# Multinomial symmetric matrices

Manifolds.MultinomialSymmetricType
MultinomialSymmetric{n} <: AbstractMultinomialDoublyStochastic{N}

The multinomial symmetric matrices manifold consists of all symmetric $n×n$ matrices with positive entries such that each column sums to one, i.e.

\begin{aligned} \mathcal{SP}(n) \coloneqq \bigl\{p ∈ ℝ^{n×n}\ \big|\ &p_{i,j} > 0 \text{ for all } i=1,…,n, j=1,…,m,\\ & p^\mathrm{T} = p,\\ & p\mathbf{1}_n = \mathbf{1}_n \bigr\}, \end{aligned}

where $\mathbf{1}_n$ is the vector of length $n$ containing ones.

It is modeled as IsIsometricEmbeddedManifold. via the AbstractMultinomialDoublyStochastic type, since it shares a few functions also with AbstractMultinomialDoublyStochastic, most and foremost projection of a point from the embedding onto the manifold.

The tangent space can be written as

$$$T_p\mathcal{SP}(n) \coloneqq \bigl\{ X ∈ ℝ^{n×n}\ \big|\ X = X^{\mathrm{T}} \text{ and } X\mathbf{1}_n = \mathbf{0}_n \bigr\},$$$

where $\mathbf{0}_n$ is the vector of length $n$ containing zeros.

More details can be found in Section IV[DouikHassibi2019].

Constructor

MultinomialSymmetric(n)

Generate the manifold of matrices $\mathbb R^{n×n}$ that are doubly stochastic and symmetric.

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ManifoldsBase.check_vectorMethod
check_vector(M::MultinomialSymmetric p, X; kwargs...)

Checks whether X is a valid tangent vector to p on the MultinomialSymmetric M. This means, that p is valid, that X is of correct dimension, symmetric, and sums to zero along any row.

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ManifoldsBase.projectMethod
project(M::MultinomialSymmetric{n}, p, Y) where {n}

Project Y onto the tangent space at p on the MultinomialSymmetric M, return the result in X. The formula reads

$$$\operatorname{proj}_p(Y) = Y - (α\mathbf{1}_n^{\mathrm{T}} + \mathbf{1}_n α^{\mathrm{T}}) ⊙ p,$$$

where $⊙$ denotes the Hadamard or elementwise product and $\mathbb{1}_n$ is the vector of length $n$ containing ones. The two vector $α ∈ ℝ^{n×n}$ is given by solving

$$$(I_n+p)α = Y\mathbf{1},$$$

where $I_n$ is teh $n×n$ unit matrix and $\mathbf{1}_n$ is the vector of length $n$ containing ones.

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ManifoldsBase.retractMethod
retract(M::MultinomialSymmetric, p, X, ::ProjectionRetraction)

compute a projection based retraction by projecting $p\odot\exp(X⨸p)$ back onto the manifold, where $⊙,⨸$ are elementwise multiplication and division, respectively. Similarly, $\exp$ refers to the elementwise exponentiation.

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## Literature

• DouikHassibi2019

A. Douik, B. Hassibi: AbstractManifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry, IEEE Transactions on Signal Processing 67(22), pp. 5761–5774, 2019. doi: 10.1109/tsp.2019.2946024, arXiv: 1802.02628.