Exponential map on the Fisher manifold for exponential family distribution












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So, I don't really understand too well Diff. Geometry and Manifolds currently. Hence, I've started studying it and it is very interesting. However, atm I just need to understand how to compute the exponential map for the Fisher manifold of exponential family distributions. Unfortunately, this is a timely manner so I'm afaraid I will need to do this before I actually learn it properly following my study.



Essentially, I'm interested in understanding given that we have a predefined Riemannian metric tensor (in this case the Fisher matrix), as I understand the exponential map will give me given a tangent vector a point $p$ the unique geodesic that is in the same direction as the tangent vector at unit speed. To me this suggest that one should be able to setup this as an optimization problem, whose solution (if anayltically exists) defines the exponential map. However, for some reason I don't seem to be able to derive this accordingly.



For instance consider a Gaussian distribution with fixed mean 0 and a covariance matrix $Sigma$. Consider the Fisher Manifold with respect to the covariance. The Fisher matrix is
$$frac{Sigma^{-1} otimes Sigma^{-1}}{2}$$
Given this information what is the actual mathematical problem that one needs to setup in order to arrive at (taken from http://www.dima.unige.it/~riccomag/Pistone-2017-01-23.pdf):
$$
Exp(tM;Sigma) = Sigma^{frac{1}{2}} exp left[Sigma^{-frac{1}{2}} (tV) Sigma^{-frac{1}{2}} right] Sigma^{frac{1}{2}}
$$










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    So, I don't really understand too well Diff. Geometry and Manifolds currently. Hence, I've started studying it and it is very interesting. However, atm I just need to understand how to compute the exponential map for the Fisher manifold of exponential family distributions. Unfortunately, this is a timely manner so I'm afaraid I will need to do this before I actually learn it properly following my study.



    Essentially, I'm interested in understanding given that we have a predefined Riemannian metric tensor (in this case the Fisher matrix), as I understand the exponential map will give me given a tangent vector a point $p$ the unique geodesic that is in the same direction as the tangent vector at unit speed. To me this suggest that one should be able to setup this as an optimization problem, whose solution (if anayltically exists) defines the exponential map. However, for some reason I don't seem to be able to derive this accordingly.



    For instance consider a Gaussian distribution with fixed mean 0 and a covariance matrix $Sigma$. Consider the Fisher Manifold with respect to the covariance. The Fisher matrix is
    $$frac{Sigma^{-1} otimes Sigma^{-1}}{2}$$
    Given this information what is the actual mathematical problem that one needs to setup in order to arrive at (taken from http://www.dima.unige.it/~riccomag/Pistone-2017-01-23.pdf):
    $$
    Exp(tM;Sigma) = Sigma^{frac{1}{2}} exp left[Sigma^{-frac{1}{2}} (tV) Sigma^{-frac{1}{2}} right] Sigma^{frac{1}{2}}
    $$










    share|cite|improve this question

























      0












      0








      0







      So, I don't really understand too well Diff. Geometry and Manifolds currently. Hence, I've started studying it and it is very interesting. However, atm I just need to understand how to compute the exponential map for the Fisher manifold of exponential family distributions. Unfortunately, this is a timely manner so I'm afaraid I will need to do this before I actually learn it properly following my study.



      Essentially, I'm interested in understanding given that we have a predefined Riemannian metric tensor (in this case the Fisher matrix), as I understand the exponential map will give me given a tangent vector a point $p$ the unique geodesic that is in the same direction as the tangent vector at unit speed. To me this suggest that one should be able to setup this as an optimization problem, whose solution (if anayltically exists) defines the exponential map. However, for some reason I don't seem to be able to derive this accordingly.



      For instance consider a Gaussian distribution with fixed mean 0 and a covariance matrix $Sigma$. Consider the Fisher Manifold with respect to the covariance. The Fisher matrix is
      $$frac{Sigma^{-1} otimes Sigma^{-1}}{2}$$
      Given this information what is the actual mathematical problem that one needs to setup in order to arrive at (taken from http://www.dima.unige.it/~riccomag/Pistone-2017-01-23.pdf):
      $$
      Exp(tM;Sigma) = Sigma^{frac{1}{2}} exp left[Sigma^{-frac{1}{2}} (tV) Sigma^{-frac{1}{2}} right] Sigma^{frac{1}{2}}
      $$










      share|cite|improve this question













      So, I don't really understand too well Diff. Geometry and Manifolds currently. Hence, I've started studying it and it is very interesting. However, atm I just need to understand how to compute the exponential map for the Fisher manifold of exponential family distributions. Unfortunately, this is a timely manner so I'm afaraid I will need to do this before I actually learn it properly following my study.



      Essentially, I'm interested in understanding given that we have a predefined Riemannian metric tensor (in this case the Fisher matrix), as I understand the exponential map will give me given a tangent vector a point $p$ the unique geodesic that is in the same direction as the tangent vector at unit speed. To me this suggest that one should be able to setup this as an optimization problem, whose solution (if anayltically exists) defines the exponential map. However, for some reason I don't seem to be able to derive this accordingly.



      For instance consider a Gaussian distribution with fixed mean 0 and a covariance matrix $Sigma$. Consider the Fisher Manifold with respect to the covariance. The Fisher matrix is
      $$frac{Sigma^{-1} otimes Sigma^{-1}}{2}$$
      Given this information what is the actual mathematical problem that one needs to setup in order to arrive at (taken from http://www.dima.unige.it/~riccomag/Pistone-2017-01-23.pdf):
      $$
      Exp(tM;Sigma) = Sigma^{frac{1}{2}} exp left[Sigma^{-frac{1}{2}} (tV) Sigma^{-frac{1}{2}} right] Sigma^{frac{1}{2}}
      $$







      manifolds riemannian-geometry fisher-information information-geometry






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      asked Nov 21 '18 at 20:06









      Alex BotevAlex Botev

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          "Geodesic shooting" method for computating gesodesic distance between 2 multivariate densities of different means aand different covariance matrices:
          https://ieeexplore.ieee.org/abstract/document/7497346






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            1 Answer
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            active

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            "Geodesic shooting" method for computating gesodesic distance between 2 multivariate densities of different means aand different covariance matrices:
            https://ieeexplore.ieee.org/abstract/document/7497346






            share|cite|improve this answer


























              1














              "Geodesic shooting" method for computating gesodesic distance between 2 multivariate densities of different means aand different covariance matrices:
              https://ieeexplore.ieee.org/abstract/document/7497346






              share|cite|improve this answer
























                1












                1








                1






                "Geodesic shooting" method for computating gesodesic distance between 2 multivariate densities of different means aand different covariance matrices:
                https://ieeexplore.ieee.org/abstract/document/7497346






                share|cite|improve this answer












                "Geodesic shooting" method for computating gesodesic distance between 2 multivariate densities of different means aand different covariance matrices:
                https://ieeexplore.ieee.org/abstract/document/7497346







                share|cite|improve this answer












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                answered Nov 22 '18 at 20:22









                Frederic BarbarescoFrederic Barbaresco

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