Methods of information geometry

Information geometry of the EM and em algorithms for neural networks

A Riemannian metric g on a flat manifold M with flat connection D is called a Hessian metric if it is locally expressed by the Hessian of local functions ϕ with respect to the affine coordinate systems, that is, g = Ddϕ Such pair (D, g), g, and M are called a Hessian structure, a Hessian metric, and a Hessian manifold, respectively [S7].

The geometry of Hessian structures

From a smooth, strictly convex function Φ: Rn → R, a parametric family of divergence function DΦ(α) may be introduced: DΦ(α)(x, y) = 4/1 - α2(1 - α/2 Φ(x)+ 1 + α/2 Φ(y) - Φ (1 - α/2x + 1 + α/2y)) for x, y ∈ int dom(Φ) ⊂ Rn, and for α ∈ R, with DΦ(±1) defined through taking the limit of α. Each member is shown to induce an α-independent Riemannian metric, as well as a pair of dual α-connections, which are generally nonflat, except for α = ±1. In the latter case, DΦ(±1) reduces to the (nonparametric) Bregman divergence, which is representable using Φ and its convex conjugate Φ* and becomes the canonical divergence for dually flat spaces (Amari, 1982, 1985; Amari & Nagaoka, 2000). This formulation based on convex analysis naturally extends the information-geometric interpretation of divergence functions (Eguchi, 1983) to allow the distinction between two different kinds of duality: referential duality (α ↔ -α) and representational duality (Φ ↔ Φ*). When applied to (not necessarily normalized) probability densities, the concept of conjugated representations of densities is introduced, so that ±α-connections defined on probability densities embody both referential and representational duality and are hence themselves bidual. When restricted to a finite-dimensional affine submanifold, the natural parameters of a certain representation of densities and the expectation parameters under its conjugate representation form biorthogonal coordinates. The alpha representation (indexed by β now, β ∈ [-1, 1]) is shown to be the only measure-invariant representation. The resulting two-parameter family of divergence functionals D(α,β), (α,β) ∈ [-1, 1] × [-1, 1] induces identical Fisher information but bidual alpha-connection pairs; it reduces in form to Amari's alpha-divergence family when α = ±1 or when β = 1, but to the family of Jensen difference (Rao, 1987) when β = -1.

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Divergence function, duality, and convex analysis

The monograph deals with the theory of conformal, geodesic,
holomorphically projective, F-planar and others mappings and
transformations of manifolds with affine connection,
Riemannian, Kahler and Riemann-Finsler manifolds. Concretely,
the monograph treats the following: basic concepts of
topological spaces, the theory of manifolds with affine
connection (particularly, the problem of semigeodesic
coordinates), Riemannian and Kahler manifolds (reconstruction
of a metric, equidistant spaces, variational problems in
Riemannian spaces, SO(3)-structure as a model of statistical
manifolds, decomposition of tensors), the theory of
differentiable mappings and transformations of manifolds (the
problem of metrization of affine connection, harmonic
diffeomorphisms), conformal mappings and transformations
(especially conformal mappings onto Einstein spaces, conformal
transformations of Riemannian manifolds), geodesic mappings
(GM; especially geodesic equivalence of a manifold with affine
connection to an equiaffine manifold), GM onto Riemannian
manifolds, GM between Riemannian manifolds (GM of equidistant
spaces, GM of Vn(B) spaces, its field of symmetric linear
endomorphisms), GM of special spaces, particularly Einstein,
Kahler, pseudosymmetric manifolds and their generalizations,
global geodesic mappings and deformations, GM between
Riemannian manifolds of different dimensions, global GM,
geodesic deformations of hypersurfaces in Riemannian spaces,
some applications of GM to general relativity, namely three
invariant classes of the Einstein equations and geodesic
mappings, F-planar mappings of spaces with affine connection,
holomorphically projective mappings (HPM) of Kahler manifolds
(fundamental equations of HPM, HPM of special Kahler manifolds,
HPM of parabolic Kahler manifolds, almost geodesic mappings,
which generalize geodesic mappings, Riemann-Finsler spaces and
their geodesic mappings, geodesic mappings of Berwald spaces
onto Riemannian spaces.

Differential Geometry of Special Mappings

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On the divergences of 1-conformally flat statistical manifolds

On etudie des immersions affines de varietes statistiques dans l'espace affine. Plus particulierement on etablit deux resultats sur le rapport entre l'existence de telles immersions et le fait que les varietes statistiques connexes en question ont des courbures constantes

Dual connections and affine geometry

Conformal-projective geometry of statistical manifolds, a natural generalization of conformal geometry of Riemannian manifolds, is studied in this paper. In particular, several fundamental results in the geometry are given: a geometric criterion for two statistical manifolds to be conformally-projectively equivalent; conditions for a statistical manifold to be conformally-projectively flat; properties of umbilical hypersurfaces of a conformally-projectively flat statistical manifold.

Conformal-Projective Geometry of Statistical Manifolds

We study the higher KdV flows on the space of closed complex equicentroaffine curves as Hamiltonian systems. Using a suitable presymplectic structure of the space, we give the Hamiltonian flows associated with the higher KdV equations and a map between the space and the space of closed curves in the complex plane, which induces the Miura transformation between the higher KdV equations and the higher mKdV equations.

Hamiltonian Formalism for the Higher KdV Flows on the Space of Closed Complex Equicentroaffine Curves

We prove that a bounded, complete hypersurface in hyperbolic space with normal curvatures greater than −1 is diffeomorphic to a sphere. The completeness condition is relaxed when the normal curvatures are bounded away from −1. The diffeomorphism is constructed via the Gauss map of some parallel hypersurface. We also give bounds for the total curvature of this parallel hypersurface.

Takashi Kurose

Papers

On the divergences of 1-conformally flat statistical manifolds

Dual connections and affine geometry

Conformal-Projective Geometry of Statistical Manifolds

Hamiltonian Formalism for the Higher KdV Flows on the Space of Closed Complex Equicentroaffine Curves

Hadamard-type theorems for hypersurfaces in hyperbolic spaces