add GPU log!, inverse_retract_polar!, parallel_transport_direction!, distance for Grassmann

[zazabap]

Summary

Add batched GPU overrides for Grassmann log!, inverse_retract_polar!, parallel_transport_direction!, and distance on PowerManifold{ℝ, <:Grassmann{ℝ}} with CuArray{T,3}. Covers steps 3 + 4 from the roadmap (#5).

Batched right-division q / (p'q) is computed as q * inv(p'q) via existing _batched_inv_gpu (safe for small, well-conditioned k×k matrices on Grassmann).

Operations added

Method	Algorithm
inverse_retract_polar!	q * inv(p'q) - p via batched LU inverse + GEMM
inverse_retract!	Dispatcher for PolarInverseRetraction
log!	Inverse retract → batched SVD → U * diag(atan(S)) * V'
parallel_transport_direction!	SVD of direction + sin/cos rotation + projection (avoids n×n identity)
vector_transport_direction!	Dispatcher for ParallelTransport
distance	Inverse retract → batched SVD → norm(atan.(S))

Files changed

File	Change
ext/ManifoldsGPUCUDAExt/Grassmann.jl	+104 lines: 6 new methods
ext/ManifoldsGPUCUDAExt/ManifoldsGPUCUDAExt.jl	+6 lines: imports for new functions
test/cuda/test_grassmann.jl	+231 lines: 14 test blocks (Float64/Float32 + large-matrix fallback)
benchmarks/utils.jl	+32 lines: _benchmark_distance helper
benchmarks/Grassmann.jl	+4 lines: distance benchmark entry

Benchmarks

Grassmann(32, 16), batch=2048, Float32 — within gesvdj! limit:

Operation	CPU (ms)	GPU (ms)	Speedup	Relative error
exp	167.38	5.99	27.93x	7.02e-5
log!	151.02	3.76	40.12x	1.25e-5
inner	0.59	0.10	5.88x	2.64e-6
norm	1.52	0.09	17.43x	1.85e-7
project!	2.25	0.19	12.01x	1.32e-7
retract_fused!(PolarRetraction)	91.29	3.25	28.05x	1.33e-6
distance	146.90	3.62	40.53x	0.0

Grassmann(64, 32), batch=2048, Float32 — exceeds gesvdj! limit, uses gesvda! fallback:

Operation	CPU (ms)	GPU (ms)	Speedup	Relative error
exp	720.21	225.22	3.20x	8.21e-5
log!	510.03	122.64	4.16x	9.38e-5
inner	2.21	0.12	18.40x	3.76e-4
norm	5.75	0.11	53.52x	1.31e-7
project!	9.67	0.56	17.31x	1.96e-7
retract_fused!(PolarRetraction)	327.76	116.23	2.82x	6.84e-7
distance	500.25	126.54	3.95x	6.55e-8

SVD-heavy operations (log!, distance, exp, polar retract) get ~40x speedup within the gesvdj! 32×32 limit and ~4x with the gesvda! fallback for larger matrices.

Tests

• 14 new CUDA testsets: Float64/Float32 for each operation + large-matrix (64×32) fallback tests for log!, parallel_transport_direction!, and distance
• All 243 existing JLArray tests pass

[mateuszbaran]

Thanks, this looks good again 👍 . I only have one small suggestion for a small improvement 🙂 .