def projection_to_tangent_space(self, vector, base_point): """Project a vector in the tangent space. Project a vector in Minkowski space on the tangent space of the hyperbolic space at a base point. Parameters ---------- vector : array-like, shape=[n_samples, n_disks, dim + 1] base_point : array-like, shape=[n_samples, n_disks, dim + 1] Returns ------- tangent_vec : array-like, shape=[n_samples, n_disks, dim + 1] """ n_disks = base_point.shape[1] hyperbolic_space = Hyperboloid(2, self.coords_type) tangent_vec = gs.stack([ hyperbolic_space.projection_to_tangent_space( vector=vector[:, i_disk, :], base_point=base_point[:, i_disk, :]) for i_disk in range(n_disks) ], axis=1) return tangent_vec
def test_exp_and_belongs(self): H2 = Hyperboloid(dim=2) METRIC = H2.metric base_point = gs.array([1., 0., 0.]) self.assertTrue(H2.belongs(base_point)) tangent_vec = H2.projection_to_tangent_space( vector=gs.array([1., 2., 1.]), base_point=base_point) exp = METRIC.exp(tangent_vec=tangent_vec, base_point=base_point) self.assertTrue(H2.belongs(exp))
class TestHyperbolicMethods(geomstats.tests.TestCase): def setUp(self): gs.random.seed(1234) self.dimension = 3 self.space = Hyperboloid(dim=self.dimension) self.metric = self.space.metric self.ball_manifold = PoincareBall(dim=2) self.n_samples = 10 def test_random_uniform_and_belongs(self): point = self.space.random_uniform() result = self.space.belongs(point) expected = True self.assertAllClose(result, expected) def test_random_uniform(self): result = self.space.random_uniform() self.assertAllClose(gs.shape(result), (self.dimension + 1,)) def test_projection_to_tangent_space(self): base_point = gs.array([1., 0., 0., 0.]) self.assertTrue(self.space.belongs(base_point)) tangent_vec = self.space.projection_to_tangent_space( vector=gs.array([1., 2., 1., 3.]), base_point=base_point) result = self.metric.inner_product(tangent_vec, base_point) expected = 0. self.assertAllClose(result, expected) result = self.space.projection_to_tangent_space( vector=gs.array([1., 2., 1., 3.]), base_point=base_point) expected = tangent_vec self.assertAllClose(result, expected) def test_intrinsic_and_extrinsic_coords(self): """ Test that the composition of intrinsic_to_extrinsic_coords and extrinsic_to_intrinsic_coords gives the identity. """ point_int = gs.ones(self.dimension) point_ext = self.space.from_coordinates(point_int, 'intrinsic') result = self.space.to_coordinates(point_ext, 'intrinsic') expected = point_int expected = helper.to_vector(expected) self.assertAllClose(result, expected) point_ext = gs.array([2.0, 1.0, 1.0, 1.0]) point_int = self.space.to_coordinates(point_ext, 'intrinsic') result = self.space.from_coordinates(point_int, 'intrinsic') expected = point_ext expected = helper.to_vector(expected) self.assertAllClose(result, expected) def test_intrinsic_and_extrinsic_coords_vectorization(self): """ Test that the composition of intrinsic_to_extrinsic_coords and extrinsic_to_intrinsic_coords gives the identity. """ point_int = gs.array([[.1, 0., 0., .1, 0., 0.], [.1, .1, .1, .4, .1, 0.], [.1, .3, 0., .1, 0., 0.], [-0.1, .1, -.4, .1, -.01, 0.], [0., 0., .1, .1, -0.08, -0.1], [.1, .1, .1, .1, 0., -0.5]]) point_ext = self.space.from_coordinates(point_int, 'intrinsic') result = self.space.to_coordinates(point_ext, 'intrinsic') expected = point_int expected = helper.to_vector(expected) self.assertAllClose(result, expected) point_ext = gs.array([[2., 1., 1., 1.], [4., 1., 3., math.sqrt(5.)], [3., 2., 0., 2.]]) point_int = self.space.to_coordinates(point_ext, 'intrinsic') result = self.space.from_coordinates(point_int, 'intrinsic') expected = point_ext expected = helper.to_vector(expected) self.assertAllClose(result, expected) def test_log_and_exp_general_case(self): """ Test that the Riemannian exponential and the Riemannian logarithm are inverse. Expect their composition to give the identity function. """ # Riemannian Log then Riemannian Exp # General case base_point = gs.array([4.0, 1., 3.0, math.sqrt(5.)]) point = gs.array([2.0, 1.0, 1.0, 1.0]) log = self.metric.log(point=point, base_point=base_point) result = self.metric.exp(tangent_vec=log, base_point=base_point) expected = point self.assertAllClose(result, expected) def test_log_and_exp_general_case_general_dim(self): """ Test that the Riemannian exponential and the Riemannian logarithm are inverse. Expect their composition to give the identity function. """ # Riemannian Log then Riemannian Exp dim = 5 n_samples = self.n_samples h5 = Hyperboloid(dim=dim) h5_metric = h5.metric base_point = h5.random_uniform() point = h5.random_uniform() one_log = h5_metric.log(point=point, base_point=base_point) result = h5_metric.exp(tangent_vec=one_log, base_point=base_point) expected = point self.assertAllClose(result, expected) # Test vectorization of log base_point = gs.stack([base_point] * n_samples, axis=0) point = gs.stack([point] * n_samples, axis=0) expected = gs.stack([one_log] * n_samples, axis=0) log = h5_metric.log(point=point, base_point=base_point) result = log self.assertAllClose(gs.shape(result), (n_samples, dim + 1)) self.assertAllClose(result, expected) result = h5_metric.exp(tangent_vec=log, base_point=base_point) expected = point self.assertAllClose(gs.shape(result), (n_samples, dim + 1)) self.assertAllClose(result, expected) # Test vectorization of exp tangent_vec = gs.stack([one_log] * n_samples, axis=0) exp = h5_metric.exp(tangent_vec=tangent_vec, base_point=base_point) result = exp expected = point self.assertAllClose(gs.shape(result), (n_samples, dim + 1)) self.assertAllClose(result, expected) def test_exp_and_belongs(self): H2 = Hyperboloid(dim=2) METRIC = H2.metric base_point = gs.array([1., 0., 0.]) self.assertTrue(H2.belongs(base_point)) tangent_vec = H2.projection_to_tangent_space( vector=gs.array([1., 2., 1.]), base_point=base_point) exp = METRIC.exp(tangent_vec=tangent_vec, base_point=base_point) self.assertTrue(H2.belongs(exp)) @geomstats.tests.np_and_pytorch_only def test_exp_vectorization(self): n_samples = 3 dim = self.dimension + 1 one_vec = gs.array([2.0, 1.0, 1.0, 1.0]) one_base_point = gs.array([4.0, 3., 1.0, math.sqrt(5)]) n_vecs = gs.array([[2., 1., 1., 1.], [4., 1., 3., math.sqrt(5.)], [3., 2., 0., 2.]]) n_base_points = gs.array([ [2.0, 0.0, 1.0, math.sqrt(2)], [5.0, math.sqrt(8), math.sqrt(8), math.sqrt(8)], [1.0, 0.0, 0.0, 0.0]]) one_tangent_vec = self.space.projection_to_tangent_space( one_vec, base_point=one_base_point) result = self.metric.exp(one_tangent_vec, one_base_point) self.assertAllClose(gs.shape(result), (dim,)) n_tangent_vecs = self.space.projection_to_tangent_space( n_vecs, base_point=one_base_point) result = self.metric.exp(n_tangent_vecs, one_base_point) self.assertAllClose(gs.shape(result), (n_samples, dim)) expected = gs.zeros((n_samples, dim)) for i in range(n_samples): expected[i] = self.metric.exp(n_tangent_vecs[i], one_base_point) expected = helper.to_vector(gs.array(expected)) self.assertAllClose(result, expected) one_tangent_vec = self.space.projection_to_tangent_space( one_vec, base_point=n_base_points) result = self.metric.exp(one_tangent_vec, n_base_points) self.assertAllClose(gs.shape(result), (n_samples, dim)) expected = gs.zeros((n_samples, dim)) for i in range(n_samples): expected[i] = self.metric.exp(one_tangent_vec[i], n_base_points[i]) expected = helper.to_vector(gs.array(expected)) self.assertAllClose(result, expected) n_tangent_vecs = self.space.projection_to_tangent_space( n_vecs, base_point=n_base_points) result = self.metric.exp(n_tangent_vecs, n_base_points) self.assertAllClose(gs.shape(result), (n_samples, dim)) expected = gs.zeros((n_samples, dim)) for i in range(n_samples): expected[i] = self.metric.exp(n_tangent_vecs[i], n_base_points[i]) expected = helper.to_vector(gs.array(expected)) self.assertAllClose(result, expected) def test_log_vectorization(self): n_samples = 3 dim = self.dimension + 1 one_point = gs.array([2.0, 1.0, 1.0, 1.0]) one_base_point = gs.array([4.0, 3., 1.0, math.sqrt(5)]) n_points = gs.array([[2.0, 1.0, 1.0, 1.0], [4.0, 1., 3.0, math.sqrt(5)], [3.0, 2.0, 0.0, 2.0]]) n_base_points = gs.array([ [2.0, 0.0, 1.0, math.sqrt(2)], [5.0, math.sqrt(8), math.sqrt(8), math.sqrt(8)], [1.0, 0.0, 0.0, 0.0]]) result = self.metric.log(one_point, one_base_point) self.assertAllClose(gs.shape(result), (dim,)) result = self.metric.log(n_points, one_base_point) self.assertAllClose(gs.shape(result), (n_samples, dim)) result = self.metric.log(one_point, n_base_points) self.assertAllClose(gs.shape(result), (n_samples, dim)) result = self.metric.log(n_points, n_base_points) self.assertAllClose(gs.shape(result), (n_samples, dim)) def test_inner_product(self): """ Test that the inner product between two tangent vectors is the Minkowski inner product. """ minkowski_space = Minkowski(self.dimension + 1) base_point = gs.array( [1.16563816, 0.36381045, -0.47000603, 0.07381469]) tangent_vec_a = self.space.projection_to_tangent_space( vector=gs.array([10., 200., 1., 1.]), base_point=base_point) tangent_vec_b = self.space.projection_to_tangent_space( vector=gs.array([11., 20., -21., 0.]), base_point=base_point) result = self.metric.inner_product( tangent_vec_a, tangent_vec_b, base_point) expected = minkowski_space.metric.inner_product( tangent_vec_a, tangent_vec_b, base_point) self.assertAllClose(result, expected) def test_squared_norm_and_squared_dist(self): """ Test that the squared distance between two points is the squared norm of their logarithm. """ point_a = gs.array([2.0, 1.0, 1.0, 1.0]) point_b = gs.array([4.0, 1., 3.0, math.sqrt(5)]) log = self.metric.log(point=point_a, base_point=point_b) result = self.metric.squared_norm(vector=log) expected = self.metric.squared_dist(point_a, point_b) self.assertAllClose(result, expected) def test_norm_and_dist(self): """ Test that the distance between two points is the norm of their logarithm. """ point_a = gs.array([2.0, 1.0, 1.0, 1.0]) point_b = gs.array([4.0, 1., 3.0, math.sqrt(5)]) log = self.metric.log(point=point_a, base_point=point_b) result = self.metric.norm(vector=log) expected = self.metric.dist(point_a, point_b) self.assertAllClose(result, expected) def test_log_and_exp_edge_case(self): """ Test that the Riemannian exponential and the Riemannian logarithm are inverse. Expect their composition to give the identity function. """ # Riemannian Log then Riemannian Exp # Edge case: two very close points, base_point_2 and point_2, # form an angle < epsilon base_point_intrinsic = gs.array([1., 2., 3.]) base_point =\ self.space.from_coordinates(base_point_intrinsic, 'intrinsic') point_intrinsic = (base_point_intrinsic + 1e-12 * gs.array([-1., -2., 1.])) point =\ self.space.from_coordinates(point_intrinsic, 'intrinsic') log = self.metric.log(point=point, base_point=base_point) result = self.metric.exp(tangent_vec=log, base_point=base_point) expected = point self.assertAllClose(result, expected) @geomstats.tests.np_and_tf_only def test_exp_and_log_and_projection_to_tangent_space_general_case(self): """ Test that the Riemannian exponential and the Riemannian logarithm are inverse. Expect their composition to give the identity function. """ # Riemannian Exp then Riemannian Log # General case base_point = gs.array([4.0, 1., 3.0, math.sqrt(5)]) vector = gs.array([2.0, 1.0, 1.0, 1.0]) vector = self.space.projection_to_tangent_space( vector=vector, base_point=base_point) exp = self.metric.exp(tangent_vec=vector, base_point=base_point) result = self.metric.log(point=exp, base_point=base_point) expected = vector self.assertAllClose(result, expected) def test_dist(self): # Distance between a point and itself is 0. point_a = gs.array([4.0, 1., 3.0, math.sqrt(5)]) point_b = point_a result = self.metric.dist(point_a, point_b) expected = 0 self.assertAllClose(result, expected) def test_exp_and_dist_and_projection_to_tangent_space(self): base_point = gs.array([4.0, 1., 3.0, math.sqrt(5)]) vector = gs.array([0.001, 0., -.00001, -.00003]) tangent_vec = self.space.projection_to_tangent_space( vector=vector, base_point=base_point) exp = self.metric.exp( tangent_vec=tangent_vec, base_point=base_point) result = self.metric.dist(base_point, exp) sq_norm = self.metric.embedding_metric.squared_norm( tangent_vec) expected = sq_norm self.assertAllClose(result, expected, atol=1e-2) def test_geodesic_and_belongs(self): # TODO(nina): Fix this tests, as it fails when geodesic goes "too far" initial_point = gs.array([4.0, 1., 3.0, math.sqrt(5)]) n_geodesic_points = 100 vector = gs.array([1., 0., 0., 0.]) initial_tangent_vec = self.space.projection_to_tangent_space( vector=vector, base_point=initial_point) geodesic = self.metric.geodesic( initial_point=initial_point, initial_tangent_vec=initial_tangent_vec) t = gs.linspace(start=0., stop=1., num=n_geodesic_points) points = geodesic(t) result = self.space.belongs(points) expected = n_geodesic_points * [True] self.assertAllClose(result, expected) def test_exp_and_log_and_projection_to_tangent_space_edge_case(self): """ Test that the Riemannian exponential and the Riemannian logarithm are inverse. Expect their composition to give the identity function. """ # Riemannian Exp then Riemannian Log # Edge case: tangent vector has norm < epsilon base_point = gs.array([2., 1., 1., 1.]) vector = 1e-10 * gs.array([.06, -51., 6., 5.]) exp = self.metric.exp(tangent_vec=vector, base_point=base_point) result = self.metric.log(point=exp, base_point=base_point) expected = self.space.projection_to_tangent_space( vector=vector, base_point=base_point) self.assertAllClose(result, expected, atol=1e-8) @geomstats.tests.np_only def test_scaled_inner_product(self): base_point_intrinsic = gs.array([1, 1, 1]) base_point = self.space.from_coordinates( base_point_intrinsic, "intrinsic") tangent_vec_a = gs.array([1, 2, 3, 4]) tangent_vec_b = gs.array([5, 6, 7, 8]) tangent_vec_a = self.space.projection_to_tangent_space( tangent_vec_a, base_point) tangent_vec_b = self.space.projection_to_tangent_space( tangent_vec_b, base_point) scale = 2 default_space = Hyperboloid(dim=self.dimension) scaled_space = Hyperboloid(dim=self.dimension, scale=2) inner_product_default_metric = \ default_space.metric.inner_product( tangent_vec_a, tangent_vec_b, base_point) inner_product_scaled_metric = \ scaled_space.metric.inner_product( tangent_vec_a, tangent_vec_b, base_point) result = inner_product_scaled_metric expected = scale ** 2 * inner_product_default_metric self.assertAllClose(result, expected) @geomstats.tests.np_only def test_scaled_squared_norm(self): base_point_intrinsic = gs.array([1, 1, 1]) base_point = self.space.from_coordinates(base_point_intrinsic, 'intrinsic') tangent_vec = gs.array([1, 2, 3, 4]) tangent_vec = self.space.projection_to_tangent_space( tangent_vec, base_point) scale = 2 default_space = Hyperboloid(dim=self.dimension) scaled_space = Hyperboloid(dim=self.dimension, scale=2) squared_norm_default_metric = default_space.metric.squared_norm( tangent_vec, base_point) squared_norm_scaled_metric = scaled_space.metric.squared_norm( tangent_vec, base_point) result = squared_norm_scaled_metric expected = scale ** 2 * squared_norm_default_metric self.assertAllClose(result, expected) @geomstats.tests.np_only def test_scaled_distance(self): point_a_intrinsic = gs.array([1, 2, 3]) point_b_intrinsic = gs.array([4, 5, 6]) point_a = self.space.from_coordinates(point_a_intrinsic, 'intrinsic') point_b = self.space.from_coordinates(point_b_intrinsic, 'intrinsic') scale = 2 scaled_space = Hyperboloid(dim=self.dimension, scale=2) distance_default_metric = self.space.metric.dist(point_a, point_b) distance_scaled_metric = scaled_space.metric.dist(point_a, point_b) result = distance_scaled_metric expected = scale * distance_default_metric self.assertAllClose(result, expected)