Geodesic convexity

[Petrichor]

is a convex function on a eucledian space still 'geodesically convex' on a Riemannian manifold?

[Nicolas Boumal]

It might be, but it's unlikely to be the case in any interesting scenario. For example, a constant function is convex / g-convex in any Riemannian metric. But more generally, even in one dimension (f from R to R), you can imagine distorting the metric on R in such a way that f fails to be g-convex in the new metric. It's still invex though, because critical points etc. are independent on the choice of metric, so you'll preserve the property that critical points are optimal.

On Sun, Aug 4, 2024, 11:58 AM Petrichor <malou...@gmail.com> wrote:

is a convex function on a eucledian space still 'geodesically convex' on a Riemannian manifold?

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[Petrichor]

Thank you for your response.

Is there any relation between g-convexity and speed rate of convergence. In they talk about it, in this article https://arxiv.org/abs/1706.03267

But I found nothing in books.

[Petrichor]

Also, in optimization on manifolds, how does g-convexity influence optimization results? If I use Riemannian gradient descent, for example, I understand that the Riemannian gradient is just a projection of the Euclidean gradient. Therefore, the optimization will mainly occur in the tangent space (which is Euclidean). Given that my function is convex in Euclidean space, do I need to be concerned about its g-convexity?

[Ronny Bergmann]

Hi,

similar to convexity (locally) ensuring uniqueness of minimisers, that is the same for g-convexity on manifolds.

For the sphere you are right with the idea of projection for the gradient, however, this is not true in general, since the Gradient is (in general) the Riesz representer (in the corresponding metric in the tangent space) of the differential, see Sect. 3.8. of Nicolas Boumals book or the tutorial of Manopt.jl here https://manoptjl.org/stable/tutorials/AutomaticDifferentiation/#An-example-for-a-non-isometrically-embedded-manifold for a more involved case.

Also not that while the tangent vector we get from the gradient (or “direction to walk into”) is a tangent vector, the crucial step here is that we then use a retraction or the exponential map to “move” on the manifold, so the next gradient is in a different tangent space..

My idea to the g-convex vs. Euclidean convex (i.e. for the sphere) is that convexity is “measured / checked” along two different paths: If you take 2 points on the sphere and look at convexity

* g-convexity is along the great arc on the sphere

* Euclidean along the line through those two points.

If your function is convex in one of these cases it is usually not convex in the other (unless it is a constant function, then it is). This problem with convexity along two different curves (necessary given) is for example also illustrated in (a slightly different setting of) Remark 4.6 / Figure 4 (p. 23) in https://arxiv.org/pdf/1506.02409

Best

Ronny

[Petrichor]

Thank you for your response.

So is there a way to have both convexity and g-convexity? for exemple: reformulating the objective function or perhaps the whole optimization problem.

(and I am mainly working on a complex circle)

[Nicolas Boumal]

>  (and I am mainly working on a complex circle)

On compact manifolds (such as the circle), geodesically convex functions are necessarily constant. (This is standard; see for example Corollary 11.10 in my book).

So, the only function that is both g-convex and convex on the circle would be a constant function, which is not useful for optimization.