Multinomial matrices

Manifolds.MultinomialMatrices — Type
MultinomialMatrices{n,m} <: AbstractPowerManifold{ℝ}

The multinomial manifold consists of m column vectors, where each column is of length n and unit norm, i.e.

\[\mathcal{MN}(n,m) \coloneqq \bigl\{ p ∈ ℝ^{n×m}\ \big|\ p_{i,j} > 0 \text{ for all } i=1,…,n, j=1,…,m \text{ and } p^{\mathrm{T}}\mathbb{1}_m = \mathbb{1}_n\bigr\},\]

where $\mathbb{1}_k$ is the vector of length $k$ containing ones.

This yields exactly the same metric as considering the product metric of the probablity vectors, i.e. PowerManifold of the $(n-1)$-dimensional ProbabilitySimplex.

The ProbabilitySimplex is stored internally within M.manifold, such that all functions of AbstractPowerManifold can be used directly.


MultinomialMatrices(n::Int, m::Int; parameter::Symbol=:type)

Generate the manifold of matrices $ℝ^{n×m}$ such that the $m$ columns are discrete probability distributions, i.e. sum up to one.

parameter: whether a type parameter should be used to store n and m. By default size is stored in type. Value can either be :field or :type.



Most functions are directly implemented for an AbstractPowerManifold with ArrayPowerRepresentation except the following special cases:

ManifoldDiff.riemannian_gradient — Method
riemannian_gradient(M::MultinomialMatrices, p, Y; kwargs...)

Let $Y$ denote the Euclidean gradient of a function $\tilde f$ defined in the embedding neighborhood of M, then the Riemannian gradient is given by Equation 5 of [DH19] as

\[ \operatorname{grad} f(p) = \proj_{T_p\mathcal M}(Y⊙p)\]

where $⊙$ denotes the Hadamard or elementwise product.