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Information geometry is a very interesting research topic. In essence it allows to define a metric (and therefore a measure of distance) between probability dis

Information geometry is a very interesting research topic. In essence it allows to define a metric (and therefore a measure of distance) between probability distributions. As a results, it has a lot of practical application in machine learning and has been use successfully for classification in Neuroscience, Radar signal processing and computer vision. we can also note that Information Geometry can be seen as a sub-field of Riemannian Geometry, with some equivalence between metric. For example, the cannonical metric for symetric and positive definite (SPD) matrices in Riemannian geometry is actually equivalent to the metric for multivariate normal distribution obtained with Information geometry. For some application, IG is very efficient. it has been used for multivariate time-series for classification of EEG signal and was at the center of the winning solution of 3 kaggle challenges : https://github.com/alexandrebarachant https://github.com/alexandrebarachant

tMcGrath 10y ago
There's also an interesting set of applications in statistical physics: many systems can be modelled by Markov chains whose rate parameters can change through time. Changing these parameters non-quasistatically (i.e. so that the system does not relax to the appropriate steady state) gives rise to distances on parameter space that are related to the amount of energy dissipated along the path through parameter space [0,1]. What's nice about this is that the derivation of a suitable metric allows us to compute trajectories that minimise quantities we care about (e.g. minimise energy dissipated), so this has clear potential to be useful. Some cool examples are in spin systems [2] and a harmonic trap [3]. For any differential geometers reading this: it seems to me that a good geometric way to think about this is as a fibre bundle, with the parameter space being the base space and the simplex being a vector bundle over it (see [4] on the simplex being a vector space). [0] https://arxiv.org/pdf/0706.0559v2.pdf https://arxiv.org/pdf/0706.0559v2.pdf [1] https://arxiv.org/pdf/1201.4166.pdf https://arxiv.org/pdf/1201.4166.pdf [2] https://arxiv.org/pdf/1607.07425v1.pdf https://arxiv.org/pdf/1607.07425v1.pdf [3] http://journals.aps.org/pre/abstract/10.1103/PhysRevE.86.041148 http://journals.aps.org/pre/abstract/10.1103/PhysRevE.86.041... [4] https://golem.ph.utexas.edu/category/2016/06/how_the_simplex_is_a_vector_sp.html https://golem.ph.utexas.edu/category/2016/06/how_the_simplex...

tMcGrath 10y ago

There's also an interesting set of applications in statistical physics: many systems can be modelled by Markov chains whose rate parameters can change through time. Changing these parameters non-quasistatically (i.e. so that the system does not relax to the appropriate steady state) gives rise to distances on parameter space that are related to the amount of energy dissipated along the path through parameter space [0,1]. What's nice about this is that the derivation of a suitable metric allows us to compute trajectories that minimise quantities we care about (e.g. minimise energy dissipated), so this has clear potential to be useful. Some cool examples are in spin systems [2] and a harmonic trap [3]. For any differential geometers reading this: it seems to me that a good geometric way to think about this is as a fibre bundle, with the parameter space being the base space and the simplex being a vector bundle over it (see [4] on the simplex being a vector space). [0] https://arxiv.org/pdf/0706.0559v2.pdf https://arxiv.org/pdf/0706.0559v2.pdf [1] https://arxiv.org/pdf/1201.4166.pdf https://arxiv.org/pdf/1201.4166.pdf [2] https://arxiv.org/pdf/1607.07425v1.pdf https://arxiv.org/pdf/1607.07425v1.pdf [3] http://journals.aps.org/pre/abstract/10.1103/PhysRevE.86.041148 http://journals.aps.org/pre/abstract/10.1103/PhysRevE.86.041... [4] https://golem.ph.utexas.edu/category/2016/06/how_the_simplex_is_a_vector_sp.html https://golem.ph.utexas.edu/category/2016/06/how_the_simplex...