[zitongzhan]

BAE's ambient-gradient mode can produce incorrect translation derivatives with current PyPose and NaN Log derivatives at identity rotations. These failures affect chained pose residuals, including the initial Sphere big-noise Log residuals.

When BAE_USE_PYPOSE_AMBIENT_GRAD=1:

• Override SE3Type.translation with a differentiable tensor slice. PyPose's merged translation fix (Fix backward gradient calculation for SE3 and Sim3 translation pypose#422) returns a padded tangent gradient, while BAE needs ambient derivatives until the parameter-boundary retraction conversion.
• Protect the SO3 Log denominators before evaluating any branch, then select the existing regular, half-turn, or small-angle formula with torch.where. The previous masked nan_to_num expressions still differentiated divisions by zero in inactive branches. This also fixes SE3 Log, which calls SO3 Log. The principal-Log branch convention is preserved; the half-turn branch remains mathematically discontinuous.

Regressions cover unbatched/batched translation, relative-pose translation followed by BAE's parameter-Jacobian conversion, SO3/SE3 Log at both identity quaternion signs and slightly non-unit identity, mixed batches, half-turn numerical finiteness, and a chained PGO Log residual at zero error. Translation is also included in the existing compiled SE3 pipeline test.

Validation against PyPose main e37ed16688176596bfb60b1b9c525ba117e0688c, using PyTorch 2.7.1+cu126:

• All four translation finite-difference cases failed before their fix and passed afterward. All five new Log regression cases likewise failed before the Log fix and passed afterward.
• tests/autograd/test_pypose_ambient_grad.py: 14 passed on CUDA; 14 passed on CPU.
• Together with tests/autograd/test_pgo_jacobian.py: 18 passed on CUDA, including sparse/dense comparisons.
• All 8,647 initial SE3 Log Jacobians in sphere_bignoise_vertex3.g2o are finite; the previous ambient SO3 Log produced a NaN Jacobian for an identity-rotation residual.
• Direct SO3/SE3 Log helpers pass torch.compile(backend="eager", fullgraph=True) forward/backward checks at identity and half turns.
• Earlier full tests/autograd/test_pypose_compile.py run: 4 passed, 6 failed in local TorchDynamo LieTensor tracing. The unmodified release compile tests produced the same six failures in this environment; full LieTensor graph validation remains limited locally. CI uses PyTorch 2.10.
• git diff --check: passed.