Reference

AngularCentralGaussian(M; params...)

The Angular Central Gaussian (ACG) distribution on the manifold $M$.

Accepted manifolds are Sphere, ProjectiveSpace, Stiefel, Grassmann, Rotations and SpecialOrthogonal.

For manifolds with matrix points, this is also called the Matrix Angular Central Gaussian distribution.

Constructors

AngularCentralGaussian(M; P)
AngularCentralGaussian(M; L)

For a manifold $M ⊂ 𝔽^{n × k}$, construct the ACG distribution parameterized either by the inverse $P = Σ^{-1}$ of an $n × n$ positive definite matrix $Σ$ or by the lower Cholesky factor $L$ of $Σ$, such that $Σ = L L^\mathrm{H}$.

Bingham(M; params...)

The Bingham distribution on the manifold $M$.

Accepted manifolds are Sphere, ProjectiveSpace, Stiefel, Grassmann, Rotations and SpecialOrthogonal.

For manifolds with matrix points, this is also called the Matrix Bingham distribution.

Constructors

Bingham(M; A)

For a manifold $M ⊂ 𝔽^{n × k}$, construct the Bingham distribution parameterized by some positive definite matrix $A ∈ 𝔽^{n × n}$.

The density function with respect to the normalized Hausdorff measure on $M$ is

\[p(x | A) = \frac{\exp(\Re⟨x, Ax⟩)}{_1 F_1(\frac{k}{2}, \frac{n}{2}; A)},\]

where $⟨⋅,⋅⟩$ is the Frobenius inner product, and $_1 F_1(a, b; A)$ is a hypergeometric function with matrix argument $A$.

Note that $p(x | A + α I) = p(x | A)$ for all scalars $α ∈ 𝔽$. Hence, $A$ can not be uniquely identified from draws from the Bingham distribution.

Haar{G,D}

The Haar measure on a group manifold.

The Haar measure on a group $G$ is a measure that is invariant to all left- and/or right- group translations on the manifold. That is, given the $τ$-invariant Haar measure $μ$ and some subset $H ⊆ G$, then $∫_H \mathrm{d}μ(p) = ∫_H \mathrm{d}μ(τ_q(p))$ for all $q ∈ G$, where $τ_q(p)$ is group translation of $p$ by $q$ in the $τ$ direction (left- or right-).

The convenient aliases LeftHaar and RightHaar are also provided.

Constructors

Haar(G::AbstractManifold, D::ActionDirection = LeftAction())

Construct the D-invariant Haar measure on group manifold G.

Hausdorff{M,A} <: PrimitiveMeasure

The un-normalized Hausdorff measure on a manifold.

The Hausdorff measure generalizes the notion of area or volume to a manifold that is embedded in a metric space. That is, the mass of the measure over some region of the manifold is the area/volume of that region in the embedded space.

Constructors

Hausdorff(M::AbstractManifold)

Constructs the Hausdorff measure for the manifold M using the default embedding of the manifold.

LeftHaar{G<:AbstractManifold}

Alias for a left-Haar measure.

RightHaar{G<:AbstractManifold}

Alias for a right-Haar measure.

VonMises(𝔽=ℝ; params...)

The von Mises distribution on the real or complex Circle.

This is just a convenient alias for VonMisesFisher(Circle(𝔽); params...).

Constructors

VonMises(μ=π, κ=1)
VonMises(ℂ; μ=im, κ=1)
VonMises(ℂ; c=3+4im)

VonMisesFisher(M; params...)

The von Mises-Fisher (vMF) or Langevin distribution on the Sphere or Stiefel manifold M.

Given a matrix $X ∈ 𝔽^{n × k}$ with IID entries $X_{ij} ∼ \mathrm{Normal}(F_{ij}, 1)$ for $F ∈ 𝔽^{n × k}$, the restriction of the corresponding distribution in $𝔽^{n × k}$ to the Stiefel(n, k, 𝔽) manifold, that is, the matrices for which $X^\mathrm{H} X = I_k$, is the vMF distribution on the Stiefel manifold. The vMF distribution can also be specified for any submanifold of the Stiefel manifold, including the Sphere and the Circle.

Parameterizations

VonMisesFisher(M::AbstractSphere{𝔽}; params...)
VonMisesFisher(n::Int[, 𝔽]; params...)

Construct the vMF distribution on Sphere(n-1,𝔽)== $𝔽𝕊^{n-1}$`.

Implemented parameterizations are:

• (μ, κ): the modal direction $μ ∈ 𝔽𝕊^{n-1}$ and concentration $κ ∈ ℝ⁺$
• (c,): $c = κ μ ∈ 𝔽^n$, the mean of the normal distribution in the embedded space.

The density of the vMF distribution on $𝔽𝕊^{n-1}$ with respect to the normalized Hausdorff measure is

\[p(x | μ, κ) = \frac{Γ(ν + 1)κ^ν}{2^ν I_ν(κ)} \exp(κ \Re⟨μ, x⟩)),\]

where $ν = n/2-1$, $⟨⋅,⋅⟩$ is the Frobenius inner product, and $I_ν(z)$ is the modified Bessel function of the first kind.

VonMisesFisher(M::Stiefel{n,k,𝔽}; params...)
VonMisesFisher(n::Int, k::Int[, 𝔽]; params...)

Construct the matrix vMF distribution on Stiefel(n, k, 𝔽)= $\mathrm{St}(n, k, 𝔽)$.

Implemented parameterizations are:

• (F,): a parameter matrix $F ∈ 𝔽^{n × k}$, the mean of the normal distribution in the embedded space.
• (U, D, V): The SVD decomposition of $F = U D V$, where $U ∈ \mathrm{St}(n, k, 𝔽)$ and $V ∈ \mathrm{U}(k, 𝔽)$.
• (H, P): The polar decomposition of $F = H P$, where $H ∈ \mathrm{St}(n, k, 𝔽)$ is the mode, and $P ∈ 𝔽^{k × k}$ is a Hermitian positive definite matrix.

The density of the vMF distribution on Stiefel(n, k, 𝔽) with respect to the normalized Hausdorff measure is

\[p(x | F) = \frac{\exp(\Re⟨F, x⟩)}{_0 F_1(\frac{n}{2}; \frac{1}{4} F^\mathrm{H}F)},\]

where $_0 F_1(b; B)$ is a hypergeometric function with matrix argument $B$.

logmass(μ::AbstractMeasure)

Compute the logarithm of the total mass of the measure μ over its manifold M, that is $μ(M) = ∫_M \mathrm{d}μ(x)$.