SparseArrays: `mul!(W, X', V)` much slower than `mul!(V, X, W)` for `Float32` entries

[severinson]

I'm seeing surprisingly low performance for mul!(W, X', V) when X is a SparseMatrixCSC with Float64 entries (X=sprand(2504, 100000, 0.05)) and W and V are dense matrices with Float32 entries. This operation takes about an order of magnitude longer than the same operation when W and V are dense matrices with Float64 entries. However, if X has Bool entries (X=sprand(Bool, 2504, 100000, 0.05)) I don't see any performance difference between Float64 and Float32 entries for V and W.

> using SparseArrays, LinearAlgebra

# Float64 X, V and W
> X = sprand(2504, 100000, 0.05); V = randn(2504, 3); W = zeros(100000, 3);

## transposed X
> @time mul!(W, X', V);
0.082900 seconds (80.11 k allocations: 4.399 MiB)
> @time mul!(W, X', V);
0.033305 seconds (1 allocation: 48 bytes)

## non-transposed X
> @time mul!(V, X, W);
0.122455 seconds (46.03 k allocations: 2.414 MiB)
> @time mul!(V, X, W);
0.088595 seconds

# Float64 X, Float32 V and W
> X = sprand(2504, 100000, 0.05); V = randn(Float32, 2504, 3); W = zeros(Float32, 100000, 3);

## transposed X
> @time mul!(W, X', V);
0.369262 seconds (77.55 k allocations: 4.190 MiB)
> @time mul!(W, X', V);
0.324316 seconds (1 allocation: 48 bytes) # about 10x slower than the same operation with Float64 entries

## non-transposed X
> @time mul!(V, X, W);
 0.123769 seconds (46.30 k allocations: 2.425 MiB)
> @time mul!(V, X, W);
0.087341 seconds # about the same performance as the same operation with Float64 entries

> versioninfo()
Julia Version 1.5.4
Commit 69fcb5745b (2021-03-11 19:13 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: AMD Ryzen 7 1700X Eight-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-9.0.1 (ORCJIT, znver1)
> LinearAlgebra.versioninfo()
BLAS: libopenblas (OpenBLAS 0.3.9  USE64BITINT DYNAMIC_ARCH NO_AFFINITY Zen MAX_THREADS=32)
LAPACK: libopenblas64_

[dkarrasch]

I have benchmarked the hell out of different approaches to this in JuliaLang/julia#38876 (just as others before me, see the references there), but couldn't find a better way to do it. The lesson so far is that, if you can afford memory-wise, then check if materializing the adjoint/transpose before multiplication is beneficial. That may depend strongly on your specific use case, like sparsity, size etc. Materializing the adjoint is not necessarily beneficial, though:

julia> X = sprand(2504, 100000, 0.05); V = randn(Float32, 2504, 3); W = zeros(Float32, 100000, 3);

julia> @btime mul!($W, copy(($X)'), $V, true, false);
  365.290 ms (9 allocations: 191.07 MiB)

julia> @btime mul!($W, $(X'), $V, true, false);
  266.900 ms (0 allocations: 0 bytes)

Apparently, most of the time is spent on conversion:

julia> X = sprand(Float32, 2504, 100000, 0.05); V = randn(Float32, 2504, 3); W = zeros(Float32, 100000, 3);

julia> @btime mul!($W, copy(($X)'), $V, true, false);
  372.054 ms (11 allocations: 143.27 MiB)

julia> @btime mul!($W, $(X'), $V, true, false);
  44.057 ms (0 allocations: 0 bytes)

Not sure how much error it would introduce to first downscale X to Float32.

[vtjnash]

Closing as "couldn't find a better way to do it" seems to be about optimal