A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions
We present a simple method to approximate the Fisher–Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher–Rao distances between successive nearby normal distributions on the curves by the square roots of their J...
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MDPI AG
2023-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/4/654 |
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| author | Frank Nielsen |
| author_facet | Frank Nielsen |
| author_sort | Frank Nielsen |
| collection | DOAJ |
| description | We present a simple method to approximate the Fisher–Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher–Rao distances between successive nearby normal distributions on the curves by the square roots of their Jeffreys divergences. We consider experimentally the linear interpolation curves in the ordinary, natural, and expectation parameterizations of the normal distributions, and compare these curves with a curve derived from the Calvo and Oller’s isometric embedding of the Fisher–Rao <i>d</i>-variate normal manifold into the cone of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>×</mo><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> symmetric positive–definite matrices. We report on our experiments and assess the quality of our approximation technique by comparing the numerical approximations with both lower and upper bounds. Finally, we present several information–geometric properties of Calvo and Oller’s isometric embedding. |
| first_indexed | 2024-03-11T05:03:02Z |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-11T05:03:02Z |
| publishDate | 2023-04-01 |
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| series | Entropy |
| spelling | doaj.art-5a01b92c70e543b5ba600d8f8605547c2023-11-17T19:09:11ZengMDPI AGEntropy1099-43002023-04-0125465410.3390/e25040654A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal DistributionsFrank Nielsen0Sony Computer Science Laboratories, Tokyo 141-0022, JapanWe present a simple method to approximate the Fisher–Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher–Rao distances between successive nearby normal distributions on the curves by the square roots of their Jeffreys divergences. We consider experimentally the linear interpolation curves in the ordinary, natural, and expectation parameterizations of the normal distributions, and compare these curves with a curve derived from the Calvo and Oller’s isometric embedding of the Fisher–Rao <i>d</i>-variate normal manifold into the cone of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>×</mo><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> symmetric positive–definite matrices. We report on our experiments and assess the quality of our approximation technique by comparing the numerical approximations with both lower and upper bounds. Finally, we present several information–geometric properties of Calvo and Oller’s isometric embedding.https://www.mdpi.com/1099-4300/25/4/654Fisher–Rao normal manifoldsymmetric positive–definite matrix coneisometric embeddinginformation geometryexponential familyelliptical distribution |
| spellingShingle | Frank Nielsen A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions Entropy Fisher–Rao normal manifold symmetric positive–definite matrix cone isometric embedding information geometry exponential family elliptical distribution |
| title | A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions |
| title_full | A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions |
| title_fullStr | A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions |
| title_full_unstemmed | A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions |
| title_short | A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions |
| title_sort | simple approximation method for the fisher rao distance between multivariate normal distributions |
| topic | Fisher–Rao normal manifold symmetric positive–definite matrix cone isometric embedding information geometry exponential family elliptical distribution |
| url | https://www.mdpi.com/1099-4300/25/4/654 |
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