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pyMEF: a Python library for mixtures of exponential families
pyMEF is a Python framework allowing to manipulate, learn, simplify and compare mixtures of exponential families. It is designed to ease the use of various exponential families in mixture models.
See also jMEF for a Java implementation of the same kind of library and libmef for a faster C implementation.
What are exponential families?
An exponential family is a generic set of probability distributions that admit the following canonical distribution:
Exponential families are characterized by the log normalizer function F, and include the following well-known distributions: Gaussian (generic, isotropic Gaussian, diagonal Gaussian, rectified Gaussian or Wald distributions, lognormal), Poisson, Bernoulli, binomial, multinomial, Laplacian, Gamma (incl. chi-squared), Beta, exponential, Wishart, Dirichlet, Rayleigh, probability simplex, negative binomial distribution, Weibull, von Mises, Pareto distributions, skew logistic, etc.
Mixtures of exponential families provide a generic framework for handling Gaussian mixture models (GMMs also called MoGs for mixture of Gaussians), mixture of Poisson distributions, and Laplacian mixture models as well.
More pyMEF specific tutorials are available here:
Basic manipulation of mixture models
- Olivier Schwander, Frank Nielsen, Simplification de modèles de mélange issus d’estimateur par noyau, GRETSI 2011
- Olivier Schwander and Frank Nielsen, pyMEF - A framework for Exponential Families in Python, in Proceedings of the 2011 IEEE Workshop on Statistical Signal Processing
- Vincent Garcia, Frank Nielsen, and Richard Nock, Levels of details for Gaussian mixture models, in Proceedings of the Asian Conference on Computer Vision, Xi’an, China, September 2009
- Frank Nielsen and Vincent Garcia, Statistical exponential families: A digest with flash cards, arXiV, http://arxiv.org/abs/0911.4863, November 2009
- Frank Nielsen and Richard Nock, Sided and symmetrized Bregman centroids, in IEEE Transactions on Information Theory, 2009, 55, 2048-2059
- Frank Nielsen, Jean-Daniel Boissonnat and Richard Nock, On Bregman Voronoi diagrams, in ACM-SIAM Symposium on Data Mining, 2007, 746-755
- A. Banerjee, S. Merugu, I. Dhillon, and J. Ghosh, Clustering with Bregman divergences, in Journal of Machine Learning Research, 2005, 6, 234-245
“Intelligence is the faculty of manufacturing artificial objects, especially tools to make tools, and of indefinitely varying the manufacture.” Henri Bergson
GSI forge presents and lists packages and softwares usually opensource (Python and associated Github depositories, R and associated CRAN-R depositories) that can be useful in the statistical and informational analysis of data with a geometrical or topological approach.
Venus at the Forge of Vulcan, Le Nain Brothers, Musée Saint-Denis, Reims (Vulcan is the god of fire and god of metalworking and the forge, often depicted with a blacksmith’s hammer)
Cartan's father Joseph (1837-1917) was born in the village of Saint Victor de Morestel, which is 13 kilometers from Dolomieu. After he married Anne Cottaz (1841-1927) the family settled in Dolomieu, where Anne had lived. Joseph Cartan was the village blacksmith. Elie Cartan recalled that his childhood had passed under "blows of the anvil, which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning-wheel".
Diffeomorphic Demons is an efficient algorithm for the diffeomorphic registration of N dimensional images. It is based on Thirion’s demons algorithm but works on a Lie group structure on diffeomorphic transformations. Typical 3D medical images can be registered in less than three minutes on a 2 x 2.8 GHz quad-core Intel Xeon Apple Mac pro computer. Diffeomorphic demons is now included in MedINRIA‘s image fusion module. The source code has been integrated into ITK since version 3.8. A command-line software can be found on the Insight Journal. Additionally, some standalone binaries and tutorials may be found on the Stark Lab website.
The Computational Geometry Algorithms Library
CGAL (Computational Geometry Algorithms Library) is a comprehensive library of geometric algorithms. The goal of CGAL is to advance the state of the art of geometric computing and to offer robust and efficient programs for research purpose and industrial applications. The initial development of CGAL is a joint effort of six groups in Europe partially funded by European Projects. The library consists of about 1,000,000 lines of C++ code with users all over the world. Since november 2003, CGAL is an Open Source Project. The spin-off Geometry Factory sells CGAL commercial licenses, support for CGAL and customized developments based on CGAL.
The library offers data structures and algorithms like triangulations, Voronoi diagrams, Boolean operations on polygons and polyhedra, point set processing, arrangements of curves, surface and volume mesh generation, geometry processing, alpha shapes, convex hull algorithms, shape analysis, AABB and KD trees...
Learn more about CGAL by browsing through the Package Overview.
GUDHI – Geometry Understanding in Higher Dimensions
The GUDHI library is a generic open source C++ library, with a Python interface, for Topological Data Analysis (TDA) and Higher Dimensional Geometry Understanding. The library offers state-of-the-art data structures and algorithms to construct simplicial complexes and compute persistent homology.
The library comes with data sets, demos, examples and test suites.
The INFOTOPO library is a generic open source suite of Python Programs (compatible with Python 3.4.x, on Linux, windows, or mac) for Information Topological Data Analysis. It is distrubuted freely under opensource GNU GPL V3 Licence and available on Github depository. The library offers state-of-the-art statistical high dimensional data structures analysis and algorithms to detect covarying patterns and clusters, multiscale data analysis.
INFOTOPO version 1.2
It computes all multivariate information functions: entropy, joint entropy between k random variables (Hk), mutual informations between k random variables (Ik), conditional entropies and mutual informations and provides their cohomological (and homotopy) visualisation in the form of information landscapes and information paths together with an approximation of the minimum information energy complex . It is applicable on any set of empirical data that is data with several trials-repetitions-essays (parameter m), and also allows to compute the undersampling regime, the degree k above which the sample size m is to small to provide good estimations of the information functions . The computational exploration is restricted to the simplicial sublattice of random variable (all the subsets of k=n random variables) and has hence a complexity in O(2^n). In this simplicial setting we can exhaustively estimate information functions on the simplicial information structure, that is joint-entropy Hk and mutual-informations Ik at all degrees k=<n and for every k-tuple, with a standard commercial personal computer (a laptop with processor Intel Core i7-4910MQ CPU @ 2.90GHz * up to k=n=21 in reasonable time (about 3 hours). The mathematical formalism can be found in [1,2,3,6], and its application as a neuroscience and data analysis method can be found in [1,4,5,6].
Baudot, Tapia, Goaillard, Topological Information Data Analysis: Poincare-Shannon Machine and Statistical Physic of Finite Heterogeneous Systems. PDF
 M. Tapia, P. Baudot, M. Dufour, C. Formisano-Tréziny, S. Temporal, M. Lasserre, J. Gabert, K. Kobayashi, JM. Goaillard . Information topology of gene expression profile in dopaminergic neurons PDF
 Baudot P., Bennequin D., The homological nature of entropy. Entropy, 2015, 17, 1-66; doi:10.3390. PDF
 Categories and Physics 2011. Classic and quantum Information topos.
 Information Topology: Statistical Physic of Complex Systems and Data Analysis -Topological and geometrical structures of information, CIRM LuminyFrance. 27-1 sept VIDEO-SLIDE
The INFOTOPO library is developed as part of the Channelomics project supported by the European Research Council, developped at UNIS Inserm 1072, and thanks previously to supports and hostings since 2007 of Max Planck Institute for Mathematic in the Sciences (MPI-MIS) and Complex System Instititute Paris-Ile-de-France (ISC-PIF) and Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG)
A Matlab toolbox for optimization on manifolds
Optimization on manifolds is a powerful paradigm to address nonlinear optimization problems. With Manopt, it is easy to deal with various types of symmetries and constraints which arise naturally in applications, such as orthonormality and low rank.
Manifolds are mathematical sets with a smooth geometry, such as spheres. If you are facing a nonlinear (and possibly nonconvex) optimization problem with nice-looking constraints, symmetries or invariance properties, Manopt may just be the tool for you. Check out the manifolds library to find out!
Manopt comes with a large library of manifolds and ready-to-use Riemannian optimization algorithms. It is well documented and includes diagnostics tools to help you get started quickly. It provides flexibility in describing your cost function and incorporates an optional caching system for more efficiency.
It's open source
Check out the license and let us know how you use Manopt. Please cite this paper if you publish work using Manopt (bibtex).
Python package that performs computations on manifolds such as hyperspheres, hyperbolic spaces, spaces of symmetric positive definite matrices and Lie groups of transformations.
- Nina Miolane, Johan Mathe, Claire Donnat, Mikael Jorda, Xavier Pennec, geomstats: a Python Package for Riemannian Geometry in Machine Learning 2018 PDF
- Benjamin Hou , Nina Miolane, Bishesh Khanal, Matthew C.H. Lee , Amir Alansary,
Steven McDonagh, Joseph Hajnal, Daniel Rueckert, Ben Glocker, Bernhard Kainz, Deep Pose Estimation for Image-Based Registration PDF