Source code for geomstats.geometry.stiefel

"""Stiefel manifold St(n,p).

A set of all orthonormal p-frames in n-dimensional space, where p <= n.

Lead author: Oleg Kachan.
"""

import geomstats.backend as gs
import geomstats.errors
import geomstats.vectorization
from geomstats import algebra_utils
from geomstats.geometry.base import LevelSet
from geomstats.geometry.euclidean import EuclideanMetric
from geomstats.geometry.matrices import Matrices
from geomstats.geometry.riemannian_metric import RiemannianMetric
from geomstats.geometry.symmetric_matrices import SymmetricMatrices

EPSILON = 1e-6


[docs]class Stiefel(LevelSet): """Class for Stiefel manifolds St(n,p). A set of all orthonormal p-frames in n-dimensional space, where p <= n. Parameters ---------- n : int Dimension of the ambient vector space. p : int Number of basis vectors in the orthonormal frame. """ def __init__(self, n, p, **kwargs): geomstats.errors.check_integer(n, "n") geomstats.errors.check_integer(p, "p") if p > n: raise ValueError("p needs to be smaller than n.") dim = int(p * n - (p * (p + 1) / 2)) matrices = Matrices(n, p) canonical_metric = StiefelCanonicalMetric(n, p) kwargs.setdefault("metric", canonical_metric) super(Stiefel, self).__init__( dim=dim, embedding_space=matrices, submersion=lambda x: matrices.mul(matrices.transpose(x), x), value=gs.eye(p), tangent_submersion=lambda v, x: 2 * matrices.to_symmetric(matrices.mul(matrices.transpose(x), v)), **kwargs ) self.canonical_metric = canonical_metric self.n = n self.p = p
[docs] @staticmethod def to_grassmannian(point): r"""Project a point of St(n, p) to Gr(n, p). If :math:`U \in St(n, p)` is an orthonormal frame, return the orthogonal projector :math:`P = U U^T` onto the subspace of :math:`\mathbb{R}^n` spanned by :math:`U`. Parameters ---------- point : array-like, shape=[..., n, p] Point. Returns ------- projected : array-like, shape=[..., n, n] Projected point. """ return Matrices.mul(point, Matrices.transpose(point))
[docs] def random_uniform(self, n_samples=1): r"""Sample on St(n,p) from the uniform distribution. If :math:`Z(p,n) \sim N(0,1)`, then :math:`St(n,p) \sim U`, according to Haar measure: :math:`St(n,p) := Z(Z^TZ)^{-1/2}`. Parameters ---------- n_samples : int Number of samples. Optional, default: 1. Returns ------- samples : array-like, shape=[..., n, p] Samples on the Stiefel manifold. """ n, p = self.n, self.p size = (n_samples, n, p) if n_samples != 1 else (n, p) std_normal = gs.random.normal(size=size) std_normal_transpose = Matrices.transpose(std_normal) aux = Matrices.mul(std_normal_transpose, std_normal) inv_sqrt_aux = SymmetricMatrices.powerm(aux, -1.0 / 2) samples = Matrices.mul(std_normal, inv_sqrt_aux) return samples
[docs] def random_point(self, n_samples=1, bound=1.0): r"""Sample on St(n,p) from the uniform distribution. If :math:`Z(p,n) \sim N(0,1)`, then :math:`St(n,p) \sim U`, according to Haar measure: :math:`St(n,p) := Z(Z^TZ)^{-1/2}`. Parameters ---------- n_samples : int Number of samples. Optional, default: 1. bound : float Unused here. Returns ------- samples : array-like, shape=[..., n, p] Samples on the Stiefel manifold. """ return self.random_uniform(n_samples)
[docs] def to_tangent(self, vector, base_point): """Project a vector to a tangent space of the manifold. Inspired by the method of Pymanopt. Parameters ---------- vector : array-like, shape=[..., n, p] Vector. base_point : array-like, shape=[..., n, p] Point on the manifold. Returns ------- tangent_vec : array-like, shape=[..., n, p] Tangent vector at base point. """ aux = Matrices.mul(Matrices.transpose(base_point), vector) sym_aux = Matrices.to_symmetric(aux) return vector - Matrices.mul(base_point, sym_aux)
[docs] def projection(self, point): """Project a close enough matrix to the Stiefel manifold. A singular value decomposition is used, and all singular values are set to 1 [Absil]_ Parameters ---------- point : array-like, shape=[..., n, p] Point in embedding manifold. Returns ------- projected : array-like, shape=[..., n, p] References ---------- ..[Absil] Absil, Pierre-Antoine, and Jérôme Malick. “Projection-like Retractions on Matrix Manifolds.” SIAM Journal on Optimization 22, no. 1 (January 2012): 135–58. https://doi.org/10.1137/100802529. """ mat_u, _, mat_v = gs.linalg.svd(point) return Matrices.mul(mat_u[..., :, : self.p], mat_v)
[docs]class StiefelCanonicalMetric(RiemannianMetric): """Class that defines the canonical metric for Stiefel manifolds. Parameters ---------- n : int Dimension of the ambient vector space. p : int Number of basis vectors in the orthonormal frames. """ def __init__(self, n, p): dim = int(p * n - (p * (p + 1) / 2)) super(StiefelCanonicalMetric, self).__init__( dim=dim, default_point_type="matrix", signature=(dim, 0, 0) ) self.embedding_metric = EuclideanMetric(n * p) self.n = n self.p = p
[docs] def inner_product(self, tangent_vec_a, tangent_vec_b, base_point): r"""Compute the inner-product of two tangent vectors at a base point. Canonical inner-product on the tangent space at `base_point`, which is different from the inner-product induced by the embedding (see [RLSMRZ2017]_). .. math:: \langle\Delta, \tilde{\Delta}\rangle_{U}=\operatorname{tr} \left(\Delta^{T}\left(I-\frac{1}{2} U U^{T}\right) \tilde{\Delta}\right) References ---------- .. [RLSMRZ2017] R Zimmermann. A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric. SIAM Journal on Matrix Analysis and Applications 38 (2), 322-342, 2017. https://epubs.siam.org/doi/pdf/10.1137/16M1074485 Parameters ---------- tangent_vec_a : array-like, shape=[..., n, p] First tangent vector at base point. tangent_vec_b : array-like, shape=[..., n, p] Second tangent vector at base point. base_point : array-like, shape=[..., n, p] Point in the Stiefel manifold. Returns ------- inner_prod : array-like, shape=[..., 1] Inner-product of the two tangent vectors. """ base_point_transpose = Matrices.transpose(base_point) aux = gs.matmul( Matrices.transpose(tangent_vec_a), gs.eye(self.n) - 0.5 * gs.matmul(base_point, base_point_transpose), ) inner_prod = Matrices.trace_product(aux, tangent_vec_b) return inner_prod
[docs] def exp(self, tangent_vec, base_point, **kwargs): """Compute the Riemannian exponential of a tangent vector. Parameters ---------- tangent_vec : array-like, shape=[..., n, p] Tangent vector at a base point. base_point : array-like, shape=[..., n, p] Point in the Stiefel manifold. Returns ------- exp : array-like, shape=[..., n, p] Point in the Stiefel manifold equal to the Riemannian exponential of tangent_vec at the base point. """ p = self.p matrix_a = Matrices.mul(Matrices.transpose(base_point), tangent_vec) matrix_k = tangent_vec - Matrices.mul(base_point, matrix_a) matrix_q, matrix_r = gs.linalg.qr(matrix_k) matrix_ar = gs.concatenate([matrix_a, -Matrices.transpose(matrix_r)], axis=-1) zeros = gs.zeros_like(tangent_vec)[..., :p, :p] matrix_rz = gs.concatenate([matrix_r, zeros], axis=-1) block = gs.concatenate([matrix_ar, matrix_rz], axis=-2) matrix_mn_e = gs.linalg.expm(block) exp = Matrices.mul(base_point, matrix_mn_e[..., :p, :p]) + Matrices.mul( matrix_q, matrix_mn_e[..., p:, :p] ) return exp
@staticmethod def _normal_component_qr(point, base_point, matrix_m): """Compute the QR decomposition of the normal component of a point. Parameters ---------- point : array-like, shape=[..., n, p] base_point : array-like, shape=[..., n, p] matrix_m : array-like Returns ------- matrix_q : array-like matrix_n : array-like """ matrix_k = point - gs.matmul(base_point, matrix_m) matrix_q, matrix_n = gs.linalg.qr(matrix_k) return matrix_q, matrix_n @staticmethod def _orthogonal_completion(matrix_m, matrix_n): """Orthogonal matrix completion. Parameters ---------- matrix_m : array-like matrix_n : array-like Returns ------- matrix_v : array-like """ matrix_w = gs.concatenate([matrix_m, matrix_n], axis=-2) matrix_v, _ = gs.linalg.qr(matrix_w, mode="complete") return matrix_v @staticmethod def _procrustes_preprocessing(p, matrix_v, matrix_m, matrix_n): """Procrustes preprocessing. Parameters ---------- matrix_v : array-like matrix_m : array-like matrix_n : array-like Returns ------- matrix_v : array-like """ [matrix_d, _, matrix_r] = gs.linalg.svd(matrix_v[..., p:, p:]) matrix_v_final = gs.copy(matrix_v) for i in range(1, p + 1): matrix_rd = Matrices.mul(matrix_r, Matrices.transpose(matrix_d)) sub_matrix_v = gs.matmul(matrix_v[..., :, p:], matrix_rd) matrix_v_final = gs.concatenate( [gs.concatenate([matrix_m, matrix_n], axis=-2), sub_matrix_v], axis=-1 ) det = gs.linalg.det(matrix_v_final) if gs.all(det > 0): break ones = gs.ones(p) reflection_vec = gs.concatenate([ones[:-i], gs.array([-1.0] * i)], axis=0) mask = gs.cast(det < 0, matrix_v.dtype) sign = mask[..., None] * reflection_vec + (1.0 - mask)[..., None] * ones matrix_d = gs.einsum( "...ij,...i->...ij", Matrices.transpose(matrix_d), sign ) return matrix_v_final
[docs] def log(self, point, base_point, max_iter=30, tol=gs.atol, **kwargs): """Compute the Riemannian logarithm of a point. When p=n, the space St(n,n)~O(n) has two non connected sheets: the log is only defined for data from the same sheet. For p<n, the space St(n,p)~O(n)/O(n-p)~SO(n)/SO(n-p) is connected. Based on [ZR2017]_. References ---------- .. [ZR2017] Zimmermann, Ralf. "A Matrix-Algebraic Algorithm for the Riemannian Logarithm on the Stiefel Manifold under the Canonical Metric" SIAM J. Matrix Anal. & Appl., 38(2), 322–342, 2017. https://arxiv.org/pdf/1604.05054.pdf Parameters ---------- point : array-like, shape=[..., n, p] Point in the Stiefel manifold. base_point : array-like, shape=[..., n, p] Point in the Stiefel manifold. max_iter: int Maximum number of iterations to perform during the algorithm. Optional, default: 30. tol: float Tolerance to reach convergence. The matrix 2-norm is used as criterion. Optional, default: 1e-6. Returns ------- log : array-like, shape=[..., dim + 1] Tangent vector at the base point equal to the Riemannian logarithm of point at the base point. """ n, p = self.n, self.p if p == n: det_point = gs.linalg.det(point) det_base_point = gs.linalg.det(base_point) if not gs.all(det_point * det_base_point > 0.0): raise ValueError("Points from different sheets in log") transpose_base_point = Matrices.transpose(base_point) matrix_m = gs.matmul(transpose_base_point, point) matrix_q, matrix_n = self._normal_component_qr(point, base_point, matrix_m) matrix_v = self._orthogonal_completion(matrix_m, matrix_n) matrix_v = self._procrustes_preprocessing(p, matrix_v, matrix_m, matrix_n) matrix_v = gs.to_ndarray(matrix_v, to_ndim=3) result = [] for x in matrix_v: result.append(self._iter_log(p, x, max_iter, tol)) result = gs.stack(result) matrix_lv = ( result[0] if (point.ndim == 2) and (base_point.ndim == 2) else result ) matrix_xv = gs.matmul(base_point, matrix_lv[..., :p, :p]) matrix_qv = gs.matmul(matrix_q, matrix_lv[..., p:, :p]) return matrix_xv + matrix_qv
@staticmethod def _iter_log(p, matrix_v, max_iter, tol): matrix_lv = gs.zeros_like(matrix_v) for _ in range(max_iter): matrix_lv = gs.linalg.logm(matrix_v) matrix_c = matrix_lv[..., p:, p:] norm_matrix_c = gs.linalg.norm(matrix_c) if norm_matrix_c <= tol: break matrix_phi = gs.linalg.expm(-Matrices.to_skew_symmetric(matrix_c)) aux_matrix = gs.matmul(matrix_v[..., :, p:], matrix_phi) matrix_v = gs.concatenate([matrix_v[..., :, :p], aux_matrix], axis=-1) return matrix_lv
[docs] @staticmethod def retraction(tangent_vec, base_point): """Compute the retraction of a tangent vector. This computation is based on the QR-decomposition. e.g. :math:`P_x(V) = qf(X + V)`. Parameters ---------- tangent_vec : array-like, shape=[..., n, p] Tangent vector at a base point. base_point : array-like, shape=[..., n, p] Point in the Stiefel manifold. Returns ------- exp : array-like, shape=[..., n, p] Point in the Stiefel manifold equal to the retraction of tangent_vec at the base point. """ matrix_q, matrix_r = gs.linalg.qr(base_point + tangent_vec) diagonal = gs.diagonal(matrix_r, axis1=-2, axis2=-1) sign = gs.sign(gs.sign(diagonal) + 0.5) diag = algebra_utils.from_vector_to_diagonal_matrix(sign) result = Matrices.mul(matrix_q, diag) return result
@staticmethod @geomstats.vectorization.decorator(["matrix", "matrix"]) def lifting(point, base_point): """Compute the lifting of a point. This computation is based on the QR-decomposion. e.g. :math:`P_x^{-1}(Q) = QR - X`. Parameters ---------- point : array-like, shape=[..., n, p] Point in the Stiefel manifold. base_point : array-like, shape=[..., n, p] Point in the Stiefel manifold. Returns ------- log : array-like, shape=[..., dim + 1] Tangent vector at the base point equal to the lifting of point at the base point. """ n_points, _, _ = point.shape n_base_points, _, n = base_point.shape if not (n_points == n_base_points or n_points == 1 or n_base_points == 1): raise NotImplementedError n_liftings = gs.maximum(n_base_points, n_points) def _make_minor(i, matrix): return matrix[: i + 1, : i + 1] def _make_column_r(i, matrix): if i == 0: if matrix[0, 0] <= 0: raise ValueError("M[0,0] <= 0") return gs.array([1.0 / matrix[0, 0]]) matrix_m_i = _make_minor(i, matrix_m_k) inv_matrix_m_i = gs.linalg.inv(matrix_m_i) b_i = _make_b(i, matrix_m_k, columns_list) column_r_i = gs.matmul(inv_matrix_m_i, b_i) if column_r_i[i] <= 0: raise ValueError("(r_i)_i <= 0") return column_r_i def _make_b(i, matrix, list_matrices_r): b = gs.ones(i + 1) for j in range(i): b[j] = -gs.matmul(matrix[i, : j + 1], list_matrices_r[j]) return b matrix_r = gs.zeros((n_liftings, n, n)) matrix_m = gs.matmul(Matrices.transpose(base_point), point) for k in range(n_liftings): columns_list = [] matrix_m_k = matrix_m[k] for j in range(n): column_r_j = _make_column_r(j, matrix_m_k) columns_list.append(column_r_j) matrix_r[k, : len(column_r_j), j] = gs.array(column_r_j) return gs.matmul(point, matrix_r) - base_point
[docs] def injectivity_radius(self, base_point): """Compute the radius of the injectivity domain. This is is the supremum of radii r for which the exponential map is a diffeomorphism from the open ball of radius r centered at the base point onto its image. In this case the exact injectivity radius is not known, and we use here a lower bound given by [Rentmeesters2015]_. Parameters ---------- base_point : array-like, shape=[..., n, p] Point on the manifold. Returns ------- radius : float Injectivity radius. References ---------- .. [Rentmeesters2015] Rentmeesters, Quentin. “Algorithms for Data Fitting on Some Common Homogeneous Spaces.” UCL - Université Catholique de Louvain, 2013. https://dial.uclouvain.be/pr/boreal/object/boreal:132587. """ return 0.89 * gs.pi