Apologies in advance for being somewhat vague. I'm trying to get pointers to establish a connection between a common trick used in practice in optimization, and the exponential map in differential geometry.

In optimization, we'll sometimes choose to optimise $y = \log x$ instead of $x$ (for positive $x$); one of the perks that this gives is the extra precision and smaller-sized steps around $0$. The idea reminds me of optimising in hyperbolic space - the distance between points grows (as compared to Euclidean) as we move towards zero. The $e^y$ is supposed to map the Euclidean space we optimised in back to the one we care about, a hyperbolic one. That sounds awfully close to the exponential map, a map from the tangent space back to the manifold.

Is there a connection? Does this sound like there must be a connection?