optimization of a manifold visualisation

[Petrichor]

If you perform optimization on a smooth manifold, and your objective function is convex in Euclidean space, is this convexity preserved when the function is transferred to the manifold? Since optimization on manifolds is performed using tangent spaces, which are locally Euclidean, does that mean the function is optimized in a Euclidean setting, and that it does not go to the manifold space?!

I am just trying to visualise 

Thanks

[Nicolas Boumal]

No. If f : R^n -> R is convex and M is a Riemannian submanifold of R^n, then the restriction of f to M is not necessarily geodesically convex. For example, if M is a sphere, then the restriction necessarily has both a maximum and a minimum. For a geodesically convex function, that only happens if the function is constant. However, it is true the restriction of f to a tangent space of M is convex. Sometimes this is helpful for designing algorithms. 

[Ronny Bergmann]

Informally phrased “convexity is measured along other curves” on manifolds that can be embedded. For example on symmetric positive definite matrices or the Poincaré disc, you could always “walk the straight line” (to look for Euclidean convexity) or along geodesics (for the Riemannian/geodesic convexity). 

Then, even more general than Nicolas remark, if a function is convex w.r.t. both in a certain area – then it is constant.

If it is not constant, it is hence non convex in one of the two cases. We considered such an example for a function on a Hadamard manifold in https://arxiv.org/pdf/1506.02409 Remark 4.6, see Figure 4 for an illustration on the Poincare disc.

The cool part is, that there are functions, that are not Euclidean-convex in the embedding on a manifold, but turn out to be geodetically-convex on the manifold.