geometric_kernels.utils.manifold_utils
======================================

.. py:module:: geometric_kernels.utils.manifold_utils

.. autoapi-nested-parse::

   Utilities for dealing with manifolds.  





Module Contents
---------------

.. py:function:: manifold_laplacian(x, manifold, egrad, ehess)

   Computes the manifold Laplacian of a given function at a given point x.
   The manifold Laplacian equals the trace of the manifold Hessian, i.e.,
   $\Delta_M f(x) = \sum_{i=1}^{d} \nabla^2 f(x_i, x_i)$, where
   $[x_i]_{i=1}^{d}$ is an orthonormal basis of the tangent space at x.

   .. warning::
       This function only works for hyperspheres out of the box. We will
       need to change that in the future.

   .. todo::
       See warning above.

   :param x:
       A point on the manifold at which to compute the Laplacian.
   :param manifold:
       A geomstats manifold.
   :param egrad:
       Euclidean gradient of the given function at x.
   :param ehess:
       Euclidean Hessian of the given function at x.

   :return:
       Manifold Laplacian of the given function at x.

   See :cite:t:`jost2011` (Chapter 3.1) for mathematical details.


.. py:function:: tangent_onb(manifold, x)

   Computes an orthonormal basis on the tangent space at x.

   .. warning::
       This function only works for hyperspheres out of the box. We will
       need to change that in the future.

   .. todo::
       See warning above.

   :param manifold:
       A geomstats manifold.
   :param x:
       A point on the manifold.

   :return:
       An [d, d]-shaped array containing the orthonormal basis
       on `manifold` at `x`.