qml.kernels.closest_psd_matrix

closest_psd_matrix(K, fix_diagonal=False, solver=None, **kwargs)

Return the closest positive semi-definite matrix to the given kernel matrix.

This method either fixes the diagonal entries to be 1 (fix_diagonal=True) or keeps the eigenvectors intact (fix_diagonal=False), in which case it reduces to the method threshold_matrix(). For fix_diagonal=True a semi-definite program is solved.

Parameters

• K (array[float]) – Kernel matrix, assumed to be symmetric.

• fix_diagonal (bool) – Whether to fix the diagonal to 1.

• solver (str, optional) – Solver to be used by cvxpy. Defaults to CVXOPT.

• kwargs (kwarg dict) – Passed to cvxpy.Problem.solve().

Returns

closest positive semi-definite matrix in Frobenius norm.

Return type

array[float]

Comments:

Requires cvxpy and the used solver (default CVXOPT) to be installed if fix_diagonal=True.

Reference:

This method is introduced in arXiv:2105.02276.

Example:

Consider a symmetric matrix with both positive and negative eigenvalues:

>>> K = np.array([[0.9, 1.], [1., 0.9]])
>>> np.linalg.eigvalsh(K)
array([-0.1, 1.9])

The positive semi-definite matrix that is closest to this matrix in any unitarily invariant norm is then given by the matrix with the eigenvalues thresholded at 0, as computed by threshold_matrix():

>>> K_psd = qml.kernels.closest_psd_matrix(K)
>>> K_psd
array([[0.95, 0.95],
       [0.95, 0.95]])
>>> np.linalg.eigvalsh(K_psd)
array([0., 1.9])
>>> np.allclose(K_psd, qml.kernels.threshold_matrix(K))
True

However, for quantum kernel matrices we may want to restore the value 1 on the diagonal:

>>> K_psd = qml.kernels.closest_psd_matrix(K, fix_diagonal=True)
>>> K_psd
array([[1.        , 0.99998008],
       [0.99998008, 1.        ]])
>>> np.linalg.eigvalsh(K_psd)
array([1.99162415e-05, 1.99998008e+00])

If the input matrix does not have negative eigenvalues and fix_diagonal=False, closest_psd_matrix does not have any effect.

code/api/pennylane.kernels.closest_psd_matrix

 

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