[WIP] ProductedArrays

[johnnychen94]

This introduces an intermediate Cartesian product array type that can be used for many operations, depending on how we're going to reduce the item. I'll give two examples that motivate this PR.

In simple words, ProductedArrays is the array version of Iterators.product:

julia> a = Float64[0.1 0.2; 0.3 0.4]
2×2 Matrix{Float64}:
 0.1  0.2
 0.3  0.4

julia> b = [1 2; 3 4]
2×2 Matrix{Int64}:
 1  2
 3  4

julia> c = ProductedArray(a, b)
2×2×2×2 ProductedArray{Tuple{Float64, Int64}, 4, Tuple{Matrix{Float64}, Matrix{Int64}}}:
[:, :, 1, 1] =
 (0.1, 1)  (0.2, 1)
 (0.3, 1)  (0.4, 1)

[:, :, 2, 1] =
 (0.1, 3)  (0.2, 3)
 (0.3, 3)  (0.4, 3)

[:, :, 1, 2] =
 (0.1, 2)  (0.2, 2)
 (0.3, 2)  (0.4, 2)

[:, :, 2, 2] =
 (0.1, 4)  (0.2, 4)
 (0.3, 4)  (0.4, 4)

TODO:

• performance tweak
• tests and docs

Example: ReadonlyMultiMappedArray

a = Float64[0.1 0.2; 0.3 0.4]
b = [1 2; 3 4]

plusc_1 = mappedarray(+, a, b)

p = ProductedArray(a, b)
plusc_2 = mappedarray(i->reduce(+, p[i, i]), CartesianIndices(a))

plusc_1 == plusc_2 # true

Theoretically, we could build the *MultiMappedArray functionality on top of ProductedArray and thus simplify the implementation. To minimize the change, I might take a try but won't do it in this PR.

Example: pairwise/patch-wise distances

using Distances
A = [rand(Float64, 7, 7) for _ in 1:7];
B = [rand(Float64, 7, 7) for _ in 1:7];

patchwise_d1 = [euclidean(a, b) for a in A, b in B]

p = ProductedArray(A, B);
patchwise_d2 = mappedarray(x->reduce(euclidean, x), p)

patchwise_d1 == patchwise_d2 # true

[codecov]

Codecov Report

Merging #42 (1c29d4a) into master (ca07b29) will decrease coverage by 4.75%.
The diff coverage is 0.00%.

@@            Coverage Diff             @@
##           master      #42      +/-   ##
==========================================
- Coverage   82.78%   78.03%   -4.76%     
==========================================
  Files           1        2       +1     
  Lines         122      132      +10     
==========================================
+ Hits          101      103       +2     
- Misses         21       29       +8     

Impacted Files	Coverage Δ
src/MappedArrays.jl	83.06% <ø> (+0.27%)	⬆️
src/ProductedArrays.jl	0.00% <0.00%> (ø)

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[FelixBenning]

@johnnychen94 since this PR aged a bit - are you still interested in this project? I wasn't aware of this effort when I created https://github.com/lazyLibraries/ProductArrays.jl (its publishing is on hold: JuliaRegistries/General#84683) you seem to have thought about the "product of arrays" edge cases too - I have written some tests already but made no performance considerations. Maybe we could combine this effort?

EDIT: I would also be very willing to move the ProductArrays.jl project into the JuliaArrays group. This might potentially make more sense than doing this under mappedarrays where few will find them. It also allows for selective import.