von Mises–Fisher distribution

In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the -sphere in . If the distribution reduces to the von Mises distribution on the circle.

Definition

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The probability density function of the von Mises–Fisher distribution for the random p-dimensional unit vector   is given by:

 

where   and the normalization constant   is equal to

 

where   denotes the modified Bessel function of the first kind at order  . If  , the normalization constant reduces to

 

The parameters   and   are called the mean direction and concentration parameter, respectively. The greater the value of  , the higher the concentration of the distribution around the mean direction  . The distribution is unimodal for  , and is uniform on the sphere for  .

The von Mises–Fisher distribution for   is also called the Fisher distribution.[1][2] It was first used to model the interaction of electric dipoles in an electric field.[3] Other applications are found in geology, bioinformatics, and text mining.

Note on the normalization constant

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In the textbook, Directional Statistics [3] by Mardia and Jupp, the normalization constant given for the Von Mises Fisher probability density is apparently different from the one given here:  . In that book, the normalization constant is specified as:

 

where   is the gamma function. This is resolved by noting that Mardia and Jupp give the density "with respect to the uniform distribution", while the density here is specified in the usual way, with respect to Lebesgue measure. The density (w.r.t. Lebesgue measure) of the uniform distribution is the reciprocal of the surface area of the (p-1)-sphere, so that the uniform density function is given by the constant:

 

It then follows that:

 

While the value for   was derived above via the surface area, the same result may be obtained by setting   in the above formula for  . This can be done by noting that the series expansion for   divided by   has but one non-zero term at  . (To evaluate that term, one needs to use the definition  .)

Support

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The support of the Von Mises–Fisher distribution is the hypersphere, or more specifically, the  -sphere, denoted as

 

This is a  -dimensional manifold embedded in  -dimensional Euclidean space,  .

Relation to normal distribution

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Starting from a normal distribution with isotropic covariance   and mean   of length  , whose density function is:

 

the Von Mises–Fisher distribution is obtained by conditioning on  . By expanding

 

and using the fact that the first two right-hand-side terms are fixed, the Von Mises-Fisher density,   is recovered by recomputing the normalization constant by integrating   over the unit sphere. If  , we get the uniform distribution, with density  .

More succinctly, the restriction of any isotropic multivariate normal density to the unit hypersphere, gives a Von Mises-Fisher density, up to normalization.

This construction can be generalized by starting with a normal distribution with a general covariance matrix, in which case conditioning on   gives the Fisher-Bingham distribution.

Estimation of parameters

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Mean direction

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A series of N independent unit vectors   are drawn from a von Mises–Fisher distribution. The maximum likelihood estimates of the mean direction   is simply the normalized arithmetic mean, a sufficient statistic:[3]

 

Concentration parameter

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Use the modified Bessel function of the first kind to define

 

Then:

 

Thus   is the solution to

 

A simple approximation to   is (Sra, 2011)

 

A more accurate inversion can be obtained by iterating the Newton method a few times

 
 

Standard error

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For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as:[4]

 

where

 

It is then possible to approximate a   a spherical confidence interval (a confidence cone) about   with semi-vertical angle:

  where  

For example, for a 95% confidence cone,   and thus  

Expected value

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The expected value of the Von Mises–Fisher distribution is not on the unit hypersphere, but instead has a length of less than one. This length is given by   as defined above. For a Von Mises–Fisher distribution with mean direction   and concentration  , the expected value is:

 .

For  , the expected value is at the origin. For finite  , the length of the expected value is strictly between zero and one and is a monotonic rising function of  .

The empirical mean (arithmetic average) of a collection of points on the unit hypersphere behaves in a similar manner, being close to the origin for widely spread data and close to the sphere for concentrated data. Indeed, for the Von Mises–Fisher distribution, the expected value of the maximum-likelihood estimate based on a collection of points is equal to the empirical mean of those points.

Entropy and KL divergence

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The expected value can be used to compute differential entropy and KL divergence.

The differential entropy of   is:

 

where the angle brackets denote expectation. Notice that the entropy is a function of   only.

The KL divergence between   and   is:

 

Transformation

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Von Mises-Fisher (VMF) distributions are closed under orthogonal linear transforms. Let   be a  -by-  orthogonal matrix. Let   and apply the invertible linear transform:  . The inverse transform is  , because the inverse of an orthogonal matrix is its transpose:  . The Jacobian of the transform is  , for which the absolute value of its determinant is 1, also because of the orthogonality. Using these facts and the form of the VMF density, it follows that:

 

One may verify that since   and   are unit vectors, then by the orthogonality, so are   and  .

Pseudo-random number generation

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General case

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An algorithm for drawing pseudo-random samples from the Von Mises Fisher (VMF) distribution was given by Ulrich[5] and later corrected by Wood.[6] An implementation in R is given by Hornik and Grün;[7] and a fast Python implementation is described by Pinzón and Jung.[8]

To simulate from a VMF distribution on the  -dimensional unitsphere,  , with mean direction  , these algorithms use the following radial-tangential decomposition for a point   :

 

where   lives in the tangential  -dimensional unit-subsphere that is centered at and perpendicular to  ; while  . To draw a sample   from a VMF with parameters   and  ,   must be drawn from the uniform distribution on the tangential subsphere; and the radial component,  , must be drawn independently from the distribution with density:

 

where  . The normalization constant for this density may be verified by using:

 

as given in Appendix 1 (A.3) in Directional Statistics.[3] Drawing the   samples from this density by using a rejection sampling algorithm is explained in the above references. To draw the uniform   samples perpendicular to  , see the algorithm in,[8] or otherwise a Householder transform can be used as explained in Algorithm 1 in.[9]

3-D sphere

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To generate a Von Mises–Fisher distributed pseudo-random spherical 3-D unit vector[10][11]   on the  sphere for a given   and  , define

 

where   is the polar angle,   the azimuthal angle, and   the distance to the center of the sphere

for   the pseudo-random triplet is then given by

 

where   is sampled from the continuous uniform distribution   with lower bound   and upper bound  

 

and

 

where   is sampled from the standard continuous uniform distribution  

 

here,  should be set to   when   and   rotated to match any other desired  .

Distribution of polar angle

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For  , the angle θ between   and   satisfies  . It has the distribution

 ,

which can be easily evaluated as

 .

For the general case,  , the distribution for the cosine of this angle:

 

is given by  , as explained above.

The uniform hypersphere distribution

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When  , the Von Mises–Fisher distribution,   on   simplifies to the uniform distribution on  . The density is constant with value  . Pseudo-random samples can be generated by generating samples in   from the standard multivariate normal distribution, followed by normalization to unit norm.

Component marginal of uniform distribution

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For  , let   be any component of  . The marginal distribution for   has the density:[12][13]

 

where   is the beta function. This distribution may be better understood by highlighting its relation to the beta distribution:

 

where the Legendre duplication formula is useful to understand the relationships between the normalization constants of the various densities above.

Note that the components of   are not independent, so that the uniform density is not the product of the marginal densities; and   cannot be assembled by independent sampling of the components.

Distribution of dot-products

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In machine learning, especially in image classification, to-be-classified inputs (e.g. images) are often compared using cosine similarity, which is the dot product between intermediate representations in the form of unitvectors (termed embeddings). The dimensionality is typically high, with   at least several hundreds. The deep neural networks that extract embeddings for classification should learn to spread the classes as far apart as possible and ideally this should give classes that are uniformly distributed on  .[14] For a better statistical understanding of across-class cosine similarity, the distribution of dot-products between unitvectors independently sampled from the uniform distribution may be helpful.


Let   be unitvectors in  , independently sampled from the uniform distribution. Define:

 

where   is the dot-product and   are transformed versions of it. Then the distribution for   is the same as the marginal component distribution given above;[13] the distribution for   is symmetric beta and the distribution for   is symmetric logistic-beta:

 

The means and variances are:

 

and

 

where   is the first polygamma function. The variances decrease, the distributions of all three variables become more Gaussian, and the final approximation gets better as the dimensionality,  , is increased.

Generalizations

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Matrix Von Mises-Fisher

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The matrix von Mises-Fisher distribution (also known as matrix Langevin distribution[15][16]) has the density

 

supported on the Stiefel manifold of   orthonormal p-frames  , where   is an arbitrary   real matrix.[17][18]

Saw distributions

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Ulrich,[5] in designing an algorithm for sampling from the VMF distribution, makes use of a family of distributions named after and explored by John G. Saw.[19] A Saw distribution is a distribution on the  -sphere,  , with modal vector   and concentration  , and of which the density function has the form:

 

where   is a non-negative, increasing function; and where   is the normalization constant. The above-mentioned radial-tangential decomposition generalizes to the Saw family and the radial compoment,   has the density:

 

where   is the beta function. Also notice that the left-hand factor of the radial density is the surface area of  .

By setting  , one recovers the VMF distribution.

Weighted Rademacher Distribution

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The definition of the Von Mises-Fisher distribution can be extended to include also the case where  , so that the support is the 0-dimensional hypersphere, which when embedded into 1-dimensional Euclidean space is the discrete set,  . The mean direction is   and the concentration is  . The probability mass function, for   is:

 

where   is the logistic sigmoid. The expected value is  . In the uniform case, at  , this distribution degenerates to the Rademacher distribution.

See also

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References

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  1. ^ Fisher, R. A. (1953). "Dispersion on a sphere". Proc. R. Soc. Lond. A. 217 (1130): 295–305. Bibcode:1953RSPSA.217..295F. doi:10.1098/rspa.1953.0064. S2CID 123166853.
  2. ^ Watson, G. S. (1980). "Distributions on the Circle and on the Sphere". J. Appl. Probab. 19: 265–280. doi:10.2307/3213566. JSTOR 3213566. S2CID 222325569.
  3. ^ a b c d Mardia, Kanti; Jupp, P. E. (1999). Directional Statistics. John Wiley & Sons Ltd. ISBN 978-0-471-95333-3.
  4. ^ Embleton, N. I. Fisher, T. Lewis, B. J. J. (1993). Statistical analysis of spherical data (1st pbk. ed.). Cambridge: Cambridge University Press. pp. 115–116. ISBN 0-521-45699-1.{{cite book}}: CS1 maint: multiple names: authors list (link)
  5. ^ a b Ulrich, Gary (1984). "Computer generation of distributions on the m-sphere". Applied Statistics. 33 (2): 158–163. doi:10.2307/2347441. JSTOR 2347441.
  6. ^ Wood, Andrew T (1994). "Simulation of the Von Mises Fisher distribution". Communications in Statistics - Simulation and Computation. 23 (1): 157–164. doi:10.1080/03610919408813161.
  7. ^ Hornik, Kurt; Grün, Bettina (2014). "movMF: An R Package for Fitting Mixtures of Von Mises-Fisher Distributions". Journal of Statistical Software. 58 (10). doi:10.18637/jss.v058.i10. S2CID 13171102.
  8. ^ a b Pinzón, Carlos; Jung, Kangsoo (2023-03-03), Fast Python sampler for the von Mises Fisher distribution, retrieved 2023-03-30
  9. ^ De Cao, Nicola; Aziz, Wilker (13 Feb 2023). "The Power Spherical distribution". arXiv:2006.04437 [stat.ML].
  10. ^ Pakyuz-Charrier, Evren; Lindsay, Mark; Ogarko, Vitaliy; Giraud, Jeremie; Jessell, Mark (2018-04-06). "Monte Carlo simulation for uncertainty estimation on structural data in implicit 3-D geological modeling, a guide for disturbance distribution selection and parameterization". Solid Earth. 9 (2): 385–402. Bibcode:2018SolE....9..385P. doi:10.5194/se-9-385-2018. ISSN 1869-9510.
  11. ^ A., Wood, Andrew T. (1992). Simulation of the Von Mises Fisher distribution. Centre for Mathematics & its Applications, Australian National University. OCLC 221030477.{{cite book}}: CS1 maint: multiple names: authors list (link)
  12. ^ Gosmann, J; Eliasmith, C (2016). "Optimizing Semantic Pointer Representations for Symbol-Like Processing in Spiking Neural Networks". PLOS ONE. 11 (2): e0149928. Bibcode:2016PLoSO..1149928G. doi:10.1371/journal.pone.0149928. PMC 4762696. PMID 26900931.
  13. ^ a b Voelker, Aaron R.; Gosmann, Jan; Stewart, Terrence C. "Efficiently sampling vectors and coordinates from the n-sphere and n-ball" (PDF). Centre for Theoretical Neuroscience – Technical Report, 2017. Retrieved 22 April 2023.
  14. ^ Wang, Tongzhou; Isola, Phillip (2020). "Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere". International Conference on Machine Learning (ICML). arXiv:2005.10242.
  15. ^ Pal, Subhadip; Sengupta, Subhajit; Mitra, Riten; Banerjee, Arunava (2020). "Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold". Bayesian Analysis. 15 (3): 871–908. doi:10.1214/19-BA1176. ISSN 1936-0975.
  16. ^ Chikuse, Yasuko (1 May 2003). "Concentrated matrix Langevin distributions". Journal of Multivariate Analysis. 85 (2): 375–394. doi:10.1016/S0047-259X(02)00065-9. ISSN 0047-259X.
  17. ^ Jupp (1979). "Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions". The Annals of Statistics. 7 (3): 599–606. doi:10.1214/aos/1176344681.
  18. ^ Downs (1972). "Orientational statistics". Biometrika. 59 (3): 665–676. doi:10.1093/biomet/59.3.665.
  19. ^ Saw, John G (1978). "A family of distributions on the m-sphere and some hypothesis tests". Biometrika. 65 (`): 69–73. doi:10.2307/2335278. JSTOR 2335278.

Further reading

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  • Dhillon, I., Sra, S. (2003) "Modeling Data using Directional Distributions". Tech. rep., University of Texas, Austin.
  • Banerjee, A., Dhillon, I. S., Ghosh, J., & Sra, S. (2005). "Clustering on the unit hypersphere using von Mises-Fisher distributions". Journal of Machine Learning Research, 6(Sep), 1345-1382.
  • Sra, S. (2011). "A short note on parameter approximation for von Mises-Fisher distributions: And a fast implementation of I_s(x)". Computational Statistics. 27: 177–190. CiteSeerX 10.1.1.186.1887. doi:10.1007/s00180-011-0232-x. S2CID 3654195.