[BlockSparseArrays] More general broadcasting and slicing

[mtfishman]

This is work in progress for defining map! for BlockSparseArrays with mismatched blocking.

This is functionality that is required to perform block fusion, as a follow-up to #1326.

So far, it performs some unnecessary copying, and also only works with one input array, which both should be relatively easy fixes.

• Add tests.
• For out-of-place expressions like a + b, currently it outputs a block sparse array with axes(a). If the axes/blocking of a and b are different, it should combine the axes (i.e. output a block sparse array with a finer blocking structure compatible with the blocking of a and b).
• map(+, a, b) outputs a dense block array, not a block sparse array.
• Fix issues with using GradedAxes as axes of block sparse arrays and performing mapping/broadcasting operations. There are some lingering bugs with that, and we need to make sure we preserve the sectors of the graded axes properly.
• Generalize new map! functionality to arbitrary number of input arrays.
• Avoid extraneous copying of source blocks (requires fixing a bug when using views of a BlockSparseArray).
• Investigate some other slicing errors I came across during this PR, either fix or add broken tests.

A demonstration of new functionality:

using BlockArrays: Block, blockedrange, blocksize
using NDTensors.BlockSparseArrays: BlockSparseArray
using Random: randn!

function blockdiagonal(f!, elt::Type, axes::Tuple)
  a = BlockSparseArray{elt}(axes)
  for i in 1:minimum(blocksize(a))
    b = Block(ntuple(Returns(i), ndims(a)))
    a[b] = f!(a[b])
  end
  return a
end

elt = Float64
d_src = blockedrange([4, 4])
d_dest = blockedrange([2, 4, 2])
a_src = blockdiagonal(randn!, elt, (d_src, d_src))
a_dest = blockdiagonal(randn!, elt, (d_dest, d_dest))

a_dest .= 10 .* a_src

which outputs:

julia> a_src
2×2-blocked 8×8 BlockSparseArray{Float64, 2, Matrix{Float64}, NDTensors.SparseArrayDOKs.SparseArrayDOK{Matrix{Float64}, 2, NDTensors.BlockSparseArrays.BlockZero{Tuple{BlockArrays.BlockedUnitRange{Vector{Int64}}, BlockArrays.BlockedUnitRange{Vector{Int64}}}}}, Tuple{BlockArrays.BlockedUnitRange{Vector{Int64}}, BlockArrays.BlockedUnitRange{Vector{Int64}}}}:
 -0.405138   0.627248   1.26039     0.152331  │   0.0        0.0        0.0        0.0     
  1.26651   -1.318      0.90538    -0.669214  │   0.0        0.0        0.0        0.0     
 -0.231649  -1.14072   -0.0183986   0.641401  │   0.0        0.0        0.0        0.0     
  0.398392   1.15866   -0.840044   -0.774931  │   0.0        0.0        0.0        0.0     
 ─────────────────────────────────────────────┼────────────────────────────────────────────
  0.0        0.0        0.0         0.0       │  -0.149512  -0.415454  -0.337988  -0.214309
  0.0        0.0        0.0         0.0       │  -1.08143   -0.88407    2.10251    0.180943
  0.0        0.0        0.0         0.0       │  -0.265952  -0.441355   1.02287   -0.343253
  0.0        0.0        0.0         0.0       │  -1.94984    0.913825  -0.61027   -0.232461

julia> a_dest
3×3-blocked 8×8 BlockSparseArray{Float64, 2, Matrix{Float64}, NDTensors.SparseArrayDOKs.SparseArrayDOK{Matrix{Float64}, 2, NDTensors.BlockSparseArrays.BlockZero{Tuple{BlockArrays.BlockedUnitRange{Vector{Int64}}, BlockArrays.BlockedUnitRange{Vector{Int64}}}}}, Tuple{BlockArrays.BlockedUnitRange{Vector{Int64}}, BlockArrays.BlockedUnitRange{Vector{Int64}}}}:
  1.89965   -0.957095   │   0.0        0.0        0.0        0.0        │   0.0       0.0      
 -0.366502   0.0968234  │   0.0        0.0        0.0        0.0        │   0.0       0.0      
 ───────────────────────┼───────────────────────────────────────────────┼──────────────────────
  0.0        0.0        │   0.366896   0.337547  -0.698596  -0.278881   │   0.0       0.0      
  0.0        0.0        │  -0.128684  -0.14473    0.751413  -0.561558   │   0.0       0.0      
  0.0        0.0        │  -0.369947   0.229541   0.949884  -0.996651   │   0.0       0.0      
  0.0        0.0        │  -0.422792  -0.339941   0.322883  -0.0676305  │   0.0       0.0      
 ───────────────────────┼───────────────────────────────────────────────┼──────────────────────
  0.0        0.0        │   0.0        0.0        0.0        0.0        │   0.546015  0.0357726
  0.0        0.0        │   0.0        0.0        0.0        0.0        │  -0.763948  1.39853  

julia> a_dest .= 10 .* a_src
3×3-blocked 8×8 BlockSparseArray{Float64, 2, Matrix{Float64}, NDTensors.SparseArrayDOKs.SparseArrayDOK{Matrix{Float64}, 2, NDTensors.BlockSparseArrays.BlockZero{Tuple{BlockArrays.BlockedUnitRange{Vector{Int64}}, BlockArrays.BlockedUnitRange{Vector{Int64}}}}}, Tuple{BlockArrays.BlockedUnitRange{Vector{Int64}}, BlockArrays.BlockedUnitRange{Vector{Int64}}}}:
 -4.05138    6.27248  │  12.6039     1.52331     0.0        0.0       │   0.0       0.0    
 12.6651   -13.18     │   9.0538    -6.69214     0.0        0.0       │   0.0       0.0    
 ─────────────────────┼───────────────────────────────────────────────┼────────────────────
 -2.31649  -11.4072   │  -0.183986   6.41401    -0.698596  -0.278881  │   0.0       0.0    
  3.98392   11.5866   │  -8.40044   -7.74931     0.751413  -0.561558  │   0.0       0.0    
  0.0        0.0      │  -0.369947   0.229541   -1.49512   -4.15454   │  -3.37988  -2.14309
  0.0        0.0      │  -0.422792  -0.339941  -10.8143    -8.8407    │  21.0251    1.80943
 ─────────────────────┼───────────────────────────────────────────────┼────────────────────
  0.0        0.0      │   0.0        0.0        -2.65952   -4.41355   │  10.2287   -3.43253
  0.0        0.0      │   0.0        0.0       -19.4984     9.13825   │  -6.1027   -2.32461

so it can handle cases where the blocking doesn't line up, while before this would have errored.

It works by first creating axes with a finer blocking structure that lines up with the blocking structure of the destination and source arrays, and then applies block-wise operations using that finer blocking structure.

[codecov-commenter]

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[emstoudenmire]

Very cool!

[mtfishman]

@ogauthe @emstoudenmire I'm merging this.

The status is that you should be able to perform arbitrary broadcasting operations (adding, scalar multiplication, permutations) of block sparse arrays, as well as perform pretty general slicing operations, and it should preserve block structures and block labels stored on the axes (including symmetry labels). It also needs more tests, but I want to merge this now as a starting point for future work. I'm sure there will be corner cases and bugs to work out as it is put into practice but I think it is a good starting point.