manifolds

Package Contents

Euclidean

Euclidean Manifold class. Usually we refer it as R^n.

PoincareBall

PoicareBall Manifold class.
class Euclidean(**kwargs)

Bases: topmost.models.hierarchical.HyperMiner.manifolds.base.Manifold

Euclidean Manifold class. Usually we refer it as R^n.

name

The manifold name, and its value is “Euclidean”.

Type:

str

Initialize an Euclidean manifold. :param **kwargs: Description

name = 'Euclidean'
proj(x, c)

A projection function that prevents x from leaving the manifold. :param x: A point should be on the manifold, but it may not meet the manifold constraints. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

A projected point, meeting the manifold constraints.

Return type:

tensor

proj_tan(v, x, c)

A projection function that prevents v from leaving the tangent space of point x. :param v: A point should be on the tangent space, but it may not meet the manifold constraints. :type v: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

A projected point, meeting the tangent space constraints.

Return type:

tensor

proj_tan0(v, c)

A projection function that prevents v from leaving the tangent space of origin point. :param v: A point should be on the tangent space, but it may not meet the manifold constraints. :type v: tensor :param c: The manifold curvature. :type c: tensor

Returns:

A projected point, meeting the tangent space constraints.

Return type:

tensor

expmap(v, x, c)

Map a point v in the tangent space of point x to the manifold. :param v: A point in the tangent space of point x. :type v: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping tangent point v to the manifold.

Return type:

tensor

expmap0(v, c)

Map a point v in the tangent space of origin point to the manifold. :param v: A point in the tangent space of origin point. :type v: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping tangent point v to the manifold.

Return type:

tensor

logmap(y, x, c)

Map a point y on the manifold to the tangent space of x. :param y: A point on the manifold. :type y: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping y to the tangent space of x.

Return type:

tensor

logmap0(y, c)

Map a point y on the manifold to the tangent space of origin point. :param y: A point on the manifold. :type y: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping y to the tangent space of origin point.

Return type:

tensor

ptransp(v, x, y, c)

Parallel transport function, used to move point v in the tangent space of x to the tangent space of y. :param v: A point in the tangent space of x. :type v: tensor :param x: A point on the manifold. :type x: tensor :param y: A point on the manifold. :type y: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of transporting v from the tangent space at x to the tangent space at y.

Return type:

tensor

ptransp0(v, x, c)

Parallel transport function, used to move point v in the tangent space of origin point to the tangent space of y. :param v: A point in the tangent space of origin point. :type v: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of transporting v from the tangent space at origin point to the tangent space at y.

Return type:

tensor

dist(x, y, c)

Calculate the squared geodesic/distance between x and y. :param x: A point on the manifold. :type x: tensor :param y: A point on the manifold. :type y: tensor :param c: The manifold curvature. :type c: tensor

Returns:

the geodesic/distance between x and y.

Return type:

tensor

egrad2rgrad(grad, x, c)

Computes Riemannian gradient from the Euclidean gradient, typically used in Riemannian optimizers. :param grad: Euclidean gradient at x. :type grad: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

Riemannian gradient at x.

Return type:

tensor

inner(v1, v2, x, c, keep_shape=False)

Computes the inner product of a pair of tangent vectors v1 and v2 at x. :param v1: A tangent point at x. :type v1: tensor :param v2: A tangent point at x. :type v2: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor :param keep_shape: Whether the output tensor keeps shape or not. :type keep_shape: bool, optional

Returns:

The inner product of v1 and v2 at x.

Return type:

tensor

class PoincareBall

Bases: topmost.models.hierarchical.HyperMiner.manifolds.base.Manifold

PoicareBall Manifold class.

We use the following convention: x0^2 + x1^2 + … + xd^2 < 1 / c. (c < 0)

So that the Poincare ball radius will be 1 / sqrt(-c). Notice that the more close c is to 0, the more flat space will be.

Initialize a manifold.

name = 'PoincareBall'
truncate_c
proj(x, c)

A projection function that prevents x from leaving the manifold. :param x: A point should be on the manifold, but it may not meet the manifold constraints. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

A projected point, meeting the manifold constraints.

Return type:

tensor

proj_tan(v, x, c)

A projection function that prevents v from leaving the tangent space of point x. :param v: A point should be on the tangent space, but it may not meet the manifold constraints. :type v: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

A projected point, meeting the tangent space constraints.

Return type:

tensor

proj_tan0(v, c)

A projection function that prevents v from leaving the tangent space of origin point. :param v: A point should be on the tangent space, but it may not meet the manifold constraints. :type v: tensor :param c: The manifold curvature. :type c: tensor

Returns:

A projected point, meeting the tangent space constraints.

Return type:

tensor

expmap(v, x, c)

Map a point v in the tangent space of point x to the manifold. :param v: A point in the tangent space of point x. :type v: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping tangent point v to the manifold.

Return type:

tensor

expmap0(v, c)

Map a point v in the tangent space of origin point to the manifold. :param v: A point in the tangent space of origin point. :type v: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping tangent point v to the manifold.

Return type:

tensor

logmap(y, x, c)

Map a point y on the manifold to the tangent space of x. :param y: A point on the manifold. :type y: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping y to the tangent space of x.

Return type:

tensor

logmap0(y, c)

Map a point y on the manifold to the tangent space of origin point. :param y: A point on the manifold. :type y: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping y to the tangent space of origin point.

Return type:

tensor

ptransp(v, x, y, c)

Parallel transport function, used to move point v in the tangent space of x to the tangent space of y. :param v: A point in the tangent space of x. :type v: tensor :param x: A point on the manifold. :type x: tensor :param y: A point on the manifold. :type y: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of transporting v from the tangent space at x to the tangent space at y.

Return type:

tensor

ptransp0(v, x, c)

Parallel transport function, used to move point v in the tangent space of origin point to the tangent space of y. :param v: A point in the tangent space of origin point. :type v: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of transporting v from the tangent space at origin point to the tangent space at y.

Return type:

tensor

dist(x, y, c)

Calculate the squared geodesic/distance between x and y. :param x: A point on the manifold. :type x: tensor :param y: A point on the manifold. :type y: tensor :param c: The manifold curvature. :type c: tensor

Returns:

the geodesic/distance between x and y.

Return type:

tensor

egrad2rgrad(grad, x, c)

Computes Riemannian gradient from the Euclidean gradient, typically used in Riemannian optimizers. :param grad: Euclidean gradient at x. :type grad: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

Riemannian gradient at x.

Return type:

tensor

inner(v1, v2, x, c, keep_shape=False)

Computes the inner product of a pair of tangent vectors v1 and v2 at x. :param v1: A tangent point at x. :type v1: tensor :param v2: A tangent point at x. :type v2: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor :param keep_shape: Whether the output tensor keeps shape or not. :type keep_shape: bool, optional

Returns:

The inner product of v1 and v2 at x.

Return type:

tensor

retraction(v, x, c)

Retraction is a continuous map function from tangent space to the manifold, typically used in Riemannian optimizers. The exp map is one of retraction functions. :param v: A tangent point at x. :type v: tensor :param x: A point on the manifold. :type x: tensor :param c: The manifold curvature. :type c: tensor

Returns:

The result of mapping tangent point v at x to the manifold.

Return type:

tensor

_mobius_add(x, y, c)
_mobius_mul(x, a, c)
_mobius_matvec(x, a, c)
_lambda_x(x, c)
_gyration(x, y, v, c)