Automatic Differentiation

Cost functions, gradients and Hessian-vector products (hvps) in Pymanopt must be defined as Python callables annotated with one of the backend decorators below. Decorating a callable with a backend decorator will wrap it in an instance of the pymanopt.autodiff.Function class that provides a backend-agnostic API to the pymanopt.core.problem.Problem class to compute derivatives.

If an autodiff backend is used via one of the provided decorators, the signature of the decorated callable must match the point layout of the manifold it is defined on. For instance, for memory efficiency points on the pymanopt.manifolds.fixed_rank.FixedRankEmbedded manifold are not represented as m x n matrices in the ambient space but as a singular value decomposition. As such a cost function defined on the manifold must accept three arguments u, s and vt. Refer to the documentation of the respective manifold on how points are represented.

New backends can be created by inheriting from the pymanopt.autodiff.backends._backend.Backend class, and creating a backend decorator using pymanopt.autodiff.backend_decorator_factory().

class pymanopt.autodiff.Function(*, function, manifold, backend)

Bases: object

get_gradient_operator()

get_hessian_operator()

pymanopt.autodiff.backend_decorator_factory(backend_cls)

Create function decorator for a backend.

Function to create a backend decorator that is used to annotate a callable:

decorator = backend_decorator_factory(backend_cls)

@decorator(manifold)
def function(x):
    ...

Parameters

backend_cls – a class implementing the backend interface defined by pymanopt.autodiff.backend._backend._Backend.

Returns

A new backend decorator.

Return type

Callable[[pymanopt.manifolds.manifold.Manifold], Callable[[Callable], pymanopt.autodiff.Function]]

Backends

pymanopt.function.autograd(manifold)

Parameters

manifold (pymanopt.manifolds.manifold.Manifold) –

Return type

Callable

pymanopt.function.jax(manifold)

Parameters

manifold (pymanopt.manifolds.manifold.Manifold) –

Return type

Callable

pymanopt.function.numpy(manifold)

Parameters

manifold (pymanopt.manifolds.manifold.Manifold) –

Return type

Callable

pymanopt.function.pytorch(manifold)

Parameters

manifold (pymanopt.manifolds.manifold.Manifold) –

Return type

Callable

pymanopt.function.tensorflow(manifold)

Parameters

manifold (pymanopt.manifolds.manifold.Manifold) –

Return type

Callable