  The likelihood model of high dimensional data $X_n$ can often be expressed as
$p(X_n|Z_n,\theta)$, where $\theta\mathrel{\mathop:}=(\theta_k)_{k\in[K]}$ is a
collection of hidden features shared across objects, indexed by $n$, and $Z_n$
is a non-negative factor loading vector with $K$ entries where $Z_{nk}$
indicates the strength of $\theta_k$ used to express $X_n$. In this paper, we
introduce random function priors for $Z_n$ for modeling correlations among its
$K$ dimensions $Z_{n1}$ through $Z_{nK}$, which we call \textit{population
random measure embedding} (PRME). Our model can be viewed as a generalized
paintbox model~\cite{Broderick13} using random functions, and can be learned
efficiently with neural networks via amortized variational inference. We derive
our Bayesian nonparametric method by applying a representation theorem on
separately exchangeable discrete random measures.
