# Developer Guide¶

## Base Manifold¶

The common base class for all manifolds is geoopt.manifolds.base.Manifold.

class geoopt.manifolds.base.Manifold(**kwargs)[source]
_assert_check_shape(shape: Tuple[int], name: str)[source]

Util to check shape and raise an error if needed.

Exhaustive implementation for checking if a given point has valid dimension size, shape, etc. It will raise a ValueError if check is not passed

Parameters: shape (tuple) – shape of point on the manifold name (str) – name to be present in errors ValueError
_check_point_on_manifold(x: torch.Tensor, *, atol=1e-05, rtol=1e-05) → Union[Tuple[bool, Optional[str]], bool][source]

Util to check point lies on the manifold.

Exhaustive implementation for checking if a given point lies on the manifold. It should return boolean and a reason of failure if check is not passed. You can assume assert_check_point is already passed beforehand

Parameters: torch.Tensor (x) – point on the manifold atol (float) – absolute tolerance as in numpy.allclose() rtol (float) – relative tolerance as in numpy.allclose() check result and the reason of fail if any bool, str or None
_check_shape(shape: Tuple[int], name: str) → Union[Tuple[bool, Optional[str]], bool][source]

Util to check shape.

Exhaustive implementation for checking if a given point has valid dimension size, shape, etc. It should return boolean and a reason of failure if check is not passed

Parameters: shape (Tuple[int]) – shape of point on the manifold name (str) – name to be present in errors check result and the reason of fail if any bool, str or None
_check_vector_on_tangent(x: torch.Tensor, u: torch.Tensor, *, atol=1e-05, rtol=1e-05) → Union[Tuple[bool, Optional[str]], bool][source]

Util to check a vector belongs to the tangent space of a point.

Exhaustive implementation for checking if a given point lies in the tangent space at x of the manifold. It should return a boolean indicating whether the test was passed and a reason of failure if check is not passed. You can assume assert_check_point is already passed beforehand

Parameters: torch.Tensor (u) – torch.Tensor – atol (float) – absolute tolerance rtol – relative tolerance check result and the reason of fail if any bool, str or None
assert_check_point(x: torch.Tensor)[source]

Check if point is valid to be used with the manifold and raise an error with informative message on failure.

Parameters: x (torch.Tensor) – point on the manifold

Notes

This check is compatible to what optimizer expects, last dimensions are treated as manifold dimensions

assert_check_point_on_manifold(x: torch.Tensor, *, atol=1e-05, rtol=1e-05)[source]

Check if point :mathx is lying on the manifold and raise an error with informative message on failure.

Parameters: x (torch.Tensor) – point on the manifold atol (float) – absolute tolerance as in numpy.allclose() rtol (float) – relative tolerance as in numpy.allclose()
assert_check_vector(u: torch.Tensor)[source]

Check if vector is valid to be used with the manifold and raise an error with informative message on failure.

Parameters: u (torch.Tensor) – vector on the tangent plane

Notes

This check is compatible to what optimizer expects, last dimensions are treated as manifold dimensions

assert_check_vector_on_tangent(x: torch.Tensor, u: torch.Tensor, *, ok_point=False, atol=1e-05, rtol=1e-05)[source]

Check if u $$u$$ is lying on the tangent space to x and raise an error on fail.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – vector on the tangent space to $$x$$ atol (float) – absolute tolerance as in numpy.allclose() rtol (float) – relative tolerance as in numpy.allclose() ok_point (bool) – is a check for point required?
check_point(x: torch.Tensor, *, explain=False) → Union[Tuple[bool, Optional[str]], bool][source]

Check if point is valid to be used with the manifold.

Parameters: x (torch.Tensor) – point on the manifold explain (bool) – return an additional information on check boolean indicating if tensor is valid and reason of failure if False bool

Notes

This check is compatible to what optimizer expects, last dimensions are treated as manifold dimensions

check_point_on_manifold(x: torch.Tensor, *, explain=False, atol=1e-05, rtol=1e-05) → Union[Tuple[bool, Optional[str]], bool][source]

Check if point $$x$$ is lying on the manifold.

Parameters: x (torch.Tensor) – point on the manifold atol (float) – absolute tolerance as in numpy.allclose() rtol (float) – relative tolerance as in numpy.allclose() explain (bool) – return an additional information on check boolean indicating if tensor is valid and reason of failure if False bool

Notes

This check is compatible to what optimizer expects, last dimensions are treated as manifold dimensions

check_vector(u: torch.Tensor, *, explain=False)[source]

Check if vector is valid to be used with the manifold.

Parameters: u (torch.Tensor) – vector on the tangent plane explain (bool) – return an additional information on check boolean indicating if tensor is valid and reason of failure if False bool

Notes

This check is compatible to what optimizer expects, last dimensions are treated as manifold dimensions

check_vector_on_tangent(x: torch.Tensor, u: torch.Tensor, *, ok_point=False, explain=False, atol=1e-05, rtol=1e-05) → Union[Tuple[bool, Optional[str]], bool][source]

Check if $$u$$ is lying on the tangent space to x.

Parameters: x (torch.Tensor) – point on the manifold u (torch.Tensor) – vector on the tangent space to $$x$$ atol (float) – absolute tolerance as in numpy.allclose() rtol (float) – relative tolerance as in numpy.allclose() explain (bool) – return an additional information on check ok_point (bool) – is a check for point required? boolean indicating if tensor is valid and reason of failure if False bool
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.

device

Manifold device.

Returns: Optional[torch.device]
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
dtype

Manifold dtype.

Returns: Optional[torch.dtype]
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
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
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) → 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: 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
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
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
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(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) → 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.base.ScalingStorage[source]

Helper class to make implementation transparent.

This is just a dictionary with additional overriden __call__ for more explicit and elegant API to declare members. A usage example may be found in Manifold.

Methods that require rescaling when wrapped into Scaled should be defined as follows

1. Regular methods like dist, dist2, expmap, retr etc. that are already present in the base class do not require registration, it has already happened in the base Manifold class.

1. New methods (like in PoincareBall) should be treated with care.
class PoincareBall(Manifold):
# make a class copy of __scaling__ info. Default methods are already present there
__scaling__ = Manifold.__scaling__.copy()
... # here come regular implementation of the required methods

@__scaling__(ScalingInfo(1))  # rescale output according to rule out * scaling ** 1
def dist0(self, x: torch.Tensor, *, dim=-1, keepdim=False):
return math.dist0(x, c=self.c, dim=dim, keepdim=keepdim)

@__scaling__(ScalingInfo(u=-1))  # rescale argument u according to the rule out * scaling ** -1
def expmap0(self, u: torch.Tensor, *, dim=-1, project=True):
res = math.expmap0(u, c=self.c, dim=dim)
if project:
return math.project(res, c=self.c, dim=dim)
else:
return res
... # other special methods implementation

1. Some methods are not compliant with the above rescaling rules. We should mark them as NotCompatible
# continuation of the PoincareBall definition
@__scaling__(ScalingInfo.NotCompatible)
def mobius_fn_apply(
self, fn: callable, x: torch.Tensor, *args, dim=-1, project=True, **kwargs
):
res = math.mobius_fn_apply(fn, x, *args, c=self.c, dim=dim, **kwargs)
if project:
return math.project(res, c=self.c, dim=dim)
else:
return res

copy() → a shallow copy of D[source]
class geoopt.manifolds.base.ScalingInfo(*results, **kwargs)[source]

Scaling info for each argument that requires rescaling.

scaled_value = value * scaling ** power if power != 0 else value


For results it is not always required to set powers of scaling, then it is no-op.

The convention for this info is the following. The output of a function is either a tuple or a single object. In any case, outputs are treated as positionals. Function inputs, in contrast, are treated by keywords. It is a common practice to maintain function signature when overriding, so this way may be considered as a sufficient in this particular scenario. The only required info for formula above is power.