feat: implement Toeplitz and Hankel Procrustes

[Ao-chuba]

This PR implements Toeplitz and Hankel matrix Procrustes algorithms using a least squares approach over the structured parameter space.
The changes made are :

• Added procrustes/toeplitz.py with toeplitz() and hankel() solvers.
• Added procrustes/test/test_toeplitz.py with comprehensive tests for exact recovery and structure validation.
• Updated procrustes/__init__.py to export the new functions.

Using the reference Yang, J., & Deng, Y. (2013). "Procrustes Problems for General, Triangular, and Symmetric Toeplitz Matrices."
Journal of Applied Mathematics, 2013.
closes #84

[codecov]

Codecov Report

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Additional details and impacted files

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procrustes/toeplitz.py	100.00% <100.00%> (ø)

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[Ao-chuba]

@FanwangM here's my PR please look into it, whenever you get time.

[Copilot]

Pull request overview

This PR implements one-sided Toeplitz and Hankel Procrustes transformations by reformulating the constrained optimization problem as a linear least-squares problem over the structured parameter space. The implementation follows the approach described in Yang & Deng (2013).

Changes:

• Added procrustes/toeplitz.py with toeplitz() and hankel() solvers that find optimal Toeplitz/Hankel transformation matrices
• Added comprehensive test suite in procrustes/test/test_toeplitz.py covering exact recovery, structure validation, and error bounds
• Updated procrustes/__init__.py to export the new functions

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 4 comments.

File	Description
procrustes/toeplitz.py	Implements Toeplitz and Hankel Procrustes solvers using linear operator construction and least-squares optimization
procrustes/test/test_toeplitz.py	Comprehensive test suite validating exact recovery, structural properties, and comparison with generic Procrustes
procrustes/init.py	Exports new toeplitz module functions

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[Ao-chuba]

@FanwangM ,
I've implemented rectangular matrix test as suggested by copilot and all test pass. Regarding the spelling suggestion Copilot flagged that this PR uses "initial" while other files use "intial". "intial" seems like a typo so I've decided not to go along with copilot's suggestion here.

[Copilot]

Pull request overview

Copilot reviewed 3 out of 3 changed files in this pull request and generated 9 comments.

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[Ao-chuba]

As per the suggestions given by co-pilot, All four np.zeros initializations in _build_toeplitz_operator, _build_hankel_operator, _params_to_toeplitz, and _params_to_hankel have been updated to carry dtype=a.dtype or dtype=params.dtype respectively, ensuring complex inputs are never silently truncated at any stage of the pipeline. Two test names were also corrected test_toeplitz_error_leq_generic and test_hankel_error_leq_generic were renamed to _geq_generic to match the >= assertion they actually make. Also the Notes sections in both toeplitz() and hankel() docstrings were updated to replace "solving the normal equations" with "found via least-squares (using an SVD-based solver)", since np.linalg.lstsq uses SVD internally, not the normal equations method.

Regarding the suggestion to replace the dense operator M with a scipy.sparse matrix and an iterative solver, this is a valid point, as the current dense formulation has O(n³) memory for square inputs. However, this represents a significant scope change, requires new dependencies, and is inconsistent with how the rest of the procrustes library handles solvers. It would be better if a separate issue will be opened for this issue rather than including it in this PR.