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Riemannian metric

Riemannian metric

Manifolds and Tangent

Manifolds and Tangent Spaces

Tangent vectors under

Tangent vectors under changes of coordinates

Repeated observations and

Repeated observations and maximum likelihood estimators

Mutual information

Mutual information

Maximum entropy principle

Maximum entropy principle and e-projection

Bregman divergence and

Bregman divergence and exponential family

Kernel exponential family

Kernel exponential family of probability distributions

Infinite-dimensional manifold of

Infinite-dimensional manifold of probability distributions: KL-divergence

Infinite-dimensional manifold of

Infinite-dimensional manifold of probability distributions

Flat structure: e-flat

Flat structure: e-flat and m-flat

Mixture family of

Mixture family of probability distributions

Example of Exponential

Example of Exponential Family: Discrete Distributions

Example of Exponential

Example of Exponential Family: Gaussian Distributions

Exponential family of

Exponential family of probability distributions

Divergence between submanifolds

Divergence between submanifolds

Projection Theorem relating

Projection Theorem relating dual orthogonal projection and divergence minimizing projection

Generalized Pythagorean Theorem

Generalized Pythagorean Theorem

Parallel Transport of

Parallel Transport of Vectors

Tangent space, basis

Tangent space, basis vectors, and Riemannian metric

Affine and dual

Affine and dual affine coordinate systems

Legendre transform of

Legendre transform of convex functions and Bregman divergence

Convex functions and

Convex functions and Bregman divergence: Examples of Bregman divergence

Convex functions and

Convex functions and Bregman divergence: Bregman divergence