[Feature]: Add Benchmarks for kornia-algebra Optimization and Lie Group Modules

🚀 Feature Description

This feature request proposes the implementation of a comprehensive benchmarking suite for the kornia-algebra crate. Specifically, it targets the newly introduced optimization module (optim) and expands the existing Lie Group benchmarks to provide comparative performance metrics against established libraries like nalgebra and sophus.

📂 Feature Category

Rust Core Library

💡 Motivation

• Performance Tracking: Currently, there are limited benchmarks for the optimization engine. As the library grows, it is crucial to track performance regressions.
• Comparative Analysis: To ensure kornia-rs remains a high-performance choice for computer vision in Rust, we need to baseline its geometric operations against standard ecosystems like nalgebra and sophus.
• Optimization Validation: Developers need a way to quantitatively verify that changes to the solver (e.g., different linear system solvers, trust-region strategies) actually improve runtime performance.

💭 Proposed Solution

I propose adding the following benchmark files and updates:

1. bench_optim.rs:
   • Implement the Rosenbrock Function as a Factor trait.
   • Use this function to benchmark the convergence speed and iteration time of the LevenbergMarquardt solver.
   • Compare execution time for fixed iteration counts.
2. bench_lie_group.rs:
   • Add sophus as a dev-dependency to facilitate direct comparison.
   • Implement side-by-side benchmarks for:
     • SO3::exp and SO3::log.
     • SE3 composition (multiplication) and inversion.
     • Compare kornia-algebra vs nalgebra vs sophus.
3. bench_linalg.rs:
   • Clean up existing benchmarks to ensure fair comparisons (e.g., removing redundant allocations or iterator conversions).

📚 Library Reference

The implementation will reference the following libraries and concepts:

• Nalgebra: The standard linear algebra library in Rust for geometric types (Isometry3, UnitQuaternion).
• Sophus: A C++ (and Rust) Lie Group library often used as a gold standard in robotics.
• Rosenbrock Function: A non-convex function used as a performance test problem for optimization algorithms.

🔄 Alternatives Considered

No response

🎯 Use Cases

1. CI Regression Testing: Running cargo bench in CI to detect performance drops in PRs.
2. Solver Development: When a developer implements a new LinearSolver (e.g., sparse Cholesky), they can use bench_optim to prove its speedup on non-trivial problems.
3. End-User Confidence: Users evaluating kornia-rs for real-time SLAM systems can see transparent performance comparisons against libraries they already know.

📝 Additional Context

No response

🤝 Contribution Intent

• I plan to submit a PR to implement this feature
• I'm requesting this feature but not planning to implement it