def _assess_dimension_(spectrum, rank, n_samples, n_features):
"""Compute the likelihood of a rank ``rank`` dataset
The dataset is assumed to be embedded in gaussian noise of shape(n,
dimf) having spectrum ``spectrum``.
Parameters
----------
spectrum : array of shape (n)
Data spectrum.
rank : int
Tested rank value.
n_samples : int
Number of samples.
n_features : int
Number of features.
Returns
-------
ll : float,
The log-likelihood
Notes
-----
This implements the method of `Thomas P. Minka:
Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604`
"""
if rank > len(spectrum):
raise ValueError("The tested rank cannot exceed the rank of the"
" dataset")
pu = -rank * log(2.)
for i in range(rank):
pu += (gammaln(
(n_features - i) / 2.) - log(gs.pi) * (n_features - i) / 2.)
pl = gs.sum(gs.log(spectrum[:rank]))
pl = -pl * n_samples / 2.
if rank == n_features:
pv = 0
v = 1
else:
v = gs.sum(spectrum[rank:]) / (n_features - rank)
pv = -gs.log(v) * n_samples * (n_features - rank) / 2.
m = n_features * rank - rank * (rank + 1.) / 2.
pp = log(2. * gs.pi) * (m + rank + 1.) / 2.
pa = 0.
spectrum_ = spectrum.copy()
spectrum_[rank:n_features] = v
for i in range(rank):
for j in range(i + 1, len(spectrum)):
pa += log((spectrum[i] - spectrum[j]) *
(1. / spectrum_[j] - 1. / spectrum_[i])) + log(n_samples)
ll = pu + pl + pv + pp - pa / 2. - rank * log(n_samples) / 2.
return ll
def squared_dist(self, point_a, point_b, **kwargs):
"""Compute the Cholesky Metric squared distance.
Compute the Riemannian squared distance between point_a and point_b.
Parameters
----------
point_a : array-like, shape=[..., n, n]
Point.
point_b : array-like, shape=[..., n, n]
Point.
Returns
-------
_ : array-like, shape=[...]
Riemannian squared distance.
"""
log_diag_a = gs.log(Matrices.diagonal(point_a))
log_diag_b = gs.log(Matrices.diagonal(point_b))
diag_diff = log_diag_a - log_diag_b
squared_dist_diag = gs.sum((diag_diff) ** 2, axis=-1)
sl_a = Matrices.to_strictly_lower_triangular(point_a)
sl_b = Matrices.to_strictly_lower_triangular(point_b)
sl_diff = sl_a - sl_b
squared_dist_sl = Matrices.frobenius_product(sl_diff, sl_diff)
return squared_dist_sl + squared_dist_diag
def random_von_mises_fisher(self, kappa=10, n_samples=1):
"""Sample in the 2-sphere with the von Mises distribution.
Sample in the 2-sphere with the von Mises distribution centered in the
north pole.
Parameters
----------
kappa : int, optional
n_samples : int, optional
Returns
-------
point : array-like
"""
if self.dimension != 2:
raise NotImplementedError(
'Sampling from the von Mises Fisher distribution'
'is only implemented in dimension 2.')
angle = 2. * gs.pi * gs.random.rand(n_samples)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
unit_vector = gs.hstack((gs.cos(angle), gs.sin(angle)))
scalar = gs.random.rand(n_samples)
coord_z = 1. + 1. / kappa * gs.log(scalar + (1. - scalar) *
gs.exp(gs.array(-2. * kappa)))
coord_z = gs.to_ndarray(coord_z, to_ndim=2, axis=1)
coord_xy = gs.sqrt(1. - coord_z**2) * unit_vector
point = gs.hstack((coord_xy, coord_z))
return point
def test_differential_log(self):
base_point = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., 4.]])
tangent_vec = gs.array([[1., 1., 3.], [1., 1., 3.], [3., 3., 4.]])
result = self.space.differential_log(tangent_vec, base_point)
x = 2 * gs.log(2)
expected = gs.array([[[1., 1., x], [1., 1., x], [x, x, 1]]])
self.assertAllClose(result, expected)
def dist(self, point_a, point_b):
"""Compute the geodesic distance between two points.
Parameters
----------
point_a : array-like, shape=[..., dim]
First point in the Poincare ball.
point_b : array-like, shape=[..., dim]
Second point in the Poincare ball.
Returns
-------
dist : array-like, shape=[...,]
Geodesic distance between the two points.
"""
point_a_norm = gs.clip(gs.sum(point_a**2, -1), 0.0, 1 - EPSILON)
point_b_norm = gs.clip(gs.sum(point_b**2, -1), 0.0, 1 - EPSILON)
diff_norm = gs.sum((point_a - point_b)**2, -1)
norm_function = 1 + 2 * diff_norm / ((1 - point_a_norm) *
(1 - point_b_norm))
dist = gs.log(norm_function + gs.sqrt(norm_function**2 - 1))
dist *= self.scale
return dist
def log(self, point, base_point, **kwargs):
"""Compute the Cholesky logarithm map.
Compute the Riemannian logarithm at point base_point,
of point wrt the Cholesky metric.
This gives a tangent vector at point base_point.
Parameters
----------
point : array-like, shape=[..., n, n]
Point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
log : array-like, shape=[..., n, n]
Riemannian logarithm.
"""
sl_base_point = Matrices.to_strictly_lower_triangular(base_point)
sl_point = Matrices.to_strictly_lower_triangular(point)
diag_base_point = Matrices.diagonal(base_point)
diag_point = Matrices.diagonal(point)
diag_product_logm = gs.log(gs.divide(diag_point, diag_base_point))
sl_log = sl_point - sl_base_point
diag_log = gs.vec_to_diag(diag_base_point * diag_product_logm)
log = sl_log + diag_log
return log
def dist(self, point_a, point_b):
"""Compute the geodesic distance between two points.
Parameters
----------
point_a : array-like, shape=[n_samples, dim]
First point in hyperbolic space.
point_b : array-like, shape=[n_samples, dim]
Second point in hyperbolic space.
Returns
-------
dist : array-like, shape=[n_samples, 1]
Geodesic distance between the two points.
"""
point_a = gs.to_ndarray(point_a, to_ndim=2)
point_b = gs.to_ndarray(point_b, to_ndim=2)
point_a_norm = gs.clip(gs.sum(point_a**2, -1), 0., 1 - EPSILON)
point_b_norm = gs.clip(gs.sum(point_b**2, -1), 0., 1 - EPSILON)
diff_norm = gs.sum((point_a - point_b)**2, -1)
norm_function = 1 + 2 * \
diff_norm / ((1 - point_a_norm) * (1 - point_b_norm))
dist = gs.log(norm_function + gs.sqrt(norm_function**2 - 1))
dist = gs.to_ndarray(dist, to_ndim=1)
dist = gs.to_ndarray(dist, to_ndim=2, axis=1)
dist *= self.scale
return dist
def aux_differential_power(power, tangent_vec, base_point):
"""Compute the differential of the matrix power.
Auxiliary function to the functions differential_power and
inverse_differential_power.
Parameters
----------
power : float
Power function to differentiate.
tangent_vec : array_like, shape=[..., n, n]
Tangent vector at base point.
base_point : array_like, shape=[..., n, n]
Base point.
Returns
-------
eigvectors : array-like, shape=[..., n, n]
transp_eigvectors : array-like, shape=[..., n, n]
numerator : array-like, shape=[..., n, n]
denominator : array-like, shape=[..., n, n]
temp_result : array-like, shape=[..., n, n]
"""
eigvalues, eigvectors = gs.linalg.eigh(base_point)
if power == 0:
powered_eigvalues = gs.log(eigvalues)
elif power == math.inf:
powered_eigvalues = gs.exp(eigvalues)
else:
powered_eigvalues = eigvalues ** power
denominator = eigvalues[..., :, None] - eigvalues[..., None, :]
numerator = powered_eigvalues[..., :, None] - powered_eigvalues[..., None, :]
if power == 0:
numerator = gs.where(denominator == 0, gs.ones_like(numerator), numerator)
denominator = gs.where(
denominator == 0, eigvalues[..., :, None], denominator
)
elif power == math.inf:
numerator = gs.where(
denominator == 0, powered_eigvalues[..., :, None], numerator
)
denominator = gs.where(
denominator == 0, gs.ones_like(numerator), denominator
)
else:
numerator = gs.where(
denominator == 0, power * powered_eigvalues[..., :, None], numerator
)
denominator = gs.where(
denominator == 0, eigvalues[..., :, None], denominator
)
transp_eigvectors = Matrices.transpose(eigvectors)
temp_result = Matrices.mul(transp_eigvectors, tangent_vec, eigvectors)
return (eigvectors, transp_eigvectors, numerator, denominator, temp_result)
def test_inverse_differential_log(self):
"""Test of inverse_differential_log method."""
base_point = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., 4.]])
x = 2 * gs.log(2.)
tangent_vec = gs.array([[1., 1., x], [1., 1., x], [x, x, 1]])
result = self.space.inverse_differential_log(tangent_vec, base_point)
expected = gs.array([[1., 1., 3.], [1., 1., 3.], [3., 3., 4.]])
self.assertAllClose(result, expected)
def test_differential_log(self):
"""Test of differential_log method."""
base_point = gs.array([[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 4.0]])
tangent_vec = gs.array([[1.0, 1.0, 3.0], [1.0, 1.0, 3.0], [3.0, 3.0, 4.0]])
result = self.space.differential_log(tangent_vec, base_point)
x = 2 * gs.log(2.0)
expected = gs.array([[1.0, 1.0, x], [1.0, 1.0, x], [x, x, 1]])
self.assertAllClose(result, expected)
def test_log_euclidean_inner_product(self):
base_point = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., 4.]])
tangent_vec = gs.array([[1., 1., 3.], [1., 1., 3.], [3., 3., 4.]])
metric = self.metric_logeuclidean
result = metric.inner_product(tangent_vec, tangent_vec, base_point)
x = 2 * gs.log(2)
expected = 5. + 4. * x**2
self.assertAllClose(result, expected)
def dist_broadcast(self, point_a, point_b):
"""Compute the geodesic distance between points.
If n_samples_a == n_samples_b then dist is the element-wise
distance result of a point in points_a with the point from
points_b of the same index. If n_samples_a not equal to
n_samples_b then dist is the result of applying geodesic
distance for each point from points_a to all points from
points_b.
Parameters
----------
point_a : array-like, shape=[n_samples_a, dim]
Set of points in hyperbolic space.
point_b : array-like, shape=[n_samples_b, dim]
Second set of points in hyperbolic space.
Returns
-------
dist : array-like,
shape=[n_samples_a, dim] or [n_samples_a, n_samples_b, dim]
Geodesic distance between the two points.
"""
if point_a.shape[-1] != point_b.shape[-1]:
raise ValueError('Manifold dimensions not equal')
if point_a.shape[0] != point_b.shape[0]:
point_a_broadcast, point_b_broadcast = gs.broadcast_arrays(
point_a[:, None], point_b[None, ...])
point_a_flatten = gs.reshape(point_a_broadcast,
(-1, point_a_broadcast.shape[-1]))
point_b_flatten = gs.reshape(point_b_broadcast,
(-1, point_b_broadcast.shape[-1]))
point_a_norm = gs.clip(gs.sum(point_a_flatten**2, -1), 0.,
1 - EPSILON)
point_b_norm = gs.clip(gs.sum(point_b_flatten**2, -1), 0.,
1 - EPSILON)
square_diff = (point_a_flatten - point_b_flatten)**2
diff_norm = gs.sum(square_diff, -1)
norm_function = 1 + 2 * \
diff_norm / ((1 - point_a_norm) * (1 - point_b_norm))
dist = gs.log(norm_function + gs.sqrt(norm_function**2 - 1))
dist *= self.scale
dist = gs.reshape(dist, (point_a.shape[0], point_b.shape[0]))
dist = gs.squeeze(dist)
elif point_a.shape == point_b.shape:
dist = self.dist(point_a, point_b)
return dist
def test_log_euclidean_inner_product(self):
"""Test of SPDMetricLogEuclidean.inner_product method."""
base_point = gs.array([[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 4.0]])
tangent_vec = gs.array([[1.0, 1.0, 3.0], [1.0, 1.0, 3.0], [3.0, 3.0, 4.0]])
metric = self.metric_logeuclidean
result = metric.inner_product(tangent_vec, tangent_vec, base_point)
x = 2 * gs.log(2.0)
expected = 5.0 + 4.0 * x ** 2
self.assertAllClose(result, expected)
def test_euclidean_frechet_mean(self):
"""Test if the frechet mean of the samples is close to mean"""
mean = gs.zeros(self.n)
cov = gs.eye(self.n)
data = LogNormal(self.euclidean, mean, cov).sample(5000)
log_data = gs.log(data)
fm = gs.mean(log_data, axis=0)
expected = mean
result = fm
self.assertAllClose(result, expected, atol=5 * 1e-2)
def log_sigmoid(vector):
"""Logsigmoid function.
Apply log sigmoid function.
Parameters
----------
vector : array-like, shape=[n_samples, dim]
Returns
-------
result : array-like, shape=[n_samples, dim]
"""
return gs.log((1 / (1 + gs.exp(-vector))))
def dist(self, point_a, point_b):
"""Compute the geodesic distance between two points.
Parameters
----------
point_a : array-like, shape=[n_samples, dimension + 1]
First point in hyperbolic space.
point_b : array-like, shape=[n_samples, dimension + 1]
Second point in hyperbolic space.
Returns
-------
dist : array-like, shape=[n_samples, 1]
Geodesic distance between the two points.
"""
if self.point_type == 'extrinsic':
sq_norm_a = self.embedding_metric.squared_norm(point_a)
sq_norm_b = self.embedding_metric.squared_norm(point_b)
inner_prod = self.embedding_metric.inner_product(point_a, point_b)
cosh_angle = -inner_prod / gs.sqrt(sq_norm_a * sq_norm_b)
cosh_angle = gs.clip(cosh_angle, 1.0, 1e24)
dist = gs.arccosh(cosh_angle)
return self.scale * dist
elif self.point_type == 'ball':
point_a_norm = gs.clip(gs.sum(point_a**2, -1), 0., 1 - EPSILON)
point_b_norm = gs.clip(gs.sum(point_b**2, -1), 0., 1 - EPSILON)
diff_norm = gs.sum((point_a - point_b)**2, -1)
norm_function = 1 + 2 * \
diff_norm / ((1 - point_a_norm) * (1 - point_b_norm))
dist = gs.log(norm_function + gs.sqrt(norm_function**2 - 1))
dist = gs.to_ndarray(dist, to_ndim=1)
dist = gs.to_ndarray(dist, to_ndim=2, axis=1)
return self.scale * dist
else:
raise NotImplementedError(
'dist is only implemented for ball and extrinsic')
def squared_dist(self, point_a, point_b, **kwargs):
"""Compute squared distance associated with the exponential Fisher Rao metric.
Parameters
----------
point_a : array-like, shape=[...,]
Point representing an exponential distribution (scale parameter).
point_b : array-like, shape=[...,] (same shape as point_a)
Point representing a exponential distribution (scale parameter).
Returns
-------
squared_dist : array-like, shape=[...,]
Squared distance between points point_a and point_b.
"""
return gs.log(point_a / point_b)**2
def random_von_mises_fisher(self, kappa=10, n_samples=1):
"""
Sample in the 2-sphere with the von Mises distribution
centered in the north pole.
"""
if self.dimension != 2:
raise NotImplementedError(
'Sampling from the von Mises Fisher distribution'
'is only implemented in dimension 2.')
angle = 2 * gs.pi * gs.random.rand(n_samples)
unit_vector = gs.vstack((gs.cos(angle), gs.sin(angle)))
scalar = gs.random.rand(n_samples)
coord_z = 1 + 1/kappa*gs.log(scalar + (1-scalar)*gs.exp(-2*kappa))
coord_xy = gs.sqrt(1 - coord_z**2) * unit_vector
point = gs.vstack((coord_xy, coord_z))
return point.T
def random_von_mises_fisher(self, kappa=10, n_samples=1):
"""Sample in the 2-sphere with the von Mises distribution.
Sample in the 2-sphere with the von Mises distribution centered at the
north pole.
References
----------
https://en.wikipedia.org/wiki/Von_Mises_distribution
Parameters
----------
kappa : int
Kappa parameter of the von Mises distribution.
Optional, default: 10.
n_samples : int
Number of samples.
Optional, default: 1.
Returns
-------
point : array-like, shape=[..., 3]
Points sampled on the sphere in extrinsic coordinates
in Euclidean space of dimension 3.
"""
if self.dim != 2:
raise NotImplementedError(
'Sampling from the von Mises Fisher distribution'
'is only implemented in dimension 2.')
angle = 2. * gs.pi * gs.random.rand(n_samples)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
unit_vector = gs.hstack((gs.cos(angle), gs.sin(angle)))
scalar = gs.random.rand(n_samples)
coord_z = 1. + 1. / kappa * gs.log(
scalar + (1. - scalar) * gs.exp(gs.array(-2. * kappa)))
coord_z = gs.to_ndarray(coord_z, to_ndim=2, axis=1)
coord_xy = gs.sqrt(1. - coord_z**2) * unit_vector
point = gs.hstack((coord_xy, coord_z))
if n_samples == 1:
point = gs.squeeze(point, axis=0)
return point
def group_log(sym_mat):
"""
Group logarithm of the Lie group of
all invertible matrices has a straight-forward
computation for symmetric positive definite matrices.
"""
sym_mat = gs.to_ndarray(sym_mat, to_ndim=3)
n_sym_mats, mat_dim, _ = sym_mat.shape
assert gs.all(is_symmetric(sym_mat))
sym_mat = make_symmetric(sym_mat)
[eigenvalues, vectors] = gs.linalg.eigh(sym_mat)
assert gs.all(eigenvalues > 0)
log_eigenvalues = gs.log(eigenvalues)
aux = gs.einsum('ijk,ik->ijk', vectors, log_eigenvalues)
log_mat = gs.einsum('ijk,ilk->ijl', aux, vectors)
log_mat = gs.to_ndarray(log_mat, to_ndim=3)
return log_mat
def random_von_mises_fisher(
self, mu=None, kappa=10, n_samples=1, max_iter=100):
"""Sample with the von Mises-Fisher distribution.
This distribution corresponds to the maximum entropy distribution
given a mean. In dimension 2, a closed form expression is available.
In larger dimension, rejection sampling is used according to [Wood94]_
References
----------
https://en.wikipedia.org/wiki/Von_Mises-Fisher_distribution
.. [Wood94] Wood, Andrew T. A. “Simulation of the von Mises Fisher
Distribution.” Communications in Statistics - Simulation
and Computation, June 27, 2007.
https://doi.org/10.1080/03610919408813161.
Parameters
----------
mu : array-like, shape=[dim]
Mean parameter of the distribution.
kappa : float
Kappa parameter of the von Mises distribution.
Optional, default: 10.
n_samples : int
Number of samples.
Optional, default: 1.
max_iter : int
Maximum number of trials in the rejection algorithm. In case it
is reached, the current number of samples < n_samples is returned.
Optional, default: 100.
Returns
-------
point : array-like, shape=[n_samples, dim + 1]
Points sampled on the sphere in extrinsic coordinates
in Euclidean space of dimension dim + 1.
"""
dim = self.dim
if dim == 2:
angle = 2. * gs.pi * gs.random.rand(n_samples)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
unit_vector = gs.hstack((gs.cos(angle), gs.sin(angle)))
scalar = gs.random.rand(n_samples)
coord_x = 1. + 1. / kappa * gs.log(
scalar + (1. - scalar) * gs.exp(gs.array(-2. * kappa)))
coord_x = gs.to_ndarray(coord_x, to_ndim=2, axis=1)
coord_yz = gs.sqrt(1. - coord_x ** 2) * unit_vector
sample = gs.hstack((coord_x, coord_yz))
else:
# rejection sampling in the general case
sqrt = gs.sqrt(4 * kappa ** 2. + dim ** 2)
envelop_param = (-2 * kappa + sqrt) / dim
node = (1. - envelop_param) / (1. + envelop_param)
correction = kappa * node + dim * gs.log(1. - node ** 2)
n_accepted, n_iter = 0, 0
result = []
while (n_accepted < n_samples) and (n_iter < max_iter):
sym_beta = beta.rvs(
dim / 2, dim / 2, size=n_samples - n_accepted)
sym_beta = gs.cast(sym_beta, node.dtype)
coord_x = (1 - (1 + envelop_param) * sym_beta) / (
1 - (1 - envelop_param) * sym_beta)
accept_tol = gs.random.rand(n_samples - n_accepted)
criterion = (
kappa * coord_x
+ dim * gs.log(1 - node * coord_x)
- correction) > gs.log(accept_tol)
result.append(coord_x[criterion])
n_accepted += gs.sum(criterion)
n_iter += 1
if n_accepted < n_samples:
logging.warning(
'Maximum number of iteration reached in rejection '
'sampling before n_samples were accepted.')
coord_x = gs.concatenate(result)
coord_rest = _Hypersphere(dim - 1).random_uniform(n_accepted)
coord_rest = gs.einsum(
'...,...i->...i', gs.sqrt(1 - coord_x ** 2), coord_rest)
sample = gs.concatenate([coord_x[..., None], coord_rest], axis=1)
if mu is not None:
sample = utils.rotate_points(sample, mu)
return sample if (n_samples > 1) else sample[0]
def aux_differential_power(power, tangent_vec, base_point):
"""Compute the differential of the matrix power.
Auxiliary function to the functions differential_power and
inverse_differential_power.
Parameters
----------
power : float
Power function to differentiate.
tangent_vec : array_like, shape=[..., n, n]
Tangent vector at base point.
base_point : array_like, shape=[..., n, n]
Base point.
Returns
-------
eigvectors : array-like, shape=[..., n, n]
transp_eigvectors : array-like, shape=[..., n, n]
numerator : array-like, shape=[..., n, n]
denominator : array-like, shape=[..., n, n]
temp_result : array-like, shape=[..., n, n]
"""
n_tangent_vecs, _, _ = tangent_vec.shape
n_base_points, _, n = base_point.shape
eigvalues, eigvectors = gs.linalg.eigh(base_point)
eigvalues = gs.to_ndarray(eigvalues, to_ndim=3, axis=1)
transp_eigvalues = gs.transpose(eigvalues, (0, 2, 1))
if power == 0:
powered_eigvalues = gs.log(eigvalues)
elif power == math.inf:
powered_eigvalues = gs.exp(eigvalues)
else:
powered_eigvalues = eigvalues**power
transp_powered_eigvalues = gs.transpose(powered_eigvalues, (0, 2, 1))
ones = gs.ones((n_base_points, 1, n))
transp_ones = gs.transpose(ones, (0, 2, 1))
vertical_index = gs.matmul(transp_eigvalues, ones)
horizontal_index = gs.matmul(transp_ones, eigvalues)
one_matrix = gs.matmul(transp_ones, ones)
vertical_index_power = gs.matmul(transp_powered_eigvalues, ones)
horizontal_index_power = gs.matmul(transp_ones, powered_eigvalues)
denominator = vertical_index - horizontal_index
numerator = vertical_index_power - horizontal_index_power
if power == 0:
numerator = gs.where(denominator == 0, one_matrix, numerator)
denominator = gs.where(denominator == 0, vertical_index,
denominator)
elif power == math.inf:
numerator = gs.where(denominator == 0, vertical_index_power,
numerator)
denominator = gs.where(denominator == 0, one_matrix, denominator)
else:
numerator = gs.where(denominator == 0,
power * vertical_index_power, numerator)
denominator = gs.where(denominator == 0, vertical_index,
denominator)
transp_eigvectors = gs.transpose(eigvectors, (0, 2, 1))
temp_result = gs.matmul(transp_eigvectors, tangent_vec)
temp_result = gs.matmul(temp_result, eigvectors)
if n_base_points == n_tangent_vecs == 1:
transp_eigvectors = gs.squeeze(transp_eigvectors, axis=0)
eigvectors = gs.squeeze(eigvectors, axis=0)
temp_result = gs.squeeze(temp_result, axis=0)
numerator = gs.squeeze(numerator, axis=0)
denominator = gs.squeeze(denominator, axis=0)
return (eigvectors, transp_eigvectors, numerator, denominator,
temp_result)
def aux_differential_power(self, power, tangent_vec, base_point):
"""Compute the differential of the matrix power.
Auxiliary function to the functions differential_power and
inverse_differential_power.
Parameters
----------
power : int
tangent_vec : array_like, shape=[n_samples, n, n]
Tangent vectors.
base_point : array_like, shape=[n_samples, n, n]
Base points.
Returns
-------
eigvectors : array-like, shape=[n_samples, n, n]
transp_eigvectors : array-like, shape=[n_samples, n, n]
numerator : array-like, shape=[n_samples, n, n]
denominator : array-like, shape=[n_samples, n, n]
temp_result : array-like, shape=[n_samples, n, n]
"""
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=3)
n_tangent_vecs, _, _ = tangent_vec.shape
base_point = gs.to_ndarray(base_point, to_ndim=3)
n_base_points, _, _ = base_point.shape
assert (n_tangent_vecs == n_base_points or n_base_points == 1
or n_tangent_vecs == 1)
eigvalues, eigvectors = gs.linalg.eigh(base_point)
eigvalues = gs.to_ndarray(eigvalues, to_ndim=3, axis=1)
transp_eigvalues = gs.transpose(eigvalues, (0, 2, 1))
if power == 0:
powered_eigvalues = gs.log(eigvalues)
elif power == math.inf:
powered_eigvalues = gs.exp(eigvalues)
else:
powered_eigvalues = eigvalues**power
transp_powered_eigvalues = gs.transpose(powered_eigvalues, (0, 2, 1))
ones = gs.ones((n_base_points, 1, self.n))
transp_ones = gs.transpose(ones, (0, 2, 1))
vertical_index = gs.matmul(transp_eigvalues, ones)
horizontal_index = gs.matmul(transp_ones, eigvalues)
one_matrix = gs.matmul(transp_ones, ones)
vertical_index_power = gs.matmul(transp_powered_eigvalues, ones)
horizontal_index_power = gs.matmul(transp_ones, powered_eigvalues)
denominator = vertical_index - horizontal_index
numerator = vertical_index_power - horizontal_index_power
if power == 0:
numerator = gs.where(denominator == 0, one_matrix, numerator)
denominator = gs.where(denominator == 0, vertical_index,
denominator)
elif power == math.inf:
numerator = gs.where(denominator == 0, vertical_index_power,
numerator)
denominator = gs.where(denominator == 0, one_matrix, denominator)
else:
numerator = gs.where(denominator == 0,
power * vertical_index_power, numerator)
denominator = gs.where(denominator == 0, vertical_index,
denominator)
transp_eigvectors = gs.transpose(eigvectors, (0, 2, 1))
temp_result = gs.matmul(transp_eigvectors, tangent_vec)
temp_result = gs.matmul(temp_result, eigvectors)
return (eigvectors, transp_eigvectors, numerator, denominator,
temp_result)
def random_von_mises_fisher(self,
mu=None,
kappa=10,
n_samples=1,
max_iter=100):
"""Sample with the von Mises-Fisher distribution.
This distribution corresponds to the maximum entropy distribution
given a mean. In dimension 2, a closed form expression is available.
In larger dimension, rejection sampling is used according to [Wood94]_
References
----------
https://en.wikipedia.org/wiki/Von_Mises-Fisher_distribution
.. [Wood94] Wood, Andrew T. A. “Simulation of the von Mises Fisher
Distribution.” Communications in Statistics - Simulation
and Computation, June 27, 2007.
https://doi.org/10.1080/03610919408813161.
Parameters
----------
mu : array-like, shape=[dim]
Mean parameter of the distribution.
kappa : float
Kappa parameter of the von Mises distribution.
Optional, default: 10.
n_samples : int
Number of samples.
Optional, default: 1.
Returns
-------
point : array-like, shape=[..., 3]
Points sampled on the sphere in extrinsic coordinates
in Euclidean space of dimension 3.
"""
dim = self.dim
if dim == 2:
angle = 2. * gs.pi * gs.random.rand(n_samples)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
unit_vector = gs.hstack((gs.cos(angle), gs.sin(angle)))
scalar = gs.random.rand(n_samples)
coord_z = 1. + 1. / kappa * gs.log(scalar + (1. - scalar) *
gs.exp(gs.array(-2. * kappa)))
coord_z = gs.to_ndarray(coord_z, to_ndim=2, axis=1)
coord_xy = gs.sqrt(1. - coord_z**2) * unit_vector
sample = gs.hstack((coord_xy, coord_z))
if mu is not None:
rot_vec = gs.cross(gs.array([0., 0., 1.]), mu)
rot_vec *= gs.arccos(mu[-1]) / gs.linalg.norm(rot_vec)
rot = SpecialOrthogonal(
3, 'vector').matrix_from_rotation_vector(rot_vec)
sample = gs.matmul(sample, gs.transpose(rot))
else:
if mu is None:
mu = gs.array([0.] * dim + [1.])
# rejection sampling in the general case
sqrt = gs.sqrt(4 * kappa**2. + dim**2)
envelop_param = (-2 * kappa + sqrt) / dim
node = (1. - envelop_param) / (1. + envelop_param)
correction = kappa * node + dim * gs.log(1. - node**2)
n_accepted, n_iter = 0, 0
result = []
while (n_accepted < n_samples) and (n_iter < max_iter):
sym_beta = beta.rvs(dim / 2,
dim / 2,
size=n_samples - n_accepted)
coord_z = (1 - (1 + envelop_param) * sym_beta) / (
1 - (1 - envelop_param) * sym_beta)
accept_tol = gs.random.rand(n_samples - n_accepted)
criterion = (kappa * coord_z + dim * gs.log(1 - node * coord_z)
- correction) > gs.log(accept_tol)
result.append(coord_z[criterion])
n_accepted += gs.sum(criterion)
n_iter += 1
if n_accepted < n_samples:
logging.warning(
'Maximum number of iteration reached in rejection '
'sampling before n_samples were accepted.')
coord_z = gs.concatenate(result)
coord_rest = self.random_uniform(n_accepted)
coord_rest = self.to_tangent(coord_rest, mu)
coord_rest = self.projection(coord_rest)
coord_rest = gs.einsum('...,...i->...i', gs.sqrt(1 - coord_z**2),
coord_rest)
sample = coord_rest + coord_z[:, None] * mu[None, :]
return sample if n_samples > 1 else sample[0]