Computational Information Geometry Wonderland

I am looking for Finslerian quasi-metric distances with applications in computer vision/medical imaging. Any recommendation?

Comments welcome!
Frank.

Mahalanobis distance is one of the most famous distances in statistics, even nowadays. It is good to look back at original papers, and see the opinions 20 years later by Mahalanobis himself. Some scientific people have fought to have their ideas published.. No social scientific web at that time!.
Read the C2 vs D2 story here.

Frank.

The Earth Mover Distance is one of the key metrics in computer vision and information retrieval based on histograms. The dissimilarity measures dates back from Monge problem. It is a cross-bin measure, that means we do not have to align the bin of the first histogram to the bin of the second histogram distribution.

At ICIP, we shall present some recent work that deals with EMD on a super pixel segmentation tree:

• Takes into account the geometrical and topological structure of segments
• Takes into account unconsistent segmentation from one image to the other
• As fast as solving an EMD problem on 1-D distribution

Here, is the

• poster
• paper

Frank.

The seminal paper of Rao written before he joined Cambridge for his PhD is available online at:

Breakthroughs in Statistics
page 235 is a reprint of:
Rao, Calyampudi (1945). "Information and the accuracy attainable in the estimation of statistical parameters". Bulletin of the Calcutta Mathematical Society 37: 81?89.

There we find three essential results:

• Cramer-Rao bound

• Population space and riemannian geometry using the Fisher information metric as the quadratic differential form.

• Test of significance (and classification)

Information geometry has since then spreaded, with the work of Chentsov on alpha-connections and its investigation by Amari. Historically, the space of distributions, was called the "population space".

The Rao distance for 1D normals is also given.

Frank.

Rao's Distance Measure, by Colin Atkinson and Ann F. S. Mitchell, 1981 Indian Statistical Institute.

Rao's distance is the Riemannian geodesic distance induced by the Fisher information matrix as the tensor. Computing Rao's distance for given parametric distributions (let d be the number of parameters), thus involved to compute explicitly the geodesic. The distance is the length of that geodesic, the sum of its infinitesimal elements along the shortest path curve. It is thus quite complicated to compute in practice the exact Rao's distance as we need to solve the differential equation of the geodesic stated by the Euler-Lagrange equations. In this paper, three different approaches are proposed;

• Classic Euler-Lagrange equations (d second order differential equations)
• Hamilton's equations (2d first order differential equations)
• Hamilton-Jacobi equations (nonlinear partial differential equation)

For uniparameter distribution, the geodesic length (Rao's distance) becomes easy, and the list of such distances are given for Poisson, binomial, exponential, chi-squared (gamma). For multi-parameter distributions, it becomes more difficult. The two classical examples are the normal distributions (which Rao geometry is hyperbolic geometry) and the multinomials (which Rao geometry is the spherical geometry). For same-mean multivariate normals, the Rao distance is given (an unpublished result due to Jensen in 1976). One problem we typically face with this distance computation is to know whether it admits a closed-form equation or not.

Frank.

Computing the Rao distance for Gamma distributions
by F. Reverter and J. M. Oller

The Gamma distribution belongs to the exponential families. Therefore, the Fisher information metric is $I(\theta)=\nabla^2 F(\theta)$. However, integrating the square root of the information matrix is difficult (no closed form solution). The author proceeds by characterizing the Riemannian geodesic using the differential equation relying on Christoffel symbols. Geodesics on the Gamma manifold are unique since the manifold is simply connected, complete with all sectional curvatures nonpositive. The authors come up with a Newton-like numerical optimization algorithm that depends on a good initialization. First, they show that the metric is bounded by Poincare metrics for which closed form equations of the geodesics are known. This yields a good starting tangent vector.
It is quite impressive to look at the formula of the closed-form equation of the Poincare geodesics. Those formula are surprisingly quite complicated.
The authors implemented their algorithm in FORTRAN and show that the algorithm always convergence on the domain examples, with high numerical precisions.