The probability simplex

ProbabilitySimplex{n} <: AbstractEmbeddedManifold{ℝ,DefaultEmbeddingType}

The (relative interior of) the probability simplex is the set

where $\mathbb{1}=(1,…,1)^{\mathrm{T}}∈ ℝ^{n+1}$ denotes the vector containing only ones.

This set is also called the unit simplex or standard simplex.

The tangent space is given by

The manifold is implemented assuming the Fisher-Rao metric for the multinomial distribution, which is equivalent to the induced metric from isometrically embedding the probability simplex in the $n$-sphere of radius 2. The corresponding diffeomorphism $\varphi: \mathbb Δ^n → \mathcal N$, where $\mathcal N \subset 2𝕊^n$ is given by $\varphi(p) = 2\sqrt{p}$.

This implementation follows the notation in ^{[ÅströmPetraSchmitzerSchnörr2017]}.

SoftmaxInverseRetraction <: AbstractInverseRetractionMethod

Describes an inverse retraction that is based on the softmax function.

SoftmaxRetraction <: AbstractRetractionMethod

Describes a retraction that is based on the softmax function.

exp(M::ProbabilitySimplex,p,X)

Compute the exponential map on the probability simplex.

where $X_p = \frac{X}{\sqrt{p}}$, with its division meant elementwise, as well as for the operations $X_p^2$ and $\sqrt{p}$.

log(M::ProbabilitySimplex, p, q)

Compute the logarithmic map of p and q on the ProbabilitySimplex M.

where $pq$ and $\sqrt{p}$ is meant elementwise.

check_manifold_point(M::ProbabilitySimplex, p; kwargs...)

Check whether p is a valid point on the ProbabilitySimplex M, i.e. is a point in the embedding with positive entries that sum to one The tolerance for the last test can be set using the kwargs....

check_tangent_vector(M::ProbabilitySimplex, p, X; check_base_point = true, kwargs... )

Check whether X is a tangent vector to p on the ProbabilitySimplex M, i.e. after check_manifold_point(M,p), X has to be of same dimension as p and its elements have to sum to one. The optional parameter check_base_point indicates, whether to call check_manifold_point for p or not. The tolerance for the last test can be set using the kwargs....

distance(M,p,q)

Compute the distance between two points on the ProbabilitySimplex M. The formula reads

injectivity_radius(M,p)

compute the injectivity radius on the ProbabilitySimplex M at the point p, i.e. the distanceradius to a point near/on the boundary, that could be reached by following the geodesic.

inner(M::ProbabilitySimplex,p,X,Y)

Compute the inner product of two tangent vectors X, Y from the tangent space $T_pΔ^n$ at p. The formula reads

inverse_retract(M::ProbabilitySimplex, p, q, ::SoftmaxInverseRetraction)

Compute a first order approximation by projection. The formula reads

where $\mathbb{1}^{m,n}$ is the size (m,n) matrix containing ones, and $\log$ is applied elementwise.

manifold_dimension(M::ProbabilitySimplex{n})

Returns the manifold dimension of the probability simplex in $ℝ^{n+1}$, i.e.

project(M::ProbabilitySimplex, p, Y)

project Y from the embedding onto the tangent space at p on the ProbabilitySimplex M. The formula reads

representation_size(::ProbabilitySimplex{n})

return the representation size of points in the $n$-dimensional probability simplex, i.e. an array size of (n+1,).

retract(M::ProbabilitySimplex, p, X, ::SoftmaxRetraction)

Compute a first order approximation by applying the softmax function. The formula reads

where multiplication, exponentiation and division are meant elementwise.

zero_tangent_vector(M::ProbabilitySimplex,p)

returns the zero tangent vector in the tangent space of the point p from the ProbabilitySimplex M, i.e. its representation by the zero vector in the embedding.

mean(
    M::ProbabilitySimplex,
    x::AbstractVector,
    [w::AbstractWeights,]
    method = GeodesicInterpolation();
    kwargs...,
)

Compute the Riemannian mean of x using GeodesicInterpolation.

FisherRaoMetric <: Metric

The Fisher-Rao metric or Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space.

See for example the ProbabilitySimplex.