import abc
import torch.nn
import itertools
from typing import Optional, Tuple, Union
__all__ = ["Manifold", "ScalingInfo"]
[docs]class ScalingInfo(object):
"""
Scaling info for each argument that requires rescaling.
.. code:: python
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``.
"""
# marks method to be not working with Scaled wrapper
NotCompatible = object()
__slots__ = ["kwargs", "results"]
def __init__(self, *results: float, **kwargs: float):
self.results = results
self.kwargs = kwargs
[docs]class ScalingStorage(dict):
"""
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 :class:`Manifold`.
Methods that require rescaling when wrapped into :class:`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 :class:`Manifold` class.
2. New methods (like in :class:`PoincareBall`) should be treated with care.
.. code-block:: python
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
3. Some methods are not compliant with the above rescaling rules. We should mark them as `NotCompatible`
.. code-block:: python
# 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
"""
def __call__(self, scaling_info: ScalingInfo, *aliases):
def register(fn):
self[fn.__name__] = scaling_info
for alias in aliases:
self[alias] = scaling_info
return fn
return register
[docs] def copy(self):
return self.__class__(self)
[docs]class Manifold(torch.nn.Module, metaclass=abc.ABCMeta):
__scaling__ = ScalingStorage() # will be filled along with implementation below
name = None
ndim = None
reversible = None
forward = NotImplemented
def __init__(self, **kwargs):
super().__init__()
@property
def device(self) -> Optional[torch.device]:
"""
Manifold device.
Returns
-------
Optional[torch.device]
"""
p = next(itertools.chain(self.buffers(), self.parameters()), None)
if p is not None:
return p.device
else:
return None
@property
def dtype(self) -> Optional[torch.dtype]:
"""
Manifold dtype.
Returns
-------
Optional[torch.dtype]
"""
p = next(itertools.chain(self.buffers(), self.parameters()), None)
if p is not None:
return p.dtype
else:
return None
[docs] def check_point(
self, x: torch.Tensor, *, explain=False
) -> Union[Tuple[bool, Optional[str]], bool]:
"""
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
Returns
-------
bool
boolean indicating if tensor is valid and reason of failure if False
Notes
-----
This check is compatible to what optimizer expects, last dimensions are treated as manifold dimensions
"""
ok, reason = self._check_shape(x.shape, "x")
if explain:
return ok, reason
else:
return ok
[docs] def assert_check_point(self, x: torch.Tensor):
"""
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
"""
ok, reason = self._check_shape(x.shape, "x")
if not ok:
raise ValueError(
"`x` seems to be not valid "
"tensor for {} manifold.\nerror: {}".format(self.name, reason)
)
[docs] def check_vector(self, u: torch.Tensor, *, explain=False):
"""
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
Returns
-------
bool
boolean indicating if tensor is valid and reason of failure if False
Notes
-----
This check is compatible to what optimizer expects, last dimensions are treated as manifold dimensions
"""
ok, reason = self._check_shape(u.shape, "u")
if explain:
return ok, reason
else:
return ok
[docs] def assert_check_vector(self, u: torch.Tensor):
"""
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
"""
ok, reason = self._check_shape(u.shape, "u")
if not ok:
raise ValueError(
"`u` seems to be not valid "
"tensor for {} manifold.\nerror: {}".format(self.name, reason)
)
[docs] def check_point_on_manifold(
self, x: torch.Tensor, *, explain=False, atol=1e-5, rtol=1e-5
) -> Union[Tuple[bool, Optional[str]], bool]:
"""
Check if point :math:`x` is lying on the manifold.
Parameters
----------
x : torch.Tensor
point on the manifold
atol: float
absolute tolerance as in :func:`numpy.allclose`
rtol: float
relative tolerance as in :func:`numpy.allclose`
explain: bool
return an additional information on check
Returns
-------
bool
boolean indicating if tensor is valid and reason of failure if False
Notes
-----
This check is compatible to what optimizer expects, last dimensions are treated as manifold dimensions
"""
ok, reason = self._check_shape(x.shape, "x")
if ok:
ok, reason = self._check_point_on_manifold(x, atol=atol, rtol=rtol)
if explain:
return ok, reason
else:
return ok
[docs] def assert_check_point_on_manifold(self, x: torch.Tensor, *, atol=1e-5, rtol=1e-5):
"""
Check if point :math`x` 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 :func:`numpy.allclose`
rtol: float
relative tolerance as in :func:`numpy.allclose`
"""
self.assert_check_point(x)
ok, reason = self._check_point_on_manifold(x, atol=atol, rtol=rtol)
if not ok:
raise ValueError(
"`x` seems to be a tensor "
"not lying on {} manifold.\nerror: {}".format(self.name, reason)
)
[docs] def check_vector_on_tangent(
self,
x: torch.Tensor,
u: torch.Tensor,
*,
ok_point=False,
explain=False,
atol=1e-5,
rtol=1e-5,
) -> Union[Tuple[bool, Optional[str]], bool]:
"""
Check if :math:`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 :math:`x`
atol: float
absolute tolerance as in :func:`numpy.allclose`
rtol: float
relative tolerance as in :func:`numpy.allclose`
explain: bool
return an additional information on check
ok_point: bool
is a check for point required?
Returns
-------
bool
boolean indicating if tensor is valid and reason of failure if False
"""
if not ok_point:
ok, reason = self._check_shape(x.shape, "x")
if ok:
ok, reason = self._check_shape(u.shape, "u")
if ok:
ok, reason = self._check_point_on_manifold(x, atol=atol, rtol=rtol)
else:
ok = True
reason = None
if ok:
ok, reason = self._check_vector_on_tangent(x, u, atol=atol, rtol=rtol)
if explain:
return ok, reason
else:
return ok
[docs] def assert_check_vector_on_tangent(
self, x: torch.Tensor, u: torch.Tensor, *, ok_point=False, atol=1e-5, rtol=1e-5
):
"""
Check if u :math:`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 :math:`x`
atol: float
absolute tolerance as in :func:`numpy.allclose`
rtol: float
relative tolerance as in :func:`numpy.allclose`
ok_point: bool
is a check for point required?
"""
if not ok_point:
ok, reason = self._check_shape(x.shape, "x")
if ok:
ok, reason = self._check_shape(u.shape, "u")
if ok:
ok, reason = self._check_point_on_manifold(x, atol=atol, rtol=rtol)
else:
ok = True
reason = None
if ok:
ok, reason = self._check_vector_on_tangent(x, u, atol=atol, rtol=rtol)
if not ok:
raise ValueError(
"`u` seems to be a tensor "
"not lying on tangent space to `x` for {} manifold.\nerror: {}".format(
self.name, reason
)
)
[docs] @__scaling__(ScalingInfo(1))
def dist(self, x: torch.Tensor, y: torch.Tensor, *, keepdim=False) -> torch.Tensor:
"""
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?
Returns
-------
torch.Tensor
distance between two points
"""
raise NotImplementedError
[docs] @__scaling__(ScalingInfo(2))
def dist2(self, x: torch.Tensor, y: torch.Tensor, *, keepdim=False) -> torch.Tensor:
"""
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?
Returns
-------
torch.Tensor
squared distance between two points
"""
return self.dist(x, y, keepdim=keepdim) ** 2
[docs] @abc.abstractmethod
@__scaling__(ScalingInfo(u=-1))
def retr(self, x: torch.Tensor, u: torch.Tensor) -> torch.Tensor:
"""
Perform a retraction from point :math:`x` with given direction :math:`u`.
Parameters
----------
x : torch.Tensor
point on the manifold
u : torch.Tensor
tangent vector at point :math:`x`
Returns
-------
torch.Tensor
transported point
"""
raise NotImplementedError
[docs] @abc.abstractmethod
@__scaling__(ScalingInfo(u=-1))
def expmap(self, x: torch.Tensor, u: torch.Tensor) -> torch.Tensor:
r"""
Perform an exponential map :math:`\operatorname{Exp}_x(u)`.
Parameters
----------
x : torch.Tensor
point on the manifold
u : torch.Tensor
tangent vector at point :math:`x`
Returns
-------
torch.Tensor
transported point
"""
raise NotImplementedError
[docs] @__scaling__(ScalingInfo(1))
def logmap(self, x: torch.Tensor, y: torch.Tensor) -> torch.Tensor:
r"""
Perform an logarithmic map :math:`\operatorname{Log}_{x}(y)`.
Parameters
----------
x : torch.Tensor
point on the manifold
y : torch.Tensor
point on the manifold
Returns
-------
torch.Tensor
tangent vector
"""
raise NotImplementedError
[docs] @__scaling__(ScalingInfo(u=-1))
def expmap_transp(
self, x: torch.Tensor, u: torch.Tensor, v: torch.Tensor
) -> Tuple[torch.Tensor, torch.Tensor]:
"""
Perform an exponential map and vector transport from point :math:`x` with given direction :math:`u`.
Parameters
----------
x : torch.Tensor
point on the manifold
u : torch.Tensor
tangent vector at point :math:`x`
v : torch.Tensor
tangent vector at point :math:`x` to be transported
Returns
-------
torch.Tensor
transported point
"""
y = self.expmap(x, u)
v_transp = self.transp(x, y, v)
return y, v_transp
[docs] @__scaling__(ScalingInfo(u=-1))
def retr_transp(
self, 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 :math:`x`
v : torch.Tensor
tangent vector at point :math:`x` to be transported
Returns
-------
Tuple[torch.Tensor, torch.Tensor]
transported point and vectors
Notes
-----
Sometimes this is a far more optimal way to preform retraction + vector transport
"""
y = self.retr(x, u)
v_transp = self.transp(x, y, v)
return y, v_transp
[docs] @__scaling__(ScalingInfo(u=-1))
def transp_follow_retr(
self, x: torch.Tensor, u: torch.Tensor, v: torch.Tensor
) -> torch.Tensor:
r"""
Perform vector transport following :math:`u`: :math:`\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 :math:`x`
v : torch.Tensor
tangent vector at point :math:`x` to be transported
Returns
-------
torch.Tensor
transported tensor
"""
y = self.retr(x, u)
return self.transp(x, y, v)
[docs] @__scaling__(ScalingInfo(u=-1))
def transp_follow_expmap(
self, x: torch.Tensor, u: torch.Tensor, v: torch.Tensor
) -> torch.Tensor:
r"""
Perform vector transport following :math:`u`: :math:`\mathfrak{T}_{x\to\operatorname{Exp}(x, u)}(v)`.
Here, :math:`\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 :math:`x`
v : torch.Tensor
tangent vector at point :math:`x` to be transported
Returns
-------
torch.Tensor
transported tensor
"""
y = self.expmap(x, u)
return self.transp(x, y, v)
[docs] def transp(self, x: torch.Tensor, y: torch.Tensor, v: torch.Tensor) -> torch.Tensor:
r"""
Perform vector transport :math:`\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 :math:`x`
Returns
-------
torch.Tensor
transported tensor
"""
raise NotImplementedError
[docs] @abc.abstractmethod
def inner(
self, x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None, *, keepdim=False
) -> torch.Tensor:
"""
Inner product for tangent vectors at point :math:`x`.
Parameters
----------
x : torch.Tensor
point on the manifold
u : torch.Tensor
tangent vector at point :math:`x`
v : Optional[torch.Tensor]
tangent vector at point :math:`x`
keepdim : bool
keep the last dim?
Returns
-------
torch.Tensor
inner product (broadcasted)
"""
raise NotImplementedError
[docs] def component_inner(
self, x: torch.Tensor, u: torch.Tensor, v: torch.Tensor = None
) -> torch.Tensor:
"""
Inner product for tangent vectors at point :math:`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 :math:`x`
v : Optional[torch.Tensor]
tangent vector at point :math:`x`
Returns
-------
torch.Tensor
inner product component wise (broadcasted)
Notes
-----
The purpose of this method is better adaptive properties in optimization since ProductManifold
will "hide" the structure in public API.
"""
return self.inner(x, u, v, keepdim=True)
[docs] def norm(self, x: torch.Tensor, u: torch.Tensor, *, keepdim=False) -> torch.Tensor:
"""
Norm of a tangent vector at point :math:`x`.
Parameters
----------
x : torch.Tensor
point on the manifold
u : torch.Tensor
tangent vector at point :math:`x`
keepdim : bool
keep the last dim?
Returns
-------
torch.Tensor
inner product (broadcasted)
"""
return self.inner(x, u, keepdim=keepdim) ** 0.5
[docs] @abc.abstractmethod
def proju(self, x: torch.Tensor, u: torch.Tensor) -> torch.Tensor:
"""
Project vector :math:`u` on a tangent space for :math:`x`, usually is the same as :meth:`egrad2rgrad`.
Parameters
----------
x torch.Tensor
point on the manifold
u torch.Tensor
vector to be projected
Returns
-------
torch.Tensor
projected vector
"""
raise NotImplementedError
[docs] @abc.abstractmethod
def egrad2rgrad(self, x: torch.Tensor, u: torch.Tensor) -> torch.Tensor:
"""
Transform gradient computed using autodiff to the correct Riemannian gradient for the point :math:`x`.
Parameters
----------
x torch.Tensor
point on the manifold
u torch.Tensor
gradient to be projected
Returns
-------
torch.Tensor
grad vector in the Riemannian manifold
"""
raise NotImplementedError
[docs] @abc.abstractmethod
def projx(self, x: torch.Tensor) -> torch.Tensor:
"""
Project point :math:`x` on the manifold.
Parameters
----------
x torch.Tensor
point to be projected
Returns
-------
torch.Tensor
projected point
"""
raise NotImplementedError
[docs] def _check_shape(
self, shape: Tuple[int], name: str
) -> Union[Tuple[bool, Optional[str]], bool]:
"""
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
Returns
-------
bool, str or None
check result and the reason of fail if any
"""
ok = len(shape) >= self.ndim
if not ok:
reason = "'{}' on the {} requires more than {} dim".format(
name, self, self.ndim
)
else:
reason = None
return ok, reason
[docs] def _assert_check_shape(self, shape: Tuple[int], name: str):
"""
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
Raises
------
ValueError
"""
ok, reason = self._check_shape(shape, name)
if not ok:
raise ValueError(reason)
[docs] @abc.abstractmethod
def _check_point_on_manifold(
self, x: torch.Tensor, *, atol=1e-5, rtol=1e-5
) -> Union[Tuple[bool, Optional[str]], bool]:
"""
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
----------
x torch.Tensor
point on the manifold
atol: float
absolute tolerance as in :func:`numpy.allclose`
rtol: float
relative tolerance as in :func:`numpy.allclose`
Returns
-------
bool, str or None
check result and the reason of fail if any
"""
# return True, None
raise NotImplementedError
[docs] @abc.abstractmethod
def _check_vector_on_tangent(
self, x: torch.Tensor, u: torch.Tensor, *, atol=1e-5, rtol=1e-5
) -> Union[Tuple[bool, Optional[str]], bool]:
"""
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
----------
x torch.Tensor
u torch.Tensor
atol : float
absolute tolerance
rtol :
relative tolerance
Returns
-------
bool, str or None
check result and the reason of fail if any
"""
# return True, None
raise NotImplementedError
def __repr__(self):
extra = self.extra_repr()
if extra:
return self.name + "({}) manifold".format(extra)
else:
return self.name + " manifold"
[docs] def unpack_tensor(self, tensor: torch.Tensor) -> torch.Tensor:
"""
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
Returns
-------
torch.Tensor
"""
return tensor
[docs] def pack_point(self, *tensors: torch.Tensor) -> torch.Tensor:
"""
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]
Returns
-------
torch.Tensor
"""
if len(tensors) != 1:
raise ValueError("1 tensor expected, got {}".format(len(tensors)))
return tensors[0]
[docs] def random(self, *size, dtype=None, device=None, **kwargs) -> torch.Tensor:
"""
Random sampling on the manifold.
The exact implementation depends on manifold and usually does not follow all
assumptions about uniform measure, etc.
"""
raise NotImplementedError
[docs] def origin(
self,
*size: Union[int, Tuple[int]],
dtype=None,
device=None,
seed: Optional[int] = 42,
) -> torch.Tensor:
"""
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)
Returns
-------
torch.Tensor
"""
if seed is not None:
# we promise pseudorandom behaviour but do not want to modify global seed
state = torch.random.get_rng_state()
torch.random.manual_seed(seed)
try:
return self.random(*size, dtype=dtype, device=device)
finally:
torch.random.set_rng_state(state)
else:
return self.random(*size, dtype=dtype, device=device)