def test_geometric_mean_couple():
n_features = 7
spd1 = np.ones((n_features, n_features))
spd1 = spd1.dot(spd1) + n_features * np.eye(n_features)
spd2 = np.tril(np.ones((n_features, n_features)))
spd2 = spd2.dot(spd2.T)
vals_spd2, vecs_spd2 = np.linalg.eigh(spd2)
spd2_sqrt = _form_symmetric(np.sqrt, vals_spd2, vecs_spd2)
spd2_inv_sqrt = _form_symmetric(np.sqrt, 1. / vals_spd2, vecs_spd2)
geo = spd2_sqrt.dot(_map_eigenvalues(np.sqrt, spd2_inv_sqrt.dot(spd1).dot(
spd2_inv_sqrt))).dot(spd2_sqrt)
assert_array_almost_equal(_geometric_mean([spd1, spd2]), geo)
def test_geometric_mean_couple():
n_features = 7
spd1 = np.ones((n_features, n_features))
spd1 = spd1.dot(spd1) + n_features * np.eye(n_features)
spd2 = np.tril(np.ones((n_features, n_features)))
spd2 = spd2.dot(spd2.T)
vals_spd2, vecs_spd2 = np.linalg.eigh(spd2)
spd2_sqrt = _form_symmetric(np.sqrt, vals_spd2, vecs_spd2)
spd2_inv_sqrt = _form_symmetric(np.sqrt, 1. / vals_spd2, vecs_spd2)
geo = spd2_sqrt.dot(_map_eigenvalues(np.sqrt, spd2_inv_sqrt.dot(spd1).dot(
spd2_inv_sqrt))).dot(spd2_sqrt)
assert_array_almost_equal(_geometric_mean([spd1, spd2]), geo)
def grad_geometric_mean(mats, init=None, max_iter=10, tol=1e-7):
"""Return the norm of the covariant derivative at each iteration step of
geometric_mean. See its docstring for details.
Norm is intrinsic norm on the tangent space of the manifold of symmetric
positive definite matrices.
Returns
-------
grad_norm : list of float
Norm of the covariant derivative in the tangent space at each step.
"""
mats = np.array(mats)
# Initialization
if init is None:
gmean = np.mean(mats, axis=0)
else:
gmean = init
norm_old = np.inf
step = 1.
grad_norm = []
for n in range(max_iter):
# Computation of the gradient
vals_gmean, vecs_gmean = linalg.eigh(gmean)
gmean_inv_sqrt = _form_symmetric(np.sqrt, 1. / vals_gmean, vecs_gmean)
whitened_mats = [
gmean_inv_sqrt.dot(mat).dot(gmean_inv_sqrt) for mat in mats
]
logs = [_map_eigenvalues(np.log, w_mat) for w_mat in whitened_mats]
logs_mean = np.mean(logs, axis=0) # Covariant derivative is
# - gmean.dot(logms_mean)
norm = np.linalg.norm(logs_mean) # Norm of the covariant derivative on
# the tangent space at point gmean
# Update of the minimizer
vals_log, vecs_log = linalg.eigh(logs_mean)
gmean_sqrt = _form_symmetric(np.sqrt, vals_gmean, vecs_gmean)
gmean = gmean_sqrt.dot(
_form_symmetric(np.exp, vals_log * step, vecs_log)).dot(gmean_sqrt)
# Update the norm and the step size
if norm < norm_old:
norm_old = norm
if norm > norm_old:
step = step / 2.
norm = norm_old
grad_norm.append(norm / gmean.size)
if tol is not None and norm / gmean.size < tol:
break
return grad_norm
def grad_geometric_mean(mats, init=None, max_iter=10, tol=1e-7):
"""Return the norm of the covariant derivative at each iteration step of
geometric_mean. See its docstring for details.
Norm is intrinsic norm on the tangent space of the manifold of symmetric
positive definite matrices.
Returns
-------
grad_norm : list of float
Norm of the covariant derivative in the tangent space at each step.
"""
mats = np.array(mats)
# Initialization
if init is None:
gmean = np.mean(mats, axis=0)
else:
gmean = init
norm_old = np.inf
step = 1.
grad_norm = []
for n in range(max_iter):
# Computation of the gradient
vals_gmean, vecs_gmean = linalg.eigh(gmean)
gmean_inv_sqrt = _form_symmetric(np.sqrt, 1. / vals_gmean, vecs_gmean)
whitened_mats = [gmean_inv_sqrt.dot(mat).dot(gmean_inv_sqrt)
for mat in mats]
logs = [_map_eigenvalues(np.log, w_mat) for w_mat in whitened_mats]
logs_mean = np.mean(logs, axis=0) # Covariant derivative is
# - gmean.dot(logms_mean)
norm = np.linalg.norm(logs_mean) # Norm of the covariant derivative on
# the tangent space at point gmean
# Update of the minimizer
vals_log, vecs_log = linalg.eigh(logs_mean)
gmean_sqrt = _form_symmetric(np.sqrt, vals_gmean, vecs_gmean)
gmean = gmean_sqrt.dot(
_form_symmetric(np.exp, vals_log * step, vecs_log)).dot(gmean_sqrt)
# Update the norm and the step size
if norm < norm_old:
norm_old = norm
if norm > norm_old:
step = step / 2.
norm = norm_old
grad_norm.append(norm / gmean.size)
if tol is not None and norm / gmean.size < tol:
break
return grad_norm