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Abstract
We introduce the information geometry module of the Python package Geomstats. The module first implements Fisher–Rao Riemannian manifolds of widely used parametric families of probability distributions, such as normal, gamma, beta, Dirichlet distributions, and more. The module further gives the Fisher–Rao Riemannian geometry of any parametric family of distributions of interest, given a parameterized probability density function as input. The implemented Riemannian geometry tools allow users to compare, average, interpolate between distributions inside a given family. Importantly, such capabilities open the door to statistics and machine learning on probability distributions. We present the object-oriented implementation of the module along with illustrative examples and show how it can be used to perform learning on manifolds of parametric probability distributions.
- [1] . 2017. Rao-geodesic distance on the generalized gamma manifold: Study of three sub-manifolds and application in the texture retrieval domain. Note di Matematica 37, supp1 (2017), 1–18.Google Scholar
- [2] . 2016. Information Geometry and Its Applications, Vol. 194. Springer.Google ScholarCross Ref
- [3] . 2013. A user’s guide to optimal transport. In Modelling and Optimisation of Flows on Networks. Springer, 1–155.Google Scholar
- [4] . 2014. Morphological processing of univariate gaussian distribution-valued images based on poincaré upper-half plane representation. In Geometric Theory of Information. Springer, 331–366.Google ScholarCross Ref
- [5] . 2006. Processing Data in Lie Groups: An Algebraic Approach. Application to Non-Linear Registration and Diffusion Tensor MRI. Ph.D. Dissertation. École polytechnique.Google Scholar
- [6] . 2021. InformationGeometry.jl. (2021).Google Scholar
- [7] . 2008. Information Geometry: Near Randomness and Near Independence. Springer Science and Business Media.Google ScholarCross Ref
- [8] . 1981. Rao’s distance measure. Sankhyā: The Indian Journal of Statistics, Series 43, 3 (1981), 345–365.Google Scholar
- [9] . 2003. Latent dirichlet allocation. Journal of Machine Learning Research 3, Jan (2003), 993–1022.Google ScholarDigital Library
- [10] . 2002. Some remarks on the information geometry of the gamma distribution. Communications in Statistics—Theory and Methods 31, 11 (2002), 1959–1975.Google ScholarCross Ref
- [11] . 1982. Statistical decision rules and optimal inference. transl. math. Monographs, American Mathematical Society, Providence, RI 53 (1982).Google Scholar
- [12] . 2013. The riemannian structure of the three-parameter gamma distribution. Applied Mathematics 4, 3 (2013), 514–522.Google ScholarCross Ref
- [13] . 2015. Fisher information distance: A geometrical reading. Discrete Applied Mathematics 197 (2015), 59–69.Google ScholarDigital Library
- [14] . 1992. Riemannian Geometry, Vol. 6. Springer.Google ScholarCross Ref
- [15] . 1998. Statistical Shape Analysis, with Applications in R. John Wiley and Sons, New York.Google Scholar
- [16] . 2021. POT: Python optimal transport. Journal of Machine Learning Research 22, 78 (2021), 1–8. Retrieved from http://jmlr.org/papers/v22/20-451.htmlGoogle Scholar
- [17] . 1991. Die fisher-information und symplektische strukturen. Mathematische Nachrichten 153, 1 (1991), 273–296.Google ScholarCross Ref
- [18] Nicolas Guigui, Nina Miolane, Xavier Pennec, et al. 2023. Introduction to riemannian geometry and geometric statistics: from basic theory to implementation with geomstats. Foundations and Trends® in Machine Learning, 16, 3 (2023), 329–493.Google Scholar
- [19] . 2009. Information geometry and evolutionary game theory. arXiv:0911.1383. Retrieved from https://arxiv.org/abs/0911.1383Google Scholar
- [20] . 2007. Use of the gamma distribution to represent monthly rainfall in africa for drought monitoring applications. International Journal of Climatology: A Journal of the Royal Meteorological Society 27, 7 (2007), 935–944.Google Scholar
- [21] . 1989. The geometry of asymptotic inference. Statistical Science 41, 3 (1989), 188–219.Google Scholar
- [22] . 1987. Statistical manifolds. Differential Geometry in Statistical Inference 10 (1987), 163–216.Google ScholarCross Ref
- [23] . 2021. Classifying histograms of medical data using information geometry of beta distributions. IFAC-PapersOnLine 54, 9 (2021), 514–520.Google ScholarCross Ref
- [24] . 2021. Fisher-rao geometry of dirichlet distributions. Differential Geometry and its Applications 74 (2021), 101702.Google ScholarCross Ref
- [25] Guy Lebanon. 2002. Learning riemannian metrics. In Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence (UAI’03). San Francisco, CA, Morgan Kaufmann Publishers Inc., 362–369.Google Scholar
- [26] . 2006. Theory of Point Estimation. Springer Science and Business Media.Google Scholar
- [27] . 2006. Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor MRI processing. Journal of Mathematical Imaging and Vision 25, 3 (2006), 423–444.Google ScholarDigital Library
- [28] . 2004. Detection of image structures using the fisher information and the rao metric. IEEE Transactions on Pattern Analysis and Machine Intelligence 26, 12 (2004), 1579–1589.Google ScholarDigital Library
- [29] Nina Miolane, Nicolas Guigui, Alice Le Brigant, Johan Mathe, Benjamin Hou, Yann Thanwerdas, Stefan Heyder, Olivier Peltre, Niklas Koep, Hadi Zaatiti, Hatem Hajri, Yann Cabanes, Thomas Gerald, Paul Chauchat, Christian Shewmake, Daniel Brooks, Bernhard Kainz, Claire Donnat, Susan Holmes, and Xavier Pennec. 2002. Geomstats: a Python package for Riemannian geometry in machine learning. Journal of Machine Learning Research, 21, 223 (2020), 1–9.Google Scholar
- [30] . 2023. A simple approximation method for the fisher–rao distance between multivariate normal distributions. Entropy 25, 4 (2023), 654.
DOI: Google ScholarCross Ref - [31] Felix Otto. 2001. The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations, 26, 1-2 (2001), 101–174.Google Scholar
- [32] . 2006. A riemannian framework for tensor computing. International Journal of Computer Vision 66, 1 (2006), 41–66.Google ScholarDigital Library
- [33] . 2019. On the fisher-rao information metric in the space of normal distributions. In International Conference on Geometric Science of Information. Springer, 676–684.Google ScholarDigital Library
- [34] . 1945. Information and the accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society 37 (1945).
DOI: Google ScholarCross Ref - [35] . 2019. The geometry of the generalized gamma manifold and an application to medical imaging. Mathematics 7, 8 (2019), 674.Google ScholarCross Ref
- [36] . 1979. The geometrical structure of the parameter space of the two-dimensional normal distribution. Reports on Mathematical Physics 16, 1 (1979), 111–119.Google ScholarCross Ref
- [37] . 2018. On a multiplicative multivariate gamma distribution with applications in insurance. Risks 6, 3 (2018), 79.Google ScholarCross Ref
- [38] . 2015. Diffusion-weighted imaging of prostate cancer using a statistical model based on the gamma distribution. Journal of Magnetic Resonance Imaging 42, 1 (2015), 56–62.Google ScholarCross Ref
- [39] . 1984. A riemannian geometry of the multivariate normal model. Scandinavian Journal of Statistics 11, 4 (1984), 211–223.Google Scholar
- [40] . 2016. Clustering using the fisher-rao distance. In 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM). IEEE, 1–5.Google Scholar
- [41] . 2011. Geodesics on the manifold of multivariate generalized gaussian distributions with an application to multicomponent texture discrimination. International Journal of Computer Vision 95, 3 (2011), 265–286.Google ScholarDigital Library
- [42] . 2012. Spaces and manifolds of shapes in computer vision: An overview. Image and Vision Computing 30, 6-7 (2012), 389–397.Google ScholarDigital Library
Index Terms
- Parametric Information Geometry with the Package Geomstats
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