From DPPs to $k$-DPPs: identifiability analysis via spectral decomposition

2026-05-25

AI Summary

This study systematically investigates the fundamental differences in parameter identifiability between determinantal point processes (DPPs) and their fixed-cardinality variants, $k$-DPPs. By leveraging the spectral decomposition $L = U\Lambda U^\top$, it reveals that the cardinality distribution is governed solely by the eigenvalues, while the conditional distribution depends only on the directions of the eigenvectors. The work establishes, for the first time, three explicit invariances inherent to $k$-DPPs—scaling, sign similarity, and eigenspace rotation—and proves the existence of additional continuous non-identifiabilities whenever $\binom{N}{k} < N(N+1)/2$. In contrast, standard DPPs are affected only by discrete sign symmetries. This analysis fully characterizes the identifiability gap between DPPs and $k$-DPPs, providing a rigorous theoretical foundation for modeling and learning with these processes.

Abstract

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.

Problem

• determinantal point processes
• k-DPPs
• identifiability
• spectral decomposition
• parameter identifiability

Innovation

• determinantal point processes
• k-DPPs
• spectral decomposition
• identifiability
• eigenspace rotation