def __init__(self, n, p): dimension = int(p * n - (p * (p + 1) / 2)) super(StiefelCanonicalMetric, self).__init__(dimension=dimension, signature=(dimension, 0, 0)) self.embedding_metric = EuclideanMetric(n * p) self.n = n self.p = p
def __init__(self, dimension): super(HypersphereMetric, self).__init__(dimension=dimension, signature=(dimension, 0, 0)) self.embedding_metric = EuclideanMetric(dimension + 1)
class HypersphereMetric(RiemannianMetric): def __init__(self, dimension): super(HypersphereMetric, self).__init__(dimension=dimension, signature=(dimension, 0, 0)) self.embedding_metric = EuclideanMetric(dimension + 1) def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None): """ Inner product. """ inner_prod = self.embedding_metric.inner_product( tangent_vec_a, tangent_vec_b, base_point) return inner_prod def squared_norm(self, vector, base_point=None): """ Squared norm of a vector associated to the inner product at the tangent space at a base point. """ sq_norm = self.embedding_metric.squared_norm(vector) return sq_norm def exp(self, tangent_vec, base_point): """ Riemannian exponential of a tangent vector wrt to a base point. """ tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) # TODO(johmathe): Evaluate the bias introduced by this variable norm_tangent_vec = self.embedding_metric.norm(tangent_vec) + EPSILON coef_1 = gs.cos(norm_tangent_vec) coef_2 = gs.sin(norm_tangent_vec) / norm_tangent_vec exp = (gs.einsum('ni,nj->nj', coef_1, base_point) + gs.einsum('ni,nj->nj', coef_2, tangent_vec)) return exp def log(self, point, base_point): """ Riemannian logarithm of a point wrt a base point. """ point = gs.to_ndarray(point, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) norm_base_point = self.embedding_metric.norm(base_point) norm_point = self.embedding_metric.norm(point) inner_prod = self.embedding_metric.inner_product(base_point, point) cos_angle = inner_prod / (norm_base_point * norm_point) cos_angle = gs.clip(cos_angle, -1., 1.) angle = gs.arccos(cos_angle) angle = gs.to_ndarray(angle, to_ndim=1) angle = gs.to_ndarray(angle, to_ndim=2, axis=1) mask_0 = gs.isclose(angle, 0.) mask_else = gs.equal(mask_0, gs.array(False)) mask_0_float = gs.cast(mask_0, gs.float32) mask_else_float = gs.cast(mask_else, gs.float32) coef_1 = gs.zeros_like(angle) coef_2 = gs.zeros_like(angle) coef_1 += mask_0_float * (1. + INV_SIN_TAYLOR_COEFFS[1] * angle**2 + INV_SIN_TAYLOR_COEFFS[3] * angle**4 + INV_SIN_TAYLOR_COEFFS[5] * angle**6 + INV_SIN_TAYLOR_COEFFS[7] * angle**8) coef_2 += mask_0_float * (1. + INV_TAN_TAYLOR_COEFFS[1] * angle**2 + INV_TAN_TAYLOR_COEFFS[3] * angle**4 + INV_TAN_TAYLOR_COEFFS[5] * angle**6 + INV_TAN_TAYLOR_COEFFS[7] * angle**8) # This avoids division by 0. angle += mask_0_float * 1. coef_1 += mask_else_float * angle / gs.sin(angle) coef_2 += mask_else_float * angle / gs.tan(angle) log = (gs.einsum('ni,nj->nj', coef_1, point) - gs.einsum('ni,nj->nj', coef_2, base_point)) mask_same_values = gs.isclose(point, base_point) mask_else = gs.equal(mask_same_values, gs.array(False)) mask_else_float = gs.cast(mask_else, gs.float32) mask_else_float = gs.to_ndarray(mask_else_float, to_ndim=1) mask_else_float = gs.to_ndarray(mask_else_float, to_ndim=2) mask_not_same_points = gs.sum(mask_else_float, axis=1) mask_same_points = gs.isclose(mask_not_same_points, 0.) mask_same_points = gs.cast(mask_same_points, gs.float32) mask_same_points = gs.to_ndarray(mask_same_points, to_ndim=2, axis=1) mask_same_points_float = gs.cast(mask_same_points, gs.float32) log -= mask_same_points_float * log return log def dist(self, point_a, point_b): """ Geodesic distance between two points. """ norm_a = self.embedding_metric.norm(point_a) norm_b = self.embedding_metric.norm(point_b) inner_prod = self.embedding_metric.inner_product(point_a, point_b) cos_angle = inner_prod / (norm_a * norm_b) cos_angle = gs.clip(cos_angle, -1, 1) dist = gs.arccos(cos_angle) return dist
def __init__(self, dimension): self.dimension = dimension self.signature = (dimension, 0, 0) self.embedding_metric = EuclideanMetric(dimension + 1)
class HypersphereMetric(RiemannianMetric): def __init__(self, dimension): self.dimension = dimension self.signature = (dimension, 0, 0) self.embedding_metric = EuclideanMetric(dimension + 1) def squared_norm(self, vector, base_point=None): """ Squared norm of a vector associated to the inner product at the tangent space at a base point. """ sq_norm = self.embedding_metric.squared_norm(vector) return sq_norm def exp(self, tangent_vec, base_point): """ Riemannian exponential of a tangent vector wrt to a base point. """ tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) # TODO(johmathe): Evaluate the bias introduced by this variable norm_tangent_vec = self.embedding_metric.norm(tangent_vec) + EPSILON coef_1 = gs.cos(norm_tangent_vec) coef_2 = gs.sin(norm_tangent_vec) / norm_tangent_vec exp = (gs.einsum('ni,nj->nj', coef_1, base_point) + gs.einsum('ni,nj->nj', coef_2, tangent_vec)) return exp def log(self, point, base_point): """ Riemannian logarithm of a point wrt a base point. """ point = gs.to_ndarray(point, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) norm_base_point = self.embedding_metric.norm(base_point) norm_point = self.embedding_metric.norm(point) inner_prod = self.embedding_metric.inner_product(base_point, point) cos_angle = inner_prod / (norm_base_point * norm_point) cos_angle = gs.clip(cos_angle, -1.0, 1.0) angle = gs.arccos(cos_angle) mask_0 = gs.isclose(angle, 0.0) mask_else = gs.equal(mask_0, gs.cast(gs.array(False), gs.int8)) coef_1 = gs.zeros_like(angle) coef_2 = gs.zeros_like(angle) coef_1[mask_0] = ( 1. + INV_SIN_TAYLOR_COEFFS[1] * angle[mask_0] ** 2 + INV_SIN_TAYLOR_COEFFS[3] * angle[mask_0] ** 4 + INV_SIN_TAYLOR_COEFFS[5] * angle[mask_0] ** 6 + INV_SIN_TAYLOR_COEFFS[7] * angle[mask_0] ** 8) coef_2[mask_0] = ( 1. + INV_TAN_TAYLOR_COEFFS[1] * angle[mask_0] ** 2 + INV_TAN_TAYLOR_COEFFS[3] * angle[mask_0] ** 4 + INV_TAN_TAYLOR_COEFFS[5] * angle[mask_0] ** 6 + INV_TAN_TAYLOR_COEFFS[7] * angle[mask_0] ** 8) coef_1[mask_else] = angle[mask_else] / gs.sin(angle[mask_else]) coef_2[mask_else] = angle[mask_else] / gs.tan(angle[mask_else]) log = (gs.einsum('ni,nj->nj', coef_1, point) - gs.einsum('ni,nj->nj', coef_2, base_point)) return log def dist(self, point_a, point_b): """ Geodesic distance between two points. """ # TODO(nina): case gs.dot(unit_vec, unit_vec) != 1 # if gs.all(gs.equal(point_a, point_b)): # return 0. norm_a = self.embedding_metric.norm(point_a) norm_b = self.embedding_metric.norm(point_b) inner_prod = self.embedding_metric.inner_product(point_a, point_b) cos_angle = inner_prod / (norm_a * norm_b) cos_angle = gs.clip(cos_angle, -1, 1) dist = gs.arccos(cos_angle) return dist
def setUp(self): self.dimension = 4 self.metric = EuclideanMetric(dimension=self.dimension) self.connection = LeviCivitaConnection(self.metric)
class HypersphereMetric(RiemannianMetric): def __init__(self, dimension): self.dimension = dimension self.signature = (dimension, 0, 0) self.embedding_metric = EuclideanMetric(dimension + 1) def squared_norm(self, vector, base_point=None): """ Squared norm associated to the Hyperbolic Metric. """ sq_norm = self.embedding_metric.squared_norm(vector) return sq_norm def exp(self, tangent_vec, base_point): """ Compute the Riemannian exponential at point base_point of tangent vector tangent_vec wrt the metric obtained by embedding of the n-dimensional sphere in the (n+1)-dimensional euclidean space. This gives a point on the n-dimensional sphere. :param base_point: a point on the n-dimensional sphere :param vector: (n+1)-dimensional vector :return exp: a point on the n-dimensional sphere """ tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) # TODO(johmathe): Evaluate the bias introduced by this variable norm_tangent_vec = self.embedding_metric.norm(tangent_vec) + EPSILON coef_1 = gs.cos(norm_tangent_vec) coef_2 = gs.sin(norm_tangent_vec) / norm_tangent_vec exp = (gs.einsum('ni,nj->nj', coef_1, base_point) + gs.einsum('ni,nj->nj', coef_2, tangent_vec)) return exp def log(self, point, base_point): """ Compute the Riemannian logarithm at point base_point, of point wrt the metric obtained by embedding of the n-dimensional sphere in the (n+1)-dimensional euclidean space. This gives a tangent vector at point base_point. :param base_point: point on the n-dimensional sphere :param point: point on the n-dimensional sphere :return log: tangent vector at base_point """ point = gs.to_ndarray(point, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) norm_base_point = self.embedding_metric.norm(base_point) norm_point = self.embedding_metric.norm(point) inner_prod = self.embedding_metric.inner_product(base_point, point) cos_angle = inner_prod / (norm_base_point * norm_point) cos_angle = gs.clip(cos_angle, -1.0, 1.0) angle = gs.arccos(cos_angle) mask_0 = gs.isclose(angle, 0.0) mask_else = gs.equal(mask_0, False) coef_1 = gs.zeros_like(angle) coef_2 = gs.zeros_like(angle) coef_1[mask_0] = ( 1. + INV_SIN_TAYLOR_COEFFS[1] * angle[mask_0] ** 2 + INV_SIN_TAYLOR_COEFFS[3] * angle[mask_0] ** 4 + INV_SIN_TAYLOR_COEFFS[5] * angle[mask_0] ** 6 + INV_SIN_TAYLOR_COEFFS[7] * angle[mask_0] ** 8) coef_2[mask_0] = ( 1. + INV_TAN_TAYLOR_COEFFS[1] * angle[mask_0] ** 2 + INV_TAN_TAYLOR_COEFFS[3] * angle[mask_0] ** 4 + INV_TAN_TAYLOR_COEFFS[5] * angle[mask_0] ** 6 + INV_TAN_TAYLOR_COEFFS[7] * angle[mask_0] ** 8) coef_1[mask_else] = angle[mask_else] / gs.sin(angle[mask_else]) coef_2[mask_else] = angle[mask_else] / gs.tan(angle[mask_else]) log = (gs.einsum('ni,nj->nj', coef_1, point) - gs.einsum('ni,nj->nj', coef_2, base_point)) return log def dist(self, point_a, point_b): """ Compute the Riemannian distance between points point_a and point_b. """ # TODO(nina): case gs.dot(unit_vec, unit_vec) != 1 # if gs.all(gs.equal(point_a, point_b)): # return 0. norm_a = self.embedding_metric.norm(point_a) norm_b = self.embedding_metric.norm(point_b) inner_prod = self.embedding_metric.inner_product(point_a, point_b) cos_angle = inner_prod / (norm_a * norm_b) cos_angle = gs.clip(cos_angle, -1, 1) dist = gs.arccos(cos_angle) return dist
def __init__(self, dimension): self.dimension = dimension self.metric = HypersphereMetric(dimension) self.embedding_metric = EuclideanMetric(dimension + 1)
class Hypersphere(Manifold): """Hypersphere embedded in Euclidean space.""" def __init__(self, dimension): self.dimension = dimension self.metric = HypersphereMetric(dimension) self.embedding_metric = EuclideanMetric(dimension + 1) def belongs(self, point, tolerance=TOLERANCE): """ By definition, a point on the Hypersphere has squared norm 1 in the embedding Euclidean space. Note: point must be given in extrinsic coordinates. """ point = vectorization.expand_dims(point, to_ndim=2) _, point_dim = point.shape if point_dim is not self.dimension + 1: if point_dim is self.dimension: logging.warning('Use the extrinsic coordinates to ' 'represent points on the hypersphere.') return False sq_norm = self.embedding_metric.squared_norm(point) diff = np.abs(sq_norm - 1) return diff < tolerance def projection_to_tangent_space(self, vector, base_point): """ Project the vector vector onto the tangent space: T_{base_point} S = {w | scal(w, base_point) = 0} """ assert self.belongs(base_point) sq_norm = self.embedding_metric.squared_norm(base_point) inner_prod = self.embedding_metric.inner_product(base_point, vector) tangent_vec = (vector - inner_prod / sq_norm * base_point) return tangent_vec def intrinsic_to_extrinsic_coords(self, point_intrinsic): """ From some intrinsic coordinates in the Hypersphere, to the extrinsic coordinates in Euclidean space. """ point_intrinsic = vectorization.expand_dims(point_intrinsic, to_ndim=2) n_points, _ = point_intrinsic.shape dimension = self.dimension point_extrinsic = np.zeros((n_points, dimension + 1)) point_extrinsic[:, 1:dimension + 1] = point_intrinsic[:, 0:dimension] point_extrinsic[:, 0] = np.sqrt( 1. - np.linalg.norm(point_intrinsic, axis=1)**2) assert np.all(self.belongs(point_extrinsic)) assert point_extrinsic.ndim == 2 return point_extrinsic def extrinsic_to_intrinsic_coords(self, point_extrinsic): """ From the extrinsic coordinates in Euclidean space, to some intrinsic coordinates in Hypersphere. """ point_extrinsic = vectorization.expand_dims(point_extrinsic, to_ndim=2) assert np.all(self.belongs(point_extrinsic)) point_intrinsic = point_extrinsic[:, 1:] assert point_intrinsic.ndim == 2 return point_intrinsic def random_uniform(self, n_samples=1, max_norm=1): """ Generate random elements on the Hypersphere. """ point = ((np.random.rand(n_samples, self.dimension) - .5) * max_norm) point = self.intrinsic_to_extrinsic_coords(point) assert np.all(self.belongs(point)) assert point.ndim == 2 return point
class HypersphereMetric(RiemannianMetric): def __init__(self, dimension): self.dimension = dimension self.signature = (dimension, 0, 0) self.embedding_metric = EuclideanMetric(dimension + 1) def squared_norm(self, vector, base_point=None): """ Squared norm associated to the Hyperbolic Metric. """ sq_norm = self.embedding_metric.squared_norm(vector) return sq_norm def exp_basis(self, tangent_vec, base_point): """ Compute the Riemannian exponential at point base_point of tangent vector tangent_vec wrt the metric obtained by embedding of the n-dimensional sphere in the (n+1)-dimensional euclidean space. This gives a point on the n-dimensional sphere. :param base_point: a point on the n-dimensional sphere :param vector: (n+1)-dimensional vector :return exp: a point on the n-dimensional sphere """ norm_tangent_vec = self.embedding_metric.norm(tangent_vec) if np.isclose(norm_tangent_vec, 0): coef_1 = (1. + COS_TAYLOR_COEFFS[2] * norm_tangent_vec**2 + COS_TAYLOR_COEFFS[4] * norm_tangent_vec**4 + COS_TAYLOR_COEFFS[6] * norm_tangent_vec**6 + COS_TAYLOR_COEFFS[8] * norm_tangent_vec**8) coef_2 = (1. + SIN_TAYLOR_COEFFS[3] * norm_tangent_vec**2 + SIN_TAYLOR_COEFFS[5] * norm_tangent_vec**4 + SIN_TAYLOR_COEFFS[7] * norm_tangent_vec**6 + SIN_TAYLOR_COEFFS[9] * norm_tangent_vec**8) else: coef_1 = np.cos(norm_tangent_vec) coef_2 = np.sin(norm_tangent_vec) / norm_tangent_vec exp = coef_1 * base_point + coef_2 * tangent_vec return exp def log_basis(self, point, base_point): """ Compute the Riemannian logarithm at point base_point, of point wrt the metric obtained by embedding of the n-dimensional sphere in the (n+1)-dimensional euclidean space. This gives a tangent vector at point base_point. :param base_point: point on the n-dimensional sphere :param point: point on the n-dimensional sphere :return log: tangent vector at base_point """ norm_base_point = self.embedding_metric.norm(base_point) norm_point = self.embedding_metric.norm(point) inner_prod = self.embedding_metric.inner_product(base_point, point) cos_angle = inner_prod / (norm_base_point * norm_point) if cos_angle >= 1.: angle = 0. else: angle = np.arccos(cos_angle) if np.isclose(angle, 0): coef_1 = (1. + INV_SIN_TAYLOR_COEFFS[1] * angle**2 + INV_SIN_TAYLOR_COEFFS[3] * angle**4 + INV_SIN_TAYLOR_COEFFS[5] * angle**6 + INV_SIN_TAYLOR_COEFFS[7] * angle**8) coef_2 = (1. + INV_TAN_TAYLOR_COEFFS[1] * angle**2 + INV_TAN_TAYLOR_COEFFS[3] * angle**4 + INV_TAN_TAYLOR_COEFFS[5] * angle**6 + INV_TAN_TAYLOR_COEFFS[7] * angle**8) else: coef_1 = angle / np.sin(angle) coef_2 = angle / np.tan(angle) log = coef_1 * point - coef_2 * base_point return log def dist(self, point_a, point_b): """ Compute the Riemannian distance between points point_a and point_b. """ # TODO(nina): case np.dot(unit_vec, unit_vec) != 1 if np.all(point_a == point_b): return 0. point_a = vectorization.expand_dims(point_a, to_ndim=2) point_b = vectorization.expand_dims(point_b, to_ndim=2) n_points_a, _ = point_a.shape n_points_b, _ = point_b.shape assert (n_points_a == n_points_b or n_points_a == 1 or n_points_b == 1) n_dists = np.maximum(n_points_a, n_points_b) dist = np.zeros((n_dists, 1)) norm_a = self.embedding_metric.norm(point_a) norm_b = self.embedding_metric.norm(point_b) inner_prod = self.embedding_metric.inner_product(point_a, point_b) cos_angle = inner_prod / (norm_a * norm_b) mask_cos_greater_1 = np.greater_equal(cos_angle, 1.) mask_cos_less_minus_1 = np.less_equal(cos_angle, -1.) mask_else = ~mask_cos_greater_1 & ~mask_cos_less_minus_1 dist[mask_cos_greater_1] = 0. dist[mask_cos_less_minus_1] = np.pi dist[mask_else] = np.arccos(cos_angle[mask_else]) return dist