(throughout, "DPP" denotes "Determinantal Point Process")
TL;DR: Discrete DPPs are straightforward to compute with, continuous DPPs less so. Can we approximate continuous DPPs well with suitable discrete DPPs?
Given a discrete index set $S$, the DPP with kernel $K$ over the set $S$ is the random point process $X$ such that for any $A \subseteq S$,
\begin{align} \mathbf{P} ( A \subseteq X ) = \mathrm{det} ( K_A ), \end{align}
where $K_A$ is the $|A| \times |A|$ matrix with entries $K(u, v)$ for $u, v \in A$. I will not go into details about the properties of $K$, existence and uniqueness of the process, etc.
Anyways, broadly speaking, discrete DPPs are relatively easy to work with. Among other things, simulation from discrete DPPs can be carried out with a complexity which is polynomial in the size of $S$.
Consider now a DPP with continuous index set $S$ (thinking of the interval $[0, 1]$ is a reasonable reference point), kernel $K$, and a reference probability measure $\mu \in \mathcal{P} ( S )$, where the corresponding random variable $X$ is again a finite subset of $S$, so that for any disjoint collection of sets $A_1, \cdots, A_M \subseteq S$,it holds that
\begin{align} \mathbf{E} \left[ \prod_{m \in [M]} | A_m \cap X | \right] = \int_{A_1 \times \cdots \times A_M} \mu (\mathrm{d} x_1) \cdots \mu (\mathrm{d} x_M) \cdot \mathrm{det} K(x_{[M]}), \end{align}
where $K(x_{[M]})$ is the $M \times M$ matrix with elements $K(x_i, x_j)$ for $i, j \in [M]$.
By contrast, my understanding is that computing with continuous DPPs is not quite so simple; even with nice $\mu$ and generic $K$, it is not entirely obvious how to draw a sample from a continuous DPP (at least to me!).
Anyways, I am curious about whether the following heuristic approach to sampling from such a continuous DPP { would be well-founded, has been studied before, etc. }:
- Draw many iid samples from $\mu$; call this collection $\hat{S}$.
- Simulate a draw from a discrete DPP with base set $\hat{S}$, and some kernel $\hat{K}$ which is derived from $K$, and can depend on $\hat{S}$.
I would expect that this approach will generally not generate perfect samples from the continuous DPP, but that for some version of this idea, the bias will be controllable (e.g. as I take sufficiently many samples in step 1, the approximation error decays to 0).
More broadly, my question is essentially whether approximate simulation from continuous DPPs can be systematically reduced to exact simulation from (random) discrete DPPs, in a fairly generic way.
