From DPPs to $k$-DPPs: identifiability analysis via spectral decomposition
2026-05-25 • Machine Learning
Machine Learning
AI summaryⓘ
The authors explore how a certain kind of random process called a determinantal point process (DPP) is shaped by two main parts: the sizes of groups selected and the directions of underlying vectors. They show that when you fix the group size, the usual ways to identify the key parameters become less straightforward, with some information lost or only partially recoverable. They explain these identification challenges using specific mathematical rules and show that for some group sizes, extra hidden ambiguities arise. Overall, their work clarifies exactly when and how the core features of these processes can or cannot be uniquely determined.
Determinantal Point ProcessSpectral DecompositionEigenvaluesEigenvectorsIdentifiabilityCardinalityElementary Symmetric PolynomialsMatrix MinorsInvarianceDimension Counting
Authors
Hideitsu Hino, Keisuke Yano
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.