Convex Hull of Outer Products of (Normalised) Nonnegative Vectors

If I define $$\mathcal{A} = \{ xx^T : x \in \mathbb{R}^d, \| x \|_2 \leqslant 1 \}$$, then (assuming I recall correctly) it is known that the convex hull of $$\mathcal{A}$$ is given by

\begin{align} \text{conv} (\mathcal{A}) = \{ M \in \text{Mat}_{d \times d} (\mathbf{R}): M = M^T, \| M \|_* = 1\}, \end{align}

where $$\| \cdot \|_*$$ is the nuclear norm.

Suppose I now define $$\mathcal{A}_+ = \{ xx^T : x \in \mathbb{R}_+^d, \| x \|_2 \leqslant 1\}$$, i.e. I restrict to taking outer products of nonnegative vectors $$x$$ with themselves.

My question is: Does there exist a similar characterisation of $$\text{conv} (\mathcal{A}_+)$$ ?

If context is useful: I'm interested in understanding and characterising when it is possible to write a symmetric probability distributions on two variables as a mixture of i.i.d. distributions, i.e. if $$p(x,y) = p(y, x)$$, when does there exist

• some parameter space $$\Theta$$,
• some family of distributions $$\{ p(\cdot | \theta) \}_{\theta \in \Theta}$$, and
• some (prior) distribution $$\pi$$ such that

\begin{align} p(x, y) = \int_{\Theta} p(x|\theta) \, p(y|\theta) \, \pi(\theta) \, d\theta? \end{align}

I recognise that this has some links to de Finetti's theorem, but I'm not yet sure whether that link can be turned into an answer.

Your characterization of $$\text{conv} (\mathcal{A})$$ needs one additional restriction---that $$M$$ is positive semidefinite (the equivalence of these two sets follows fairly quickly from the spectral decomposition).
For $$\text{conv} (\mathcal{A}_+)$$, the convex hull is the exact same, but with the positive semidefiniteness requirement replaced by a complete positivity requirement. Since checking complete positivity is NP-hard, don't expect an easy way of determining membership in this convex hull.
• To add to the completely positive part of the answer: while checking membership of the completely positive cone is NP-hard, you can use the set of doubly non-negative matrices (i.e. positive semidefinite and entry-wise non-negative matrices): this is a valid outer approximation which is semidefinitely representable, and agrees with the copositive cone for $n \leq 4$, see sciencedirect.com/science/article/pii/S002437950900281X. – ryanseadub May 7 at 3:52