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vector_field.py
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vector_field.py
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import abc
import haiku as hk
import jax.numpy as jnp
from hydra.utils import instantiate
from geomstats.geometry.hypersphere import Hypersphere
from geomstats.geometry.base import EmbeddedManifold
class VectorFieldGenerator(hk.Module, abc.ABC):
def __init__(self, architecture, output_shape, manifold):
"""X = fi * Xi with fi weights and Xi generators"""
super().__init__()
self.net = instantiate(architecture, output_shape=output_shape)
self.manifold = manifold
@staticmethod
@abc.abstractmethod
def output_shape(manifold):
"""Cardinality of the generating set."""
def _weights(self, x, t):
"""shape=[..., card=n]"""
return self.net(x, t)
@abc.abstractmethod
def _generators(self, x):
"""Set of generating vector fields: shape=[..., d, card=n]"""
@property
def decomposition(self):
return lambda x, t: self._weights(x, t), lambda x: self._generators(x)
def __call__(self, x, t):
fi_fn, Xi_fn = self.decomposition
fi, Xi = fi_fn(x, t), Xi_fn(x)
out = jnp.einsum("...n,...dn->...d", fi, Xi)
# NOTE: seems that extra projection is required for generator=eigen
# during the ODE solve cf tests/test_lkelihood.py
out = self.manifold.to_tangent(out, x)
return out
def div_generators(self, x):
"""Divergence of the generating vector fields: shape=[..., card=n]"""
class AmbientGenerator(VectorFieldGenerator):
def __init__(self, architecture, output_shape, manifold):
super().__init__(architecture, output_shape, manifold)
@staticmethod
def output_shape(manifold):
if isinstance(manifold, EmbeddedManifold):
output_shape = manifold.embedding_space.dim
else:
output_shape = manifold.dim
return output_shape
def _generators(self, x):
return self.manifold.eigen_generators(x)
def __call__(self, x, t):
# `to_tangent`` have an 1/sq_norm(x) term that wrongs the div
return self.manifold.to_tangent(self.net(x, t), x)
class DivFreeGenerator(VectorFieldGenerator):
def __init__(self, architecture, output_shape, manifold):
super().__init__(architecture, output_shape, manifold)
@staticmethod
def output_shape(manifold):
return manifold.isom_group.dim
def _generators(self, x):
return self.manifold.div_free_generators(x)
def div_generators(self, x):
shape = [*x.shape[:-1], self.output_shape(self.manifold)]
return jnp.zeros(shape)
class EigenGenerator(VectorFieldGenerator):
"""Gradient of laplacien eigenfunctions with eigenvalue=1"""
def __init__(self, architecture, output_shape, manifold):
super().__init__(architecture, output_shape, manifold)
assert isinstance(manifold, Hypersphere)
@staticmethod
def output_shape(manifold):
return manifold.embedding_space.dim
def _generators(self, x):
return self.manifold.eigen_generators(x)
def div_generators(self, x):
# NOTE: Empirically need this factor 2 to match AmbientGenerator but why??
return -self.manifold.dim * 2 * x
class LieAlgebraGenerator(VectorFieldGenerator):
def __init__(self, architecture, output_shape, manifold):
super().__init__(architecture, output_shape, manifold)
@staticmethod
def output_shape(manifold):
return manifold.dim
def _generators(self, x):
return self.manifold.lie_algebra.basis
def __call__(self, x, t):
x = x.reshape((x.shape[0], self.manifold.dim, self.manifold.dim))
fi_fn, Xi_fn = self.decomposition
x_input = x.reshape((*x.shape[:-2], -1))
fi, Xi = fi_fn(x_input, t), Xi_fn(x)
out = jnp.einsum("...i,ijk ->...jk", fi, Xi)
out = self.manifold.compose(x, out)
return out.reshape((x.shape[0], -1))
class TorusGenerator(VectorFieldGenerator):
def __init__(self, architecture, output_shape, manifold):
super().__init__(architecture, output_shape, manifold)
self.rot_mat = jnp.array([[0, -1], [1, 0]])
@staticmethod
def output_shape(manifold):
return manifold.dim
def _generators(self, x):
return (
self.rot_mat @ x.reshape((*x.shape[:-1], self.manifold.dim, 2))[..., None]
)[..., 0]
def __call__(self, x, t):
weights_fn, fields_fn = self.decomposition
weights = weights_fn(x, t)
fields = fields_fn(x)
return (fields * weights[..., None]).reshape(
(*x.shape[:-1], self.manifold.dim * 2)
)
class CanonicalGenerator:
def __init__(self, architecture, output_shape=None, manifold=None):
self.net = instantiate(architecture, output_shape=output_shape)
@staticmethod
def output_shape(manifold):
return manifold.dim
def __call__(self, x, t):
return self.net(x, t)
class ParallelTransportGenerator:
def __init__(self, architecture, output_shape=None, manifold=None):
self.net = instantiate(architecture, output_shape=output_shape)
self.manifold = manifold
@staticmethod
def output_shape(manifold):
return manifold.identity.shape[-1]
def __call__(self, x, t):
"""
Rescale since ||s(x, t)||^2_x = s(x, t)^t G(x) s(x, t) = \lambda(x)^2 ||s(x, t)||^2_2
with G(x)=\lambda(x)^2 Id
"""
tangent = self.net(x, t)
tangent = self.manifold.metric.transpfrom0(x, tangent)
return tangent
class ProjectionGenerator(VectorFieldGenerator):
def __init__(self, architecture, output_shape, manifold):
super().__init__(architecture, output_shape, manifold)
@staticmethod
def output_shape(manifold):
if isinstance(manifold, EmbeddedManifold):
output_shape = manifold.embedding_space.dim
else:
output_shape = manifold.dim
return output_shape
def _generators(self, x):
return self.manifold.eigen_generators(x)
def __call__(self, x, t):
return self.manifold.projection(self.net(x, t))