I'm struggling to understand the geometry of projection for infinite dimensional statistical manifolds. In finite dimensions, a strictly convex smooth function $F$ defines a Bregman divergence. From this function/divergence, an finite-dimensional information-geometric is defined, which is dually-flat. The Riemannian metric is obtained from the Hessian of the strictly convex function, and the rest of the information is obtained from partial derivatives (including the connection $\nabla^{F}$). In fact, there is a global coordinate system $[\theta]$ defined by the gradient $\nabla F$. In this case, it can be shown that unique projections from a point on the manifold to a submanifold exist, provided the submanifold is $\nabla$-flat (i.e. corresponds to a $\nabla$-affine subspace in the $[\theta]$-coordinate system). This, of course, broadly generalizes a projection to an affine subspace of Euclidean space. Does there exist a suitable analog for these facts in the infinite-dimensional information case?

I'm struggling to understand the geometry of projection for infinite dimensional statistical manifolds. In finite dimensions, a strictly convex smooth function $F$ defines a Bregman divergence. From this function/divergence, an finite-dimensional information-geometric structure is defined, which is dually-flat. The Riemannian metric is obtained from the Hessian of the strictly convex function, and the rest of the information is obtained from partial derivatives (including the connection $\nabla^{F}$). In fact, there is a global coordinate system $[\theta]$ defined by the gradient $\nabla F$. In this case, it can be shown that unique projections from a point on the manifold to a submanifold exist, provided the submanifold is $\nabla$-flat (i.e. corresponds to a $\nabla$-affine subspace in the $[\theta]$-coordinate system). This, of course, broadly generalizes a projection to an affine subspace of Euclidean space. Does there exist a suitable analog for these facts in the infinite-dimensional information case?