This pull request introduces significant improvements to the pypose optimization and Lie tensor operations, with a focus on adding sparse optimization support and refactoring mathematical operations for better performance and readability.

Sparse Optimization Support

• Added optional sparse mode to the LevenbergMarquardt optimizer (pypose/optim/optimizer.py), leveraging the BAE library for efficient sparse matrix operations. This includes new logic for sparse Jacobian computation, parameter updates, and optimization steps. The optimizer now supports both dense and sparse workflows, with runtime checks for BAE availability. [1] [2] [3] [4]

• Implemented custom sparse Jacobian calculation and parameter update logic, including handling for SE3 gradients and integration with BAE’s sparse tensor formats. [1] [2]

Refactoring and Mathematical Improvements

• Refactored several mathematical operations in pypose/lietensor/operation.py to use element-wise arithmetic and broadcasting instead of advanced indexing, improving readability and performance. This includes changes in so3_Jl_inv, calcQ, and SE3-related functions. [1] [2] [3] [4] [5]

Solver and Linear Algebra Updates

• Improved the CG solver in pypose/optim/solver.py to handle input dimensions more robustly, replaced custom matrix-vector multiplication with native PyTorch operations. [1] [2]

API and Interface Changes

• Updated optimizer and model APIs for consistency, including changes to method signatures (e.g., optional target argument in forward methods) and improved documentation for new sparse features. [1] [2]

Cleanup

• Removed unused and deprecated code in pypose/sparse/ops.py related to sparse matrix multiplication registration.