Identication and Estimation of Nonsymmetric Determinantal Point Processes

We study the geometry of determinantal point processes (DPPs) through the spectral decomposition $L=UΛU^{\top}$. The spectrum $Λ$ governs the cardinality distribution via elementary symmetric polynomials, while the eigenspace orientation $U$ governs the conditional law within each fixed-cardinality stratum. Conditioning on cardinality $k$ yields the $k$-DPP, for which the identifiability structure changes fundamentally: the spectral parameter becomes identifiable only up to a common scale, and the eigenspace rotation parameter is identifiable only through squared minors of the eigenvector matrix. We characterize the identifiability gap precisely, via three explicit invariances (scale, sign similarity, and eigenspace rotation) and a dimension-counting theorem showing the existence of additional continuous non-identifiability whenever $\binom{N}{k}<N(N+1)/2$. In contrast, for the full DPP the non-identifiability comes only from the discrete sign similarity.

Determinantal point processes (DPPs) model subset selection probability with repulsion, but many applications also exhibit attraction arising from pairwise relations or shared latent structures. Nonsymmetric DPPs (NDPPs) capture both effects, but their kernels are difficult to interpret and specify. We propose the determinant-ratio point process (DRPP), which models repulsion and attraction via a ratio of the determinants of two symmetric positive-definite kernels. This formulation preserves the feature-based interpretation of standard DPPs while allowing the two effects to be specified separately. Furthermore, we introduce an auxiliary-variable representation that enables efficient MCMC sampling and yields tractable expectation formulas for key statistics including the normalizing constant, which together facilitate maximum-likelihood learning. We theoretically characterize the interaction structures well suited to DRPPs: for clustered attraction with $C$ latent groups, DRPPs exactly realize the target pairwise interactions with an attraction rank $C$, whereas NDPPs can require a rank that grows with the number of items given the same repulsion component. Conversely, nontransitive pair-specific attraction may require compensation through the DRPP repulsion kernel. Experiments support these predictions, showing that DRPPs fit transitive attraction with substantially fewer parameters than NDPPs.

The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fdω$ relies on independent samples from $ω$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $ω$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.