Multinomial symmetric positive definite matrices

MultinomialSymmetricPositiveDefinite <: AbstractMultinomialDoublyStochastic

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

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

where $\mathbf{1}_n$ and $\mathbr{0}_n$ are the vectors of length $n$ containing ones and zeros, respectively. More details about this manifold can be found in [DH19].



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


Generate a random point on MultinomialSymmetricPositiveDefinite manifold. The steps are as follows:

  1. Generate a random totally positive matrix a. Construct a vector L of n random positive increasing real numbers. b. Construct the Vandermonde matrix V based on the sequence L. c. Perform LU factorization of V in such way that both L and U components have positive elements. d. Convert the LU factorization into LDU factorization by taking the diagonal of U and dividing U by it, V=LDU. e. Construct a new matrix R = UDL which is totally positive.
  2. Project the totally positive matrix R onto the manifold of MultinomialDoubleStochastic matrices.
  3. Symmetrize the projected matrix and return the result.

This method roughly follows the procedure described in



A. Douik and B. Hassibi. Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry. IEEE Transactions on Signal Processing 67, 5761–5774 (2019), arXiv:1802.02628.