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) angle = self.dist(base_point, point) angle = gs.to_ndarray(angle, to_ndim=1) angle = gs.to_ndarray(angle, to_ndim=2) mask_0 = gs.isclose(angle, 0) mask_else = ~mask_0 coef_1 = gs.zeros_like(angle) coef_2 = gs.zeros_like(angle) coef_1[mask_0] = (1. + INV_SINH_TAYLOR_COEFFS[1] * angle[mask_0]**2 + INV_SINH_TAYLOR_COEFFS[3] * angle[mask_0]**4 + INV_SINH_TAYLOR_COEFFS[5] * angle[mask_0]**6 + INV_SINH_TAYLOR_COEFFS[7] * angle[mask_0]**8) coef_2[mask_0] = (1. + INV_TANH_TAYLOR_COEFFS[1] * angle[mask_0]**2 + INV_TANH_TAYLOR_COEFFS[3] * angle[mask_0]**4 + INV_TANH_TAYLOR_COEFFS[5] * angle[mask_0]**6 + INV_TANH_TAYLOR_COEFFS[7] * angle[mask_0]**8) coef_1[mask_else] = angle[mask_else] / gs.sinh(angle[mask_else]) coef_2[mask_else] = angle[mask_else] / gs.tanh(angle[mask_else]) log = (gs.einsum('ni,nj->nj', coef_1, point) - gs.einsum('ni,nj->nj', coef_2, base_point)) return log
def exp(self, tangent_vec, base_point, **kwargs): """Compute the Riemannian exponential of a tangent vector. Parameters ---------- tangent_vec : array-like, shape=[..., dim] Tangent vector at a base point. base_point : array-like, shape=[..., dim] Point in the Poincare ball. Returns ------- exp : array-like, shape=[..., dim] Point in the Poincare ball equal to the Riemannian exponential of tangent_vec at the base point. """ squared_norm_bp = gs.sum(base_point**2, axis=-1) norm_tan = gs.linalg.norm(tangent_vec, axis=-1) lambda_base_point = 1 / (1 - squared_norm_bp) # This avoids dividing by 0 norm_tan_eps = gs.where(gs.isclose(norm_tan, 0.0), EPSILON, norm_tan) direction = gs.einsum("...i,...->...i", tangent_vec, 1 / norm_tan_eps) factor = gs.tanh(lambda_base_point * norm_tan) exp = self.mobius_add(base_point, gs.einsum("...,...i->...i", factor, direction)) return exp
def log(self, point, base_point): """ Riemannian logarithm of a point wrt a base point. Parameters ---------- point : array-like, shape=[n_samples, dimension + 1] or shape=[1, dimension + 1] base_point : array-like, shape=[n_samples, dimension + 1] or shape=[1, dimension + 1] Returns ------- log : array-like, shape=[n_samples, dimension + 1] or shape=[1, dimension + 1] """ point = gs.to_ndarray(point, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) angle = self.dist(base_point, point) angle = gs.to_ndarray(angle, to_ndim=1) angle = gs.to_ndarray(angle, to_ndim=2) mask_0 = gs.isclose(angle, 0.) mask_else = ~mask_0 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_SINH_TAYLOR_COEFFS[1] * angle ** 2 + INV_SINH_TAYLOR_COEFFS[3] * angle ** 4 + INV_SINH_TAYLOR_COEFFS[5] * angle ** 6 + INV_SINH_TAYLOR_COEFFS[7] * angle ** 8) coef_2 += mask_0_float * ( 1. + INV_TANH_TAYLOR_COEFFS[1] * angle ** 2 + INV_TANH_TAYLOR_COEFFS[3] * angle ** 4 + INV_TANH_TAYLOR_COEFFS[5] * angle ** 6 + INV_TANH_TAYLOR_COEFFS[7] * angle ** 8) # This avoids dividing by 0. angle += mask_0_float * 1. coef_1 += mask_else_float * (angle / gs.sinh(angle)) coef_2 += mask_else_float * (angle / gs.tanh(angle)) log = (gs.einsum('ni,nj->nj', coef_1, point) - gs.einsum('ni,nj->nj', coef_2, base_point)) return log
def log(self, point, base_point): """Compute Riemannian logarithm of a point wrt a base point. If point_type = 'poincare' then base_point belongs to the Poincare ball and point is a vector in the Euclidean space of the same dimension as the ball. Parameters ---------- point : array-like, shape=[..., dim + 1] Point in hyperbolic space. base_point : array-like, shape=[..., dim + 1] Point in hyperbolic space. Returns ------- log : array-like, shape=[..., dim + 1] Tangent vector at the base point equal to the Riemannian logarithm of point at the base point. """ angle = self.dist(base_point, point) / self.scale angle = gs.to_ndarray(angle, to_ndim=1) mask_0 = gs.isclose(angle, 0.) mask_else = ~mask_0 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_SINH_TAYLOR_COEFFS[1] * angle ** 2 + INV_SINH_TAYLOR_COEFFS[3] * angle ** 4 + INV_SINH_TAYLOR_COEFFS[5] * angle ** 6 + INV_SINH_TAYLOR_COEFFS[7] * angle ** 8) coef_2 += mask_0_float * ( 1. + INV_TANH_TAYLOR_COEFFS[1] * angle ** 2 + INV_TANH_TAYLOR_COEFFS[3] * angle ** 4 + INV_TANH_TAYLOR_COEFFS[5] * angle ** 6 + INV_TANH_TAYLOR_COEFFS[7] * angle ** 8) # This avoids dividing by 0. angle += mask_0_float * 1. coef_1 += mask_else_float * (angle / gs.sinh(angle)) coef_2 += mask_else_float * (angle / gs.tanh(angle)) log_term_1 = gs.einsum('...,...j->...j', coef_1, point) log_term_2 = - gs.einsum('...,...j->...j', coef_2, base_point) log = log_term_1 + log_term_2 return log
def exp(self, tangent_vec, base_point): """Compute the Riemannian exponential of a tangent vector. Parameters ---------- tangent_vec : array-like, shape=[n_samples, dim] Tangent vector at a base point. base_point : array-like, shape=[n_samples, dim] Point in hyperbolic space. Returns ------- exp : array-like, shape=[n_samples, dim] Point in hyperbolic space equal to the Riemannian exponential of tangent_vec at the base point. """ tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) norm_base_point =\ gs.expand_dims(gs.linalg.norm(base_point, axis=-1), axis=-1) den = 1 - norm_base_point**2 norm_tan =\ gs.expand_dims(gs.linalg.norm(tangent_vec, axis=-1), axis=-1) lambda_base_point = 1 / den zero_tan =\ gs.isclose(gs.sum(tangent_vec * tangent_vec, axis=-1), 0.) if gs.any(zero_tan): if norm_tan[zero_tan].shape[0] != 0: norm_tan[zero_tan] = EPSILON direction = gs.einsum('...i,...k->...i', tangent_vec, 1 / norm_tan) factor = gs.tanh(lambda_base_point * norm_tan) exp = self.mobius_add(base_point, direction * factor) zero_tan =\ gs.isclose(gs.sum(tangent_vec * tangent_vec, axis=-1), 0.) if gs.any(zero_tan): if exp[zero_tan].shape[0] != 0: exp[zero_tan] = base_point[zero_tan] return exp
def asymptotic_modulation(dim, theta): """Compute the asymptotic modulation factor. Parameters ---------- dim: dimension of the sphere (embedded in R^{dim+1}) theta: radius of the bubble distribution Returns ------- tuple (modulation factor, std-dev on the modulation factor) """ gamma = 1.0 / dim + (1.0 - 1.0 / dim) * theta / gs.tanh(theta) return (1.0 / gamma)**2
def exp(self, tangent_vec, base_point): """Compute the Riemannian exponential of a tangent vector. Parameters ---------- tangent_vec : array-like, shape=[..., dim] Tangent vector at a base point. base_point : array-like, shape=[..., dim] Point in hyperbolic space. Returns ------- exp : array-like, shape=[..., dim] Point in hyperbolic space equal to the Riemannian exponential of tangent_vec at the base point. """ norm_base_point = gs.linalg.norm(base_point, axis=-1) norm_tan = gs.linalg.norm(tangent_vec, axis=-1) den = 1 - norm_base_point ** 2 lambda_base_point = 1 / den zero_tan = gs.isclose(gs.sum(tangent_vec ** 2, axis=-1), 0.) if gs.any(zero_tan): norm_tan = gs.assignment(norm_tan, EPSILON, zero_tan) direction = gs.einsum('...i,...->...i', tangent_vec, 1 / norm_tan) factor = gs.tanh( gs.einsum('...,...->...', lambda_base_point, norm_tan)) exp = self.mobius_add( base_point, gs.einsum('...i,...->...i', direction, factor)) if gs.any(zero_tan): exp = gs.assignment( exp, base_point[zero_tan], zero_tan) 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 hyperbolic space in the Minkowski space. This gives a tangent vector at point base_point. :param base_point: point on the hyperbolic space :param point: point on the hyperbolic space :returns riem_log: tangent vector at base_point """ point = gs.to_ndarray(point, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) angle = self.dist(base_point, point) angle = gs.to_ndarray(angle, to_ndim=1) angle = gs.to_ndarray(angle, to_ndim=2) mask_0 = gs.isclose(angle, 0) mask_else = ~mask_0 coef_1 = gs.zeros_like(angle) coef_2 = gs.zeros_like(angle) coef_1[mask_0] = (1. + INV_SINH_TAYLOR_COEFFS[1] * angle[mask_0]**2 + INV_SINH_TAYLOR_COEFFS[3] * angle[mask_0]**4 + INV_SINH_TAYLOR_COEFFS[5] * angle[mask_0]**6 + INV_SINH_TAYLOR_COEFFS[7] * angle[mask_0]**8) coef_2[mask_0] = (1. + INV_TANH_TAYLOR_COEFFS[1] * angle[mask_0]**2 + INV_TANH_TAYLOR_COEFFS[3] * angle[mask_0]**4 + INV_TANH_TAYLOR_COEFFS[5] * angle[mask_0]**6 + INV_TANH_TAYLOR_COEFFS[7] * angle[mask_0]**8) coef_1[mask_else] = angle[mask_else] / gs.sinh(angle[mask_else]) coef_2[mask_else] = angle[mask_else] / gs.tanh(angle[mask_else]) log = (gs.einsum('ni,nj->nj', coef_1, point) - gs.einsum('ni,nj->nj', coef_2, base_point)) return log
} var_inv_tanc_close_0 = { "function": lambda x: (1 - (x / gs.tan(x))) / x ** 2, "coefficients": VAR_INV_TAN_TAYLOR_COEFFS, } sinch_close_0 = { "function": lambda x: gs.sinh(x) / x, "coefficients": SINHC_TAYLOR_COEFFS, } cosh_close_0 = {"function": gs.cosh, "coefficients": COSH_TAYLOR_COEFFS} inv_sinch_close_0 = { "function": lambda x: x / gs.sinh(x), "coefficients": INV_SINHC_TAYLOR_COEFFS, } inv_tanh_close_0 = { "function": lambda x: x / gs.tanh(x), "coefficients": INV_TANH_TAYLOR_COEFFS, } arctanh_card_close_0 = { "function": lambda x: gs.arctanh(x) / x, "coefficients": ARCTANH_CARD_TAYLOR_COEFFS, } def from_vector_to_diagonal_matrix(vector, num_diag=0): """Create diagonal matrices from rows of a matrix. Parameters ---------- vector : array-like, shape=[m, n] num_diag : int
} var_inv_tanc_close_0 = { 'function': lambda x: (1 - (x / gs.tan(x))) / x**2, 'coefficients': VAR_INV_TAN_TAYLOR_COEFFS } sinch_close_0 = { 'function': lambda x: gs.sinh(x) / x, 'coefficients': SINHC_TAYLOR_COEFFS } cosh_close_0 = {'function': gs.cosh, 'coefficients': COSH_TAYLOR_COEFFS} inv_sinch_close_0 = { 'function': lambda x: x / gs.sinh(x), 'coefficients': INV_SINHC_TAYLOR_COEFFS } inv_tanh_close_0 = { 'function': lambda x: x / gs.tanh(x), 'coefficients': INV_TANH_TAYLOR_COEFFS } arctanh_card_close_0 = { 'function': lambda x: gs.arctanh(x) / x, 'coefficients': ARCTANH_CARD_TAYLOR_COEFFS } def from_vector_to_diagonal_matrix(vector): """Create diagonal matrices from rows of a matrix. Parameters ---------- vector : array-like, shape=[m, n]
def log(self, point, base_point): """Compute Riemannian logarithm of a point wrt a base point. If point_type = 'poincare' then base_point belongs to the Poincare ball and point is a vector in the Euclidean space of the same dimension as the ball. Parameters ---------- point : array-like, shape=[n_samples, dimension + 1] Point in hyperbolic space. base_point : array-like, shape=[n_samples, dimension + 1] Point in hyperbolic space. Returns ------- log : array-like, shape=[n_samples, dimension + 1] Tangent vector at the base point equal to the Riemannian logarithm of point at the base point. """ if self.point_type == 'extrinsic': point = gs.to_ndarray(point, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) angle = self.dist(base_point, point) / self.scale angle = gs.to_ndarray(angle, to_ndim=1) angle = gs.to_ndarray(angle, to_ndim=2) mask_0 = gs.isclose(angle, 0.) mask_else = ~mask_0 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_SINH_TAYLOR_COEFFS[1] * angle**2 + INV_SINH_TAYLOR_COEFFS[3] * angle**4 + INV_SINH_TAYLOR_COEFFS[5] * angle**6 + INV_SINH_TAYLOR_COEFFS[7] * angle**8) coef_2 += mask_0_float * (1. + INV_TANH_TAYLOR_COEFFS[1] * angle**2 + INV_TANH_TAYLOR_COEFFS[3] * angle**4 + INV_TANH_TAYLOR_COEFFS[5] * angle**6 + INV_TANH_TAYLOR_COEFFS[7] * angle**8) # This avoids dividing by 0. angle += mask_0_float * 1. coef_1 += mask_else_float * (angle / gs.sinh(angle)) coef_2 += mask_else_float * (angle / gs.tanh(angle)) log = (gs.einsum('ni,nj->nj', coef_1, point) - gs.einsum('ni,nj->nj', coef_2, base_point)) return log elif self.point_type == 'ball': add_base_point = self.mobius_add(-base_point, point) norm_add = gs.to_ndarray(gs.linalg.norm(add_base_point, axis=-1), 2, -1) norm_add = gs.repeat(norm_add, base_point.shape[-1], -1) norm_base_point = gs.to_ndarray( gs.linalg.norm(base_point, axis=-1), 2, -1) norm_base_point = gs.repeat(norm_base_point, base_point.shape[-1], -1) log = (1 - norm_base_point**2) * gs.arctanh(norm_add)\ * (add_base_point / norm_add) mask_0 = gs.all(gs.isclose(norm_add, 0.)) log[mask_0] = 0 return log else: raise NotImplementedError( 'log is only implemented for ball and extrinsic')
def exp(self, tangent_vec, base_point): """Compute the Riemannian exponential of a tangent vector. Parameters ---------- tangent_vec : array-like, shape=[n_samples, dimension + 1] Tangent vector at a base point. base_point : array-like, shape=[n_samples, dimension + 1] Point in hyperbolic space. Returns ------- exp : array-like, shape=[n_samples, dimension + 1] Point in hyperbolic space equal to the Riemannian exponential of tangent_vec at the base point. """ if self.point_type == 'extrinsic': tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2) base_point = gs.to_ndarray(base_point, to_ndim=2) sq_norm_tangent_vec = self.embedding_metric.squared_norm( tangent_vec) sq_norm_tangent_vec = gs.clip(sq_norm_tangent_vec, 0, math.inf) norm_tangent_vec = gs.sqrt(sq_norm_tangent_vec) mask_0 = gs.isclose(sq_norm_tangent_vec, 0.) mask_0 = gs.to_ndarray(mask_0, to_ndim=1) mask_else = ~mask_0 mask_else = gs.to_ndarray(mask_else, to_ndim=1) mask_0_float = gs.cast(mask_0, gs.float32) mask_else_float = gs.cast(mask_else, gs.float32) coef_1 = gs.zeros_like(norm_tangent_vec) coef_2 = gs.zeros_like(norm_tangent_vec) coef_1 += mask_0_float * ( 1. + COSH_TAYLOR_COEFFS[2] * norm_tangent_vec**2 + COSH_TAYLOR_COEFFS[4] * norm_tangent_vec**4 + COSH_TAYLOR_COEFFS[6] * norm_tangent_vec**6 + COSH_TAYLOR_COEFFS[8] * norm_tangent_vec**8) coef_2 += mask_0_float * ( 1. + SINH_TAYLOR_COEFFS[3] * norm_tangent_vec**2 + SINH_TAYLOR_COEFFS[5] * norm_tangent_vec**4 + SINH_TAYLOR_COEFFS[7] * norm_tangent_vec**6 + SINH_TAYLOR_COEFFS[9] * norm_tangent_vec**8) # This avoids dividing by 0. norm_tangent_vec += mask_0_float * 1.0 coef_1 += mask_else_float * (gs.cosh(norm_tangent_vec)) coef_2 += mask_else_float * ((gs.sinh(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)) hyperbolic_space = Hyperbolic(dimension=self.dimension) exp = hyperbolic_space.regularize(exp) return exp elif self.point_type == 'ball': norm_base_point = gs.to_ndarray(gs.linalg.norm(base_point, -1), 2, -1) norm_base_point = gs.repeat(norm_base_point, base_point.shape[-1], -1) den = 1 - norm_base_point**2 norm_tan = gs.to_ndarray(gs.linalg.norm(tangent_vec, axis=-1), 2, -1) norm_tan = gs.repeat(norm_tan, base_point.shape[-1], -1) lambda_base_point = 1 / den direction = tangent_vec / norm_tan factor = gs.tanh(lambda_base_point * norm_tan) exp = self.mobius_add(base_point, direction * factor) return exp else: raise NotImplementedError( 'exp is only implemented for ball and extrinsic')