From DPPs to $k$-DPPs: identifiability analysis via spectral decomposition
For researchers using DPPs in machine learning and statistics, this work clarifies fundamental parameter estimation limitations, though the results are theoretical and incremental.
The paper analyzes identifiability of determinantal point processes (DPPs) and their cardinality-conditioned variants (k-DPPs), showing that k-DPPs suffer from additional continuous non-identifiability due to scale and eigenspace rotation invariances, unlike full DPPs which only have discrete sign ambiguity.
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.