Related papers: Information Geometry of the Probability Simplex: A…
We present a new method for learning Soft Random Geometric Graphs (SRGGs), drawn in probabilistic metric spaces, with the connection function of the graph defined as the marginal posterior probability of an edge random variable, given the…
Symmetry of information establishes a relation between the information that x has about y (denoted I(x : y)) and the information that y has about x (denoted I(y : x)). In classical information theory, the two are exactly equal, but in…
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the…
We define a class of probability distributions that we call simplicial mixture models, inspired by simplicial complexes from algebraic topology. The parameters of these distributions represent their topology and we show that it is possible…
The paper suggests a short survey of integration algorithms which evolved since 1982. These theorems and algorithms form discrete versions of the calculus theorems.
We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as…
These notes are from a 4-lecture mini-course taught by the author at the conference on von Neumann algebras as part of the ``Geometrie non commutative en mathematiques et physique'' month at CIRM in 2004.
We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts…
This note is the sequel of "Geometric structures as variational objects, I." It generalizes the main result and perspectives of that work to a class of geometric structures that includes integrable almost-complex structures.
We study the flat geometry of the least degenerate singularity of a singular surface in $\mathbb R^4$, the $I_{1}$ singularity parametrised by $(x,y)\mapsto(x,xy,y^{2},y^{3})$. This singularity appears generically when projecting a regular…
In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special…
At this point in time, two major areas of physics, statistical mechanics and quantum mechanics, rest on the foundations of probability and entropy. The last century saw several significant fundamental advances in our understanding of the…
These are course notes I wrote for my Fall 2013 graduate topics course on geometric structures, taught at ICERM. The notes rework many of proofs in William P. Thurston's beautiful but hard-to-understand paper, "Shapes of Polyhedra". A…
The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pair of torsion-free conjugate connections. In recent work,…
Parametric inference posits a statistical model that is a specified family of probability distributions. Restricted inference, e.g., restricted likelihood ratio testing, attempts to exploit the structure of a statistical submodel that is a…
Probabilistic graphical modeling is a branch of machine learning that uses probability distributions to describe the world, make predictions, and support decision-making under uncertainty. Underlying this modeling framework is an elegant…
This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a…
These are lecture notes from the IMPANGA 2010 Summer School. The lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the…
A new formalism is presented for analytically obtaining the probability density function, \( P_{n}(s) \), for the distance between two random points in an \( n \)-dimensional sphere of radius \( R \). Our formalism allows \( P_{n}(s) \) to…
This monograph is about discrete energy minimization for discrete graphical models. It considers graphical models, or, more precisely, maximum a posteriori inference for graphical models, purely as a combinatorial optimization problem.…