espnet2.enh.layers.beamformer.generalized_eigenvalue_decomposition

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espnet2.enh.layers.beamformer.generalized_eigenvalue_decomposition(a: Tensor, b: Tensor, eps=1e-06)

Solves the generalized eigenvalue decomposition through Cholesky decomposition.

This function computes the generalized eigenvalue decomposition of matrices a and b, such that:

a @ e_vec = e_val * b @ e_vec

The method utilizes Cholesky decomposition on matrix b to transform the problem into a standard eigenvalue problem.

Steps involved:

1. Perform Cholesky decomposition on b:

b = L @ L^H, where L is a lower triangular matrix.

1. Define C = L^-1 @ a @ L^-H, which is Hermitian.
2. Solve the eigenvalue problem C @ y = lambda * y.
3. Obtain the eigenvectors e_vec = L^-H @ y.

Reference: https://www.netlib.org/lapack/lug/node54.html

• Parameters:
  • a (torch.Tensor) – A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors will be computed. Shape: (…, C, C).
  • b (torch.Tensor) – A complex Hermitian or real symmetric definite positive matrix. Shape: (…, C, C).
  • eps (float , optional) – A small value for numerical stability during Cholesky decomposition. Default is 1e-6.
• Returns:
  • e_val (torch.Tensor): Generalized eigenvalues (ascending order).
  • e_vec (torch.Tensor): Generalized eigenvectors.
• Return type: Tuple[torch.Tensor, torch.Tensor]

Examples

>>> a = torch.tensor([[1, 0], [0, 2]], dtype=torch.complex64)
>>> b = torch.tensor([[1, 0], [0, 1]], dtype=torch.complex64)
>>> e_val, e_vec = generalized_eigenvalue_decomposition(a, b)
>>> print(e_val)
>>> print(e_vec)

NOTE

The input matrices a and b must be Hermitian or symmetric as per the requirements of eigenvalue decomposition.