benchmark: Horn vs. Kabsch throughput for 3D inputs

[hunter-heidenreich]

Goal

Direct throughput comparison of horn vs kabsch on identical 3D inputs across all supporting frameworks (PyTorch, JAX, TensorFlow, MLX).

Background

For 3D inputs both algorithms produce equivalent results (up to floating-point), but via different linear algebra:

• Kabsch: cross-covariance H (3×3), then SVD of H (3×3)
• Horn: cross-covariance H (3×3), construct N matrix (4×4), then eigh of N (4×4)

eigh on a symmetric 4×4 may be faster or slower than full SVD on a 3×3 depending on the framework's LAPACK dispatch. Additionally Horn involves more scalar arithmetic (9 quaternion-to-R ops expanded element-wise vs a single matmul in Kabsch). The relative performance is unknown and undocumented.

Experimental Design

• Fix D=3, sweep B over [1, 16, 256, 4096] and N over [10, 100, 1000]
• For each (B, N): measure kabsch and horn wall-clock time (median over 100 runs after warmup)
• Test in both eager and compiled modes
• Report: speedup ratio horn/kabsch (>1 means kabsch is faster)

Expected Deliverables

• Table or heatmap: speedup ratio across (B, N) for each framework
• Determination of whether Horn is ever meaningfully faster than Kabsch in practice
• A recommendation in the docs about which to prefer when both are available