# Manifolds¶

All manifolds share same API. Some manifols may have several implementations of retraction operation, every implementation has a corresponding class.

class geoopt.manifolds.Euclidean(ndim=0)[source]

Simple Euclidean manifold, every coordinate is treated as an independent element.

Parameters: ndim (int) – number of trailing dimensions treated as manifold dimensions. All the operations acting on cuch as inner products, etc will respect the ndim.
component_inner(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$ according to components of the manifold.

The result of the function is same as inner with keepdim=True for all the manifolds except ProductManifold. For this manifold it acts different way computing inner product for each component and then building an output correctly tiling and reshaping the result.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ inner product component wise (broadcasted) torch.Tensor

Notes

The purpose of this method is better adaptive properties in optimization since ProductManifold will “hide” the structure in public API.

dist(x: torch.Tensor, y: torch.Tensor, *, keepdim=False) → torch.Tensor[source]

Compute distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool) – keep the last dim? distance between two points torch.Tensor
dist2(x: torch.Tensor, y: torch.Tensor, *, keepdim=False) → torch.Tensor[source]

Compute squared distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool) – keep the last dim? squared distance between two points torch.Tensor
egrad2rgrad(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Transform gradient computed using autodiff to the correct Riemannian gradient for the point $$x$$.

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – gradient to be projected grad vector in the Riemannian manifold torch.Tensor
expmap(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
extra_repr()[source]

Set the extra representation of the module

To print customized extra information, you should re-implement this method in your own modules. Both single-line and multi-line strings are acceptable.

inner(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None, *, keepdim=False) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
logmap(x: torch.Tensor, y: torch.Tensor) → torch.Tensor[source]

Perform an logarithmic map $$\operatorname{Log}_{x}(y)$$.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold tangent vector torch.Tensor
norm(x: torch.Tensor, u: torch.Tensor, *, keepdim=False)[source]

Norm of a tangent vector at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
origin(*size, dtype=None, device=None, seed=42) → geoopt.tensor.ManifoldTensor[source]

Zero point origin.

Parameters: size (shape) – the desired shape device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype seed (int) – ignored ManifoldTensor
proju(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
projx(x: torch.Tensor) → torch.Tensor[source]

Project point $$x$$ on the manifold.

Parameters: torch.Tensor (x) – point to be projected projected point torch.Tensor
random(*size, mean=0.0, std=1.0, device=None, dtype=None) → geoopt.tensor.ManifoldTensor

Create a point on the manifold, measure is induced by Normal distribution.

Parameters: size (shape) – the desired shape mean (float|tensor) – mean value for the Normal distribution std (float|tensor) – std value for the Normal distribution device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype random point on the manifold ManifoldTensor
random_normal(*size, mean=0.0, std=1.0, device=None, dtype=None) → geoopt.tensor.ManifoldTensor[source]

Create a point on the manifold, measure is induced by Normal distribution.

Parameters: size (shape) – the desired shape mean (float|tensor) – mean value for the Normal distribution std (float|tensor) – std value for the Normal distribution device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype random point on the manifold ManifoldTensor
retr(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform a retraction from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
transp(x: torch.Tensor, y: torch.Tensor, v: torch.Tensor) → torch.Tensor[source]

Perform vector transport $$\mathfrak{T}_{x\to y}(v)$$.

Parameters: x (torch.Tensor) – start point on the manifold y (torch.Tensor) – target point on the manifold v (torch.Tensor) – tangent vector at point $$x$$ transported tensor torch.Tensor
class geoopt.manifolds.Stiefel(**kwargs)[source]

Manifold induced by the following matrix constraint:

$\begin{split}X^\top X = I\\ X \in \mathrm{R}^{n\times m}\\ n \ge m\end{split}$
Parameters: canonical (bool) – Use canonical inner product instead of euclidean one (defaults to canonical)
origin(*size, dtype=None, device=None, seed=42) → torch.Tensor[source]

Identity matrix point origin.

Parameters: size (shape) – the desired shape device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype seed (int) – ignored ManifoldTensor
projx(x: torch.Tensor) → torch.Tensor[source]

Project point $$x$$ on the manifold.

Parameters: torch.Tensor (x) – point to be projected projected point torch.Tensor
random(*size, dtype=None, device=None) → torch.Tensor

Naive approach to get random matrix on Stiefel manifold.

A helper function to sample a random point on the Stiefel manifold. The measure is non-uniform for this method, but fast to compute.

Parameters: size (shape) – the desired output shape dtype (torch.dtype) – desired dtype device (torch.device) – desired device random point on Stiefel manifold ManifoldTensor
random_naive(*size, dtype=None, device=None) → torch.Tensor[source]

Naive approach to get random matrix on Stiefel manifold.

A helper function to sample a random point on the Stiefel manifold. The measure is non-uniform for this method, but fast to compute.

Parameters: size (shape) – the desired output shape dtype (torch.dtype) – desired dtype device (torch.device) – desired device random point on Stiefel manifold ManifoldTensor
class geoopt.manifolds.CanonicalStiefel(**kwargs)[source]

Stiefel Manifold with Canonical inner product

Manifold induced by the following matrix constraint:

$\begin{split}X^\top X = I\\ X \in \mathrm{R}^{n\times m}\\ n \ge m\end{split}$
egrad2rgrad(x: torch.Tensor, u: torch.Tensor) → torch.Tensor

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
expmap(x: torch.Tensor, u: torch.Tensor) → torch.Tensor

Perform a retraction from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
expmap_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → Tuple[torch.Tensor, torch.Tensor]

Perform a retraction + vector transport at once.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point and vectors Tuple[torch.Tensor, torch.Tensor]

Notes

Sometimes this is a far more optimal way to preform retraction + vector transport

inner(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None, *, keepdim=False) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
proju(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
retr(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform a retraction from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
retr_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → Tuple[torch.Tensor, torch.Tensor][source]

Perform a retraction + vector transport at once.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point and vectors Tuple[torch.Tensor, torch.Tensor]

Notes

Sometimes this is a far more optimal way to preform retraction + vector transport

transp_follow_expmap(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → torch.Tensor

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{retr}(x, u)}(v)$$.

This operation is sometimes is much more simpler and can be optimized.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
transp_follow_retr(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → torch.Tensor[source]

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{retr}(x, u)}(v)$$.

This operation is sometimes is much more simpler and can be optimized.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
class geoopt.manifolds.EuclideanStiefel(**kwargs)[source]

Stiefel Manifold with Euclidean inner product

Manifold induced by the following matrix constraint:

$\begin{split}X^\top X = I\\ X \in \mathrm{R}^{n\times m}\\ n \ge m\end{split}$
egrad2rgrad(x: torch.Tensor, u: torch.Tensor) → torch.Tensor

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
expmap(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
inner(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None, *, keepdim=False) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
proju(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
retr(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform a retraction from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
transp(x: torch.Tensor, y: torch.Tensor, v: torch.Tensor) → torch.Tensor[source]

Perform vector transport $$\mathfrak{T}_{x\to y}(v)$$.

Parameters: x (torch.Tensor) – start point on the manifold y (torch.Tensor) – target point on the manifold v (torch.Tensor) – tangent vector at point $$x$$ transported tensor torch.Tensor
class geoopt.manifolds.EuclideanStiefelExact(**kwargs)[source]

Stiefel Manifold with Euclidean inner product

Manifold induced by the following matrix constraint:

$\begin{split}X^\top X = I\\ X \in \mathrm{R}^{n\times m}\\ n \ge m\end{split}$

Notes

The implementation of retraction is an exact exponential map, this retraction will be used in optimization

extra_repr()[source]

Set the extra representation of the module

To print customized extra information, you should re-implement this method in your own modules. Both single-line and multi-line strings are acceptable.

retr(x: torch.Tensor, u: torch.Tensor) → torch.Tensor

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
retr_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → Tuple[torch.Tensor, torch.Tensor]

Perform an exponential map and vector transport from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point torch.Tensor
transp_follow_retr(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → torch.Tensor

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{Exp}(x, u)}(v)$$.

Here, $$\operatorname{Exp}$$ is the best possible approximation of the true exponential map. There are cases when the exact variant is hard or impossible implement, therefore a fallback, non-exact, implementation is used.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
class geoopt.manifolds.Sphere(intersection: torch.Tensor = None, complement: torch.Tensor = None)[source]

Sphere manifold induced by the following constraint

$\begin{split}\|x\|=1\\ x \in \mathbb{span}(U)\end{split}$

where $$U$$ can be parametrized with compliment space or intersection.

Parameters: intersection (tensor) – shape (..., dim, K), subspace to intersect with complement (tensor) – shape (..., dim, K), subspace to compliment
dist(x: torch.Tensor, y: torch.Tensor, *, keepdim=False) → torch.Tensor[source]

Compute distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool) – keep the last dim? distance between two points torch.Tensor
egrad2rgrad(x: torch.Tensor, u: torch.Tensor) → torch.Tensor

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
expmap(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
inner(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None, *, keepdim=False) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
logmap(x: torch.Tensor, y: torch.Tensor) → torch.Tensor[source]

Perform an logarithmic map $$\operatorname{Log}_{x}(y)$$.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold tangent vector torch.Tensor
proju(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
projx(x: torch.Tensor) → torch.Tensor[source]

Project point $$x$$ on the manifold.

Parameters: torch.Tensor (x) – point to be projected projected point torch.Tensor
random(*size, dtype=None, device=None) → torch.Tensor

Uniform random measure on Sphere manifold.

Parameters: size (shape) – the desired output shape dtype (torch.dtype) – desired dtype device (torch.device) – desired device random point on Sphere manifold ManifoldTensor

Notes

In case of projector on the manifold, dtype and device are set automatically and shouldn’t be provided. If you provide them, they are checked to match the projector device and dtype

random_uniform(*size, dtype=None, device=None) → torch.Tensor[source]

Uniform random measure on Sphere manifold.

Parameters: size (shape) – the desired output shape dtype (torch.dtype) – desired dtype device (torch.device) – desired device random point on Sphere manifold ManifoldTensor

Notes

In case of projector on the manifold, dtype and device are set automatically and shouldn’t be provided. If you provide them, they are checked to match the projector device and dtype

retr(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform a retraction from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
transp(x: torch.Tensor, y: torch.Tensor, v: torch.Tensor) → torch.Tensor[source]

Perform vector transport $$\mathfrak{T}_{x\to y}(v)$$.

Parameters: x (torch.Tensor) – start point on the manifold y (torch.Tensor) – target point on the manifold v (torch.Tensor) – tangent vector at point $$x$$ transported tensor torch.Tensor
class geoopt.manifolds.SphereExact(intersection: torch.Tensor = None, complement: torch.Tensor = None)[source]

Sphere manifold induced by the following constraint

$\begin{split}\|x\|=1\\ x \in \mathbb{span}(U)\end{split}$

where $$U$$ can be parametrized with compliment space or intersection.

Parameters: intersection (tensor) – shape (..., dim, K), subspace to intersect with complement (tensor) – shape (..., dim, K), subspace to compliment

Notes

The implementation of retraction is an exact exponential map, this retraction will be used in optimization

extra_repr()[source]

Set the extra representation of the module

To print customized extra information, you should re-implement this method in your own modules. Both single-line and multi-line strings are acceptable.

retr(x: torch.Tensor, u: torch.Tensor) → torch.Tensor

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
retr_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → Tuple[torch.Tensor, torch.Tensor]

Perform an exponential map and vector transport from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point torch.Tensor
transp_follow_retr(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → torch.Tensor

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{Exp}(x, u)}(v)$$.

Here, $$\operatorname{Exp}$$ is the best possible approximation of the true exponential map. There are cases when the exact variant is hard or impossible implement, therefore a fallback, non-exact, implementation is used.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
class geoopt.manifolds.Stereographic(k=0.0, learnable=False)[source]

$$\kappa$$-Stereographic model.

Parameters: k (float|tensor) – sectional curvature $$\kappa$$ of the manifold - k<0: Poincaré ball (stereographic projection of hyperboloid) - k>0: Stereographic projection of sphere - k=0: Euclidean geometry

Notes

It is extremely recommended to work with this manifold in double precision.

References

The functions for the mathematics in gyrovector spaces are taken from the following resources:

 Ganea, Octavian, Gary Bécigneul, and Thomas Hofmann. “Hyperbolic
neural networks.” Advances in neural information processing systems. 2018.
 Bachmann, Gregor, Gary Bécigneul, and Octavian-Eugen Ganea. “Constant
Curvature Graph Convolutional Networks.” arXiv preprint arXiv:1911.05076 (2019).
 Skopek, Ondrej, Octavian-Eugen Ganea, and Gary Bécigneul.
“Mixed-curvature Variational Autoencoders.” arXiv preprint arXiv:1911.08411 (2019).
 Ungar, Abraham A. Analytic hyperbolic geometry: Mathematical
foundations and applications. World Scientific, 2005.
 Albert, Ungar Abraham. Barycentric calculus in Euclidean and
hyperbolic geometry: A comparative introduction. World Scientific, 2010.
dist(x: torch.Tensor, y: torch.Tensor, *, keepdim=False, dim=-1) → torch.Tensor[source]

Compute distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool) – keep the last dim? distance between two points torch.Tensor
dist2(x: torch.Tensor, y: torch.Tensor, *, keepdim=False, dim=-1) → torch.Tensor[source]

Compute squared distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool) – keep the last dim? squared distance between two points torch.Tensor
egrad2rgrad(x: torch.Tensor, u: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Transform gradient computed using autodiff to the correct Riemannian gradient for the point $$x$$.

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – gradient to be projected grad vector in the Riemannian manifold torch.Tensor
expmap(x: torch.Tensor, u: torch.Tensor, *, project=True, dim=-1) → torch.Tensor[source]

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
expmap_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor, *, dim=-1, project=True) → Tuple[torch.Tensor, torch.Tensor][source]

Perform an exponential map and vector transport from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point torch.Tensor
inner(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None, *, keepdim=False, dim=-1) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
logmap(x: torch.Tensor, y: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Perform an logarithmic map $$\operatorname{Log}_{x}(y)$$.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold tangent vector torch.Tensor
norm(x: torch.Tensor, u: torch.Tensor, *, keepdim=False, dim=-1) → torch.Tensor[source]

Norm of a tangent vector at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
origin(*size, dtype=None, device=None, seed=42) → geoopt.tensor.ManifoldTensor[source]

Zero point origin.

Parameters: size (shape) – the desired shape device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype seed (int) – ignored random point on the manifold ManifoldTensor
proju(x: torch.Tensor, u: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
projx(x: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Project point $$x$$ on the manifold.

Parameters: torch.Tensor (x) – point to be projected projected point torch.Tensor
random(*size, mean=0, std=1, dtype=None, device=None) → geoopt.tensor.ManifoldTensor

Create a point on the manifold, measure is induced by Normal distribution on the tangent space of zero.

Parameters: size (shape) – the desired shape mean (float|tensor) – mean value for the Normal distribution std (float|tensor) – std value for the Normal distribution dtype (torch.dtype) – target dtype for sample, if not None, should match Manifold dtype device (torch.device) – target device for sample, if not None, should match Manifold device random point on the PoincareBall manifold ManifoldTensor

Notes

The device and dtype will match the device and dtype of the Manifold

random_normal(*size, mean=0, std=1, dtype=None, device=None) → geoopt.tensor.ManifoldTensor[source]

Create a point on the manifold, measure is induced by Normal distribution on the tangent space of zero.

Parameters: size (shape) – the desired shape mean (float|tensor) – mean value for the Normal distribution std (float|tensor) – std value for the Normal distribution dtype (torch.dtype) – target dtype for sample, if not None, should match Manifold dtype device (torch.device) – target device for sample, if not None, should match Manifold device random point on the PoincareBall manifold ManifoldTensor

Notes

The device and dtype will match the device and dtype of the Manifold

retr(x: torch.Tensor, u: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Perform a retraction from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
retr_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor, *, dim=-1) → Tuple[torch.Tensor, torch.Tensor][source]

Perform a retraction + vector transport at once.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point and vectors Tuple[torch.Tensor, torch.Tensor]

Notes

Sometimes this is a far more optimal way to preform retraction + vector transport

transp(x: torch.Tensor, y: torch.Tensor, v: torch.Tensor, *, dim=-1)[source]

Perform vector transport $$\mathfrak{T}_{x\to y}(v)$$.

Parameters: x (torch.Tensor) – start point on the manifold y (torch.Tensor) – target point on the manifold v (torch.Tensor) – tangent vector at point $$x$$ transported tensor torch.Tensor
transp_follow_expmap(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor, *, dim=-1, project=True) → torch.Tensor[source]

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{Exp}(x, u)}(v)$$.

Here, $$\operatorname{Exp}$$ is the best possible approximation of the true exponential map. There are cases when the exact variant is hard or impossible implement, therefore a fallback, non-exact, implementation is used.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
transp_follow_retr(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{retr}(x, u)}(v)$$.

This operation is sometimes is much more simpler and can be optimized.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
wrapped_normal(*size, mean: torch.Tensor, std=1, dtype=None, device=None) → geoopt.tensor.ManifoldTensor[source]

Create a point on the manifold, measure is induced by Normal distribution on the tangent space of mean.

Definition is taken from  Mathieu, Emile et. al. “Continuous Hierarchical Representations with Poincaré Variational Auto-Encoders.” arXiv preprint arxiv:1901.06033 (2019).

Parameters: size (shape) – the desired shape mean (float|tensor) – mean value for the Normal distribution std (float|tensor) – std value for the Normal distribution dtype (torch.dtype) – target dtype for sample, if not None, should match Manifold dtype device (torch.device) – target device for sample, if not None, should match Manifold device random point on the PoincareBall manifold ManifoldTensor

Notes

The device and dtype will match the device and dtype of the Manifold

class geoopt.manifolds.StereographicExact(k=0.0, learnable=False)[source]

$$\kappa$$-Stereographic model.

Parameters: k (float|tensor) – sectional curvature $$\kappa$$ of the manifold - k<0: Poincaré ball (stereographic projection of hyperboloid) - k>0: Stereographic projection of sphere - k=0: Euclidean geometry

Notes

It is extremely recommended to work with this manifold in double precision.

The implementation of retraction is an exact exponential map, this retraction will be used in optimization.

extra_repr()[source]

Set the extra representation of the module

To print customized extra information, you should re-implement this method in your own modules. Both single-line and multi-line strings are acceptable.

retr(x: torch.Tensor, u: torch.Tensor, *, project=True, dim=-1) → torch.Tensor

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
retr_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor, *, dim=-1, project=True) → Tuple[torch.Tensor, torch.Tensor]

Perform an exponential map and vector transport from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point torch.Tensor
transp_follow_retr(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor, *, dim=-1, project=True) → torch.Tensor

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{Exp}(x, u)}(v)$$.

Here, $$\operatorname{Exp}$$ is the best possible approximation of the true exponential map. There are cases when the exact variant is hard or impossible implement, therefore a fallback, non-exact, implementation is used.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
class geoopt.manifolds.PoincareBall(c=1.0, learnable=False)[source]

Poincare ball model.

See more in \kappa-Stereographic Projection model

Parameters: c (float|tensor) – ball’s negative curvature. The parametrization is constrained to have positive c

Notes

It is extremely recommended to work with this manifold in double precision

class geoopt.manifolds.PoincareBallExact(c=1.0, learnable=False)[source]

Poincare ball model.

See more in \kappa-Stereographic Projection model

Parameters: c (float|tensor) – ball’s negative curvature. The parametrization is constrained to have positive c

Notes

It is extremely recommended to work with this manifold in double precision

The implementation of retraction is an exact exponential map, this retraction will be used in optimization.

class geoopt.manifolds.SphereProjection(k=1.0, learnable=False)[source]

Stereographic Projection Spherical model.

See more in \kappa-Stereographic Projection model

Parameters: k (float|tensor) – sphere’s positive curvature. The parametrization is constrained to have positive k

Notes

It is extremely recommended to work with this manifold in double precision

class geoopt.manifolds.SphereProjectionExact(k=1.0, learnable=False)[source]

Stereographic Projection Spherical model.

See more in \kappa-Stereographic Projection model

Parameters: k (float|tensor) – sphere’s positive curvature. The parametrization is constrained to have positive k

Notes

It is extremely recommended to work with this manifold in double precision

The implementation of retraction is an exact exponential map, this retraction will be used in optimization.

class geoopt.manifolds.Scaled(manifold: geoopt.manifolds.base.Manifold, scale=1.0, learnable=False)[source]

Scaled manifold.

Scales all the distances on tha manifold by a constant factor. Scaling may be learnable since the underlying representation is canonical.

Examples

Here is a simple example of radius 2 Sphere

>>> import geoopt, torch, numpy as np
>>> sphere = geoopt.Sphere()
>>> radius_2_sphere = Scaled(sphere, 2)
>>> p1 = torch.tensor([-1., 0.])
>>> p2 = torch.tensor([0., 1.])
>>> np.testing.assert_allclose(sphere.dist(p1, p2), np.pi / 2)
>>> np.testing.assert_allclose(radius_2_sphere.dist(p1, p2), np.pi)

egrad2rgrad(x: torch.Tensor, u: torch.Tensor, **kwargs) → torch.Tensor[source]

Transform gradient computed using autodiff to the correct Riemannian gradient for the point $$x$$.

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – gradient to be projected grad vector in the Riemannian manifold torch.Tensor
inner(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None, *, keepdim=False, **kwargs) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
norm(x: torch.Tensor, u: torch.Tensor, *, keepdim=False, **kwargs) → torch.Tensor[source]

Norm of a tangent vector at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
proju(x: torch.Tensor, u: torch.Tensor, **kwargs) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
projx(x: torch.Tensor, **kwargs) → torch.Tensor[source]

Project point $$x$$ on the manifold.

Parameters: torch.Tensor (x) – point to be projected projected point torch.Tensor
random(*size, dtype=None, device=None, **kwargs) → torch.Tensor[source]

Random sampling on the manifold.

The exact implementation depends on manifold and usually does not follow all assumptions about uniform measure, etc.

transp(x: torch.Tensor, y: torch.Tensor, v: torch.Tensor, **kwargs) → torch.Tensor[source]

Perform vector transport $$\mathfrak{T}_{x\to y}(v)$$.

Parameters: x (torch.Tensor) – start point on the manifold y (torch.Tensor) – target point on the manifold v (torch.Tensor) – tangent vector at point $$x$$ transported tensor torch.Tensor
class geoopt.manifolds.ProductManifold(*manifolds_with_shape)[source]

Product Manifold.

Examples

A Torus

>>> import geoopt
>>> sphere = geoopt.Sphere()
>>> torus = ProductManifold((sphere, 2), (sphere, 2))

component_inner(x: torch.Tensor, u: torch.Tensor, v=None) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$ according to components of the manifold.

The result of the function is same as inner with keepdim=True for all the manifolds except ProductManifold. For this manifold it acts different way computing inner product for each component and then building an output correctly tiling and reshaping the result.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ inner product component wise (broadcasted) torch.Tensor

Notes

The purpose of this method is better adaptive properties in optimization since ProductManifold will “hide” the structure in public API.

dist(x, y, *, keepdim=False)[source]

Compute distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool) – keep the last dim? distance between two points torch.Tensor
dist2(x: torch.Tensor, y: torch.Tensor, *, keepdim=False)[source]

Compute squared distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool) – keep the last dim? squared distance between two points torch.Tensor
egrad2rgrad(x: torch.Tensor, u: torch.Tensor)[source]

Transform gradient computed using autodiff to the correct Riemannian gradient for the point $$x$$.

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – gradient to be projected grad vector in the Riemannian manifold torch.Tensor
expmap(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
expmap_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → Tuple[torch.Tensor, torch.Tensor][source]

Perform an exponential map and vector transport from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point torch.Tensor
classmethod from_point(*parts, batch_dims=0)[source]

Construct Product manifold from given points.

Parameters: parts (tuple[geoopt.ManifoldTensor]) – Manifold tensors to construct Product manifold from batch_dims (int) – number of first dims to treat as batch dims and not include in the Product manifold ProductManifold
inner(x: torch.Tensor, u: torch.Tensor, v=None, *, keepdim=False) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
logmap(x: torch.Tensor, y: torch.Tensor) → torch.Tensor[source]

Perform an logarithmic map $$\operatorname{Log}_{x}(y)$$.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold tangent vector torch.Tensor
origin(*size, dtype=None, device=None, seed=42) → geoopt.tensor.ManifoldTensor[source]

Create some reasonable point on the manifold in a deterministic way.

For some manifolds there may exist e.g. zero vector or some analogy. In case it is possible to define this special point, this point is returned with the desired size. In other case, the returned point is sampled on the manifold in a deterministic way.

Parameters: size (Union[int, Tuple[int]]) – the desired shape device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype seed (Optional[int]) – A parameter controlling deterministic randomness for manifolds that do not provide .origin, but provide .random. (default: 42) torch.Tensor
pack_point(*tensors) → torch.Tensor[source]

Construct a tensor representation of a manifold point.

In case of regular manifolds this will return the same tensor. However, for e.g. Product manifold this function will pack all non-batch dimensions.

Parameters: tensors (Tuple[torch.Tensor]) – torch.Tensor
proju(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
projx(x: torch.Tensor) → torch.Tensor[source]

Project point $$x$$ on the manifold.

Parameters: torch.Tensor (x) – point to be projected projected point torch.Tensor
retr(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform a retraction from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
retr_transp(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor)[source]

Perform a retraction + vector transport at once.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported point and vectors Tuple[torch.Tensor, torch.Tensor]

Notes

Sometimes this is a far more optimal way to preform retraction + vector transport

take_submanifold_value(x: torch.Tensor, i: int, reshape=True) → torch.Tensor[source]

Take i’th slice of the ambient tensor and possibly reshape.

Parameters: x (tensor) – Ambient tensor i (int) – submanifold index reshape (bool) – reshape the slice? torch.Tensor
transp(x: torch.Tensor, y: torch.Tensor, v: torch.Tensor) → torch.Tensor[source]

Perform vector transport $$\mathfrak{T}_{x\to y}(v)$$.

Parameters: x (torch.Tensor) – start point on the manifold y (torch.Tensor) – target point on the manifold v (torch.Tensor) – tangent vector at point $$x$$ transported tensor torch.Tensor
transp_follow_expmap(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → torch.Tensor[source]

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{Exp}(x, u)}(v)$$.

Here, $$\operatorname{Exp}$$ is the best possible approximation of the true exponential map. There are cases when the exact variant is hard or impossible implement, therefore a fallback, non-exact, implementation is used.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
transp_follow_retr(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor) → torch.Tensor[source]

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{retr}(x, u)}(v)$$.

This operation is sometimes is much more simpler and can be optimized.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
unpack_tensor(tensor: torch.Tensor) → Tuple[torch.Tensor][source]

Construct a point on the manifold.

This method should help to work with product and compound manifolds. Internally all points on the manifold are stored in an intuitive format. However, there might be cases, when this representation is simpler or more efficient to store in a different way that is hard to use in practice.

Parameters: tensor (torch.Tensor) – torch.Tensor
class geoopt.manifolds.Lorentz(k=1.0, learnable=False)[source]

Lorentz model

Parameters: k (float|tensor) – manifold negative curvature

Notes

It is extremely recommended to work with this manifold in double precision

dist(x: torch.Tensor, y: torch.Tensor, *, keepdim=False, dim=-1) → torch.Tensor[source]

Compute distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool) – keep the last dim? distance between two points torch.Tensor
egrad2rgrad(x: torch.Tensor, u: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Transform gradient computed using autodiff to the correct Riemannian gradient for the point $$x$$.

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – gradient to be projected grad vector in the Riemannian manifold torch.Tensor
expmap(x: torch.Tensor, u: torch.Tensor, *, norm_tan=True, project=True, dim=-1) → torch.Tensor[source]

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
inner(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None, *, keepdim=False, dim=-1) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
logmap(x: torch.Tensor, y: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Perform an logarithmic map $$\operatorname{Log}_{x}(y)$$.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold tangent vector torch.Tensor
norm(u: torch.Tensor, *, keepdim=False, dim=-1) → torch.Tensor[source]

Norm of a tangent vector at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
origin(*size, dtype=None, device=None, seed=42) → geoopt.tensor.ManifoldTensor[source]

Zero point origin.

Parameters: size (shape) – the desired shape device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype seed (int) – ignored zero point on the manifold ManifoldTensor
proju(x: torch.Tensor, v: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
projx(x: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Project point $$x$$ on the manifold.

Parameters: torch.Tensor (x) – point to be projected projected point torch.Tensor
random_normal(*size, mean=0, std=1, dtype=None, device=None) → geoopt.tensor.ManifoldTensor[source]

Create a point on the manifold, measure is induced by Normal distribution on the tangent space of zero.

Parameters: size (shape) – the desired shape mean (float|tensor) – mean value for the Normal distribution std (float|tensor) – std value for the Normal distribution dtype (torch.dtype) – target dtype for sample, if not None, should match Manifold dtype device (torch.device) – target device for sample, if not None, should match Manifold device random points on Hyperboloid ManifoldTensor

Notes

The device and dtype will match the device and dtype of the Manifold

retr(x: torch.Tensor, u: torch.Tensor, *, norm_tan=True, project=True, dim=-1) → torch.Tensor

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
transp(x: torch.Tensor, y: torch.Tensor, v: torch.Tensor, *, dim=-1) → torch.Tensor[source]

Perform vector transport $$\mathfrak{T}_{x\to y}(v)$$.

Parameters: x (torch.Tensor) – start point on the manifold y (torch.Tensor) – target point on the manifold v (torch.Tensor) – tangent vector at point $$x$$ transported tensor torch.Tensor
transp_follow_expmap(x: torch.Tensor, u: torch.Tensor, v: torch.Tensor, *, dim=-1, project=True) → torch.Tensor[source]

Perform vector transport following $$u$$: $$\mathfrak{T}_{x\to\operatorname{Exp}(x, u)}(v)$$.

Here, $$\operatorname{Exp}$$ is the best possible approximation of the true exponential map. There are cases when the exact variant is hard or impossible implement, therefore a fallback, non-exact, implementation is used.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (torch.Tensor) – tangent vector at point $$x$$ to be transported transported tensor torch.Tensor
class geoopt.manifolds.SymmetricPositiveDefinite(default_metric: Union[str, geoopt.manifolds.symmetric_positive_definite.SPDMetric] = 'AIM')[source]

Manifold of symmetric positive definite matrices.

$\begin{split}A = A^T\\ \langle x, A x \rangle > 0 \quad , \forall x \in \mathrm{R}^{n}, x \neq 0 \\ A \in \mathrm{R}^{n\times m}\end{split}$

The tangent space of the manifold contains all symmetric matrices.

References

Parameters: default_metric (Union[str, SPDMetric]) – one of AIM, SM, LEM. So far only AIM is fully implemented.
dist(x: torch.Tensor, y: torch.Tensor, keepdim=False) → torch.Tensor[source]

Compute distance between 2 points on the manifold that is the shortest path along geodesics.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold keepdim (bool, optional) – keep the last dim?, by default False distance between two points torch.Tensor ValueError – if mode isn’t in _dist_metric
egrad2rgrad(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Transform gradient computed using autodiff to the correct Riemannian gradient for the point $$x$$.

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – gradient to be projected grad vector in the Riemannian manifold torch.Tensor
expmap(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform an exponential map $$\operatorname{Exp}_x(u)$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
extra_repr() → str[source]

Set the extra representation of the module

To print customized extra information, you should re-implement this method in your own modules. Both single-line and multi-line strings are acceptable.

inner(x: torch.Tensor, u: torch.Tensor, v: Optional[torch.Tensor] = None, keepdim=False) → torch.Tensor[source]

Inner product for tangent vectors at point $$x$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ v (Optional[torch.Tensor]) – tangent vector at point $$x$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor ValueError – if keepdim sine torch.trace doesn’t support keepdim
logmap(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform an logarithmic map $$\operatorname{Log}_{x}(y)$$.

Parameters: x (torch.Tensor) – point on the manifold y (torch.Tensor) – point on the manifold tangent vector torch.Tensor
origin(*size, dtype=None, device=None, seed: Optional[int] = 42) → torch.Tensor[source]

Create some reasonable point on the manifold in a deterministic way.

For some manifolds there may exist e.g. zero vector or some analogy. In case it is possible to define this special point, this point is returned with the desired size. In other case, the returned point is sampled on the manifold in a deterministic way.

Parameters: size (Union[int, Tuple[int]]) – the desired shape device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype seed (Optional[int]) – A parameter controlling deterministic randomness for manifolds that do not provide .origin, but provide .random. (default: 42) torch.Tensor
proju(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Project vector $$u$$ on a tangent space for $$x$$, usually is the same as egrad2rgrad().

Parameters: torch.Tensor (u) – point on the manifold torch.Tensor – vector to be projected projected vector torch.Tensor
projx(x: torch.Tensor) → torch.Tensor[source]

Project point $$x$$ on the manifold.

Parameters: torch.Tensor (x) – point to be projected projected point torch.Tensor
random(*size, dtype=None, device=None, **kwargs) → torch.Tensor[source]

Random sampling on the manifold.

The exact implementation depends on manifold and usually does not follow all assumptions about uniform measure, etc.

retr(x: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Perform a retraction from point $$x$$ with given direction $$u$$.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$x$$ transported point torch.Tensor
transp(x: torch.Tensor, y: torch.Tensor, v: torch.Tensor) → torch.Tensor[source]

Perform vector transport $$\mathfrak{T}_{x\to y}(v)$$.

Parameters: x (torch.Tensor) – start point on the manifold y (torch.Tensor) – target point on the manifold v (torch.Tensor) – tangent vector at point $$x$$ transported tensor torch.Tensor
class geoopt.manifolds.UpperHalf(metric: geoopt.manifolds.siegel.vvd_metrics.SiegelMetricType = <SiegelMetricType.RIEMANNIAN: 'riem'>, rank: int = None)[source]

Upper Half Space Manifold.

This model generalizes the upper half plane model of the hyperbolic plane. Points in the space are complex symmetric matrices.

$\mathcal{S}_n = \{Z = X + iY \in \operatorname{Sym}(n, \mathbb{C}) | Y >> 0 \}.$
Parameters: metric (SiegelMetricType) – one of Riemannian, Finsler One, Finsler Infinity, Finsler metric of minimum entropy, or learnable weighted sum. rank (int) – Rank of the space. Only mandatory for “fmin” and “wsum” metrics.
egrad2rgrad(z: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Transform gradient computed using autodiff to the correct Riemannian gradient for the point $$Z$$.

For a function $$f(Z)$$ on $$\mathcal{S}_n$$, the gradient is:

$\operatorname{grad}_{R}(f(Z)) = Y \cdot \operatorname{grad}_E(f(Z)) \cdot Y$

where $$Y$$ is the imaginary part of $$Z$$.

Parameters: z (torch.Tensor) – point on the manifold u (torch.Tensor) – gradient to be projected Riemannian gradient torch.Tensor
inner(z: torch.Tensor, u: torch.Tensor, v=None, *, keepdim=False) → torch.Tensor[source]

Inner product for tangent vectors at point $$Z$$.

The inner product at point $$Z = X + iY$$ of the vectors $$U, V$$ is:

$g_{Z}(U, V) = \operatorname{Tr}[ Y^{-1} U Y^{-1} \overline{V} ]$
Parameters: z (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$z$$ v (torch.Tensor) – tangent vector at point $$z$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
origin(*size, dtype=None, device=None, seed: Optional[int] = 42) → torch.Tensor[source]

Create points at the origin of the manifold in a deterministic way.

For the Upper half model, the origin is the imaginary identity. This is, a matrix whose real part is all zeros, and the identity as the imaginary part.

Parameters: size (Union[int, Tuple[int]]) – the desired shape device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype seed (Optional[int]) – A parameter controlling deterministic randomness for manifolds that do not provide .origin, but provide .random. (default: 42) torch.Tensor
projx(z: torch.Tensor) → torch.Tensor[source]

Project point $$Z$$ on the manifold.

In this space, we need to ensure that $$Y = Im(Z)$$ is positive definite. Since the matrix Y is symmetric, it is possible to diagonalize it. For a diagonal matrix the condition is just that all diagonal entries are positive, so we clamp the values that are <= 0 in the diagonal to an epsilon, and then restore the matrix back into non-diagonal form using the base change matrix that was obtained from the diagonalization.

Parameters: z (torch.Tensor) – point on the manifold Projected points torch.Tensor
random(*size, dtype=None, device=None, **kwargs) → torch.Tensor[source]

Random sampling on the manifold.

The exact implementation depends on manifold and usually does not follow all assumptions about uniform measure, etc.

class geoopt.manifolds.BoundedDomain(metric: geoopt.manifolds.siegel.vvd_metrics.SiegelMetricType = <SiegelMetricType.RIEMANNIAN: 'riem'>, rank: int = None)[source]

Bounded domain Manifold.

This model generalizes the Poincare ball model. Points in the space are complex symmetric matrices.

$\mathcal{B}_n := \{ Z \in \operatorname{Sym}(n, \mathbb{C}) | Id - Z^*Z >> 0 \}$
Parameters: metric (SiegelMetricType) – one of Riemannian, Finsler One, Finsler Infinity, Finsler metric of minimum entropy, or learnable weighted sum. rank (int) – Rank of the space. Only mandatory for “fmin” and “wsum” metrics.
dist(z1: torch.Tensor, z2: torch.Tensor, *, keepdim=False) → torch.Tensor[source]

Compute distance in the Bounded domain model.

To compute distances in the Bounded Domain Model we need to map the elements to the Upper Half Space Model by means of the Cayley Transform, and then compute distances in that domain.

Parameters: z1 (torch.Tensor) – point on the manifold z2 (torch.Tensor) – point on the manifold keepdim (bool, optional) – keep the last dim?, by default False distance between two points torch.Tensor
egrad2rgrad(z: torch.Tensor, u: torch.Tensor) → torch.Tensor[source]

Transform gradient computed using autodiff to the correct Riemannian gradient for the point $$Z$$.

For a function $$f(Z)$$ on $$\mathcal{B}_n$$, the gradient is:

$\operatorname{grad}_{R}(f(Z)) = A \cdot \operatorname{grad}_E(f(Z)) \cdot A$

where $$A = Id - \overline{Z}Z$$

Parameters: z (torch.Tensor) – point on the manifold u (torch.Tensor) – gradient to be projected Riemannian gradient torch.Tensor
inner(z: torch.Tensor, u: torch.Tensor, v=None, *, keepdim=False) → torch.Tensor[source]

Inner product for tangent vectors at point $$Z$$.

The inner product at point $$Z = X + iY$$ of the vectors $$U, V$$ is:

$g_{Z}(U, V) = \operatorname{Tr}[(Id - \overline{Z}Z)^{-1} U (Id - Z\overline{Z})^{-1} \overline{V}]$
Parameters: z (torch.Tensor) – point on the manifold u (torch.Tensor) – tangent vector at point $$z$$ v (torch.Tensor) – tangent vector at point $$z$$ keepdim (bool) – keep the last dim? inner product (broadcasted) torch.Tensor
origin(*size, dtype=None, device=None, seed: Optional[int] = 42) → torch.Tensor[source]

Create points at the origin of the manifold in a deterministic way.

For the Bounded domain model, the origin is the zero matrix. This is, a matrix whose real and imaginary parts are all zeros.

Parameters: size (Union[int, Tuple[int]]) – the desired shape device (torch.device) – the desired device dtype (torch.dtype) – the desired dtype seed (Optional[int]) – A parameter controlling deterministic randomness for manifolds that do not provide .origin, but provide .random. (default: 42) torch.Tensor
projx(z: torch.Tensor) → torch.Tensor[source]

Project point $$Z$$ on the manifold.

In the Bounded domain model, we need to ensure that $$Id - \overline(Z)Z$$ is positive definite.

Steps to project: Z complex symmetric matrix 1) Diagonalize Z: $$Z = \overline{S} D S^*$$ 2) Clamp eigenvalues: $$D' = clamp(D, max=1 - epsilon)$$ 3) Rebuild Z: $$Z' = \overline{S} D' S^*$$

Parameters: z (torch.Tensor) – point on the manifold Projected points torch.Tensor
random(*size, dtype=None, device=None, **kwargs) → torch.Tensor[source]

Random sampling on the manifold.

The exact implementation depends on manifold and usually does not follow all assumptions about uniform measure, etc.