y = rand(length(B),length(C))

z = [
sum(y[b,c]*x[a,b,c] for b=B)
for a=A, c=C
]
``````

I can see this potentially taking a very long time if my dimensions (A, B and C) get bigger, which they will eventually.

I know that in 2 dimensions I could do a vector/matrix operation which would be faster if I have a multi threaded processor (if I understand correctly), like the following:

``````x = rand(length(B), length(A))
y = rand(length(B), 1)
z = y' * x
``````

I was wondering if there’s an obvious way to do that for the first example?

This is like 4x faster on my machine, could definitely be improved still.

``````hcat([view(x,:,:,c)*view(y,:,c) for c in C]...)
``````
2 Likes

Cool! I didn’t know about the `view` function. I’ll try and adapt it to my code.

This is a tensor contraction, so maybe you can use https://github.com/Jutho/TensorOperations.jl

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Thanks for the recomendation! I implemented tbeason’s use of view and actually got a ~ 20 times speed up, so that’s good enough for me. Makes me want to go through my code and figure out what else I can improve…

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This one isn’t handled by TensorOperations:

``````TensorOperations.@tensor z[a,c] := y[b,c] * x[a,b,c] # IndexError{String}("non-matching indices
``````

but it can be done with various other packages:

``````Einsum.@einsum z1[a,c] := y[b,c] * x[a,b,c]
OMEinsum.@ein z2[a,c] := y[b,c] * x[a,b,c]
TensorCast.@reduce z3[a,c] := sum(b) y[b,c] * x[a,b,c]
Tullio.@tullio z4[a,c] := y[b,c] * x[a,b,c]

z5 = reshape(NNlib.batched_mul(x, reshape(y,200,1,300)), 100,300)
z ≈ z1 ≈ z2 ≈ z3 ≈ z4 ≈ z5
``````
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