Propose an algorithmic solution to make your classifier invariant to local factors of variation that correspond to movement on the manifold. Explain the mathematical intuition behind your approach.

Approximate each 3D shape by its tangent plane at a sample point and measure the distance between the two tangents, or between a tangent plane and a point. This can be achieved by solving a low-dimensional linear system in the dimension of the manifolds. The key insight is that this approach provides a locally invariant representation of the manifold.