Computational Information Geometry Wonderland

The web page is freshly out:

@FrnkNlsn

I have been reading a few introductory papers recently in japanese and took the opportunity to collect terms related to information geometry.
Here is a very first dictionary of english-japanese-french terms I encountered.

Frank.

Tomorrow, we shall held for 3 days a workshop on matrix information geometries. I would like to take the opportunity to advertise the nice book (I read in the airplane):

Positive Definite Matrices by R. Bhatia (2008)

http://www.informationgeometry.org/MIG/

Frank.

Let us give some examples of information manifolds:

• Statistical manifolds (parametric distributions),
• Neural manifolds (Boltzmann machines with fixed topology, i.e., number of nodes),
• ARMA(p,q) time-series manifolds (e-flat=-1-flat)

Strictly speaking, geometrizing information-theoretic problems does not provide a more powerful framework in theory. This is because synthetical and analytical geometries are equivalent. Informally, that means that we can do geometry by algebraic equations.

However, geometrizing problems help grab intuition on the problem at hand. Geometry also yields novel notions to mathematical theories. For example, let us cite the two curvature notions in statistics: exponential and mixture curvatures emanating from conjugate connections. So although synthetical geometry provides the same power as analytical geometry, the third-order asymptotic theory of statistics has been obtained so far only from synthetical information geometry.

Dual differential geometries are also useful to tackle information-theoretic problems such as

• Multiterminal problems met in information theory,
• Linear programming problems (e.g., continuous Karmarkar inner method walking along the m-geodesic),
• Clustering (negative entropy and dual Legendre log-normalizer conjugate for soft/hard clustering).

Frank.

Annals of the Institute of Statistical Mathematics has run a a special Issue onInformation Geometry and Its Applications: http://www.ism.ac.jp/editsec/aism/vol59.html No.1 March 2007

Special Issue:Information Geometry and Its Applications Preface .......... Shun-ichi Amari and Shiro Ikeda (59, 1-2) A modified EM algorithm for mixture models based on Bregman divergence .......... Yu Fujimoto and Noboru Murata (59, 3-25) Exponential statistical manifold .......... Alberto Cena and Giovanni Pistone (59, 27-56) A new algorithm of non-Gaussian component analysis with radial kernel functions .......... Motoaki Kawanabe, Masashi Sugiyama, Gilles Blanchard and Klaus-Robert Müller (59, 57-75) The geometry of proper scoring rules .......... A. P. Dawid (59, 77-93) Extending local mixture models .......... Paul Marriott (59, 95-110) Local mixtures of the exponential distribution .......... K.A. Anaya-Izquierdo and P. K. Marriott (59, 111-134) Bayesian prediction based on a class of shrinkage priors for location-scale models .......... Fumiyasu Komaki (59, 135-146) Uncertainty principle and quantum Fisher information .......... Paolo Gibilisco and Tommaso Isola (59, 147-159) A note on curvature of \alpha-connections of a statistical manifold .......... Jun Zhang (59, 161-170)

The paper:

Error Analysis of a Numerical Calculation about One-qubit Quantum Channel Capacity (ISVD'07)

investigates the numerical error in computing the Holevo channel capacity using the furthest Voronoi diagram wrt. to the von Neuman quantum divergence. They show that the error is in O(1/eps) for sampling in the latitude-longitude 1/eps points.

This algorithm allows us to study, e.g., the additivity conjecture of quantum channels.

It is a nice aspect of quantum information geometry derived from traditional computational geometry. Interestingly, the Legendre transformation for 1-qubit represented on the Bloch sphere is explicited too.

Today is rainy saturday in Tokyo and I have soon to take the train to Saitama. So here is a short post: I like very much to think of abstraction as the main creativity engine in scientific activities.

I found by browsing the www the following paper:
THE RIEMANNIAN MANIFOLD OF ALL RIEMANNIAN METRICS (1991)
I am amazed about the possibilities of studying things once they have an avatar point lying on an avatar world (ie: the differential manifold)

Multivariate normal distributions are engineer's favorite statistical distributions. They occur in numerous applications, so that it is worth taking a look at them from the viewpoint of differential geometry. This is precisely the topic of the paper:

Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space
That is short saturday's post -:) !

I recently read the article

Statistics on the Manifold of Multivariate Normal Distributions (J Math Imaging 2006)

The article gives a nice and good introduction to information geometry from the viewpoint of differential geometry. Diffusion Tensor Imaging (DTI) is a recent technique, the only non-invasive one for monitoring the white matter in brains. DTI acquires for each "grid voxel" position a probability density function characterizing the statistics of water molecule Brownian motion at that neighborhood. So basically, DTI images are 3D volumetric dense sets of zero-mean 3D-variate normal distributions given by their variance-covariance matrices.

Note that in this article references are shifted by one. A problem likely arising from LaTeX macros, I guess.

More on the DTI technique at http://en.wikipedia.org/wiki/Diffusion_tensor_imaging