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

arXiv: https://arxiv.org/abs/2605.25526 · HTML: https://mlsanity.com/paper/2605.25526/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 8 references that are in the map.
- First posted on arXiv 25 May 2026.

## Builds on (most cited first, 1 of 1)

- [Expectation-Maximization for Learning Determinantal Point Processes](https://mlsanity.com/paper/1411.1088/) (2014)

## Built on by (most cited first, 2 of 2)

- [Exact Calibration and Sharp Risk Geometry for Volume-Sampled Ridge Regression](https://mlsanity.com/paper/2610.07721/) (2026)
- [Scalable Learning of Determinantal Point Processes with Kernel-Structured Interactions](https://mlsanity.com/paper/2610.12829/) (2026)

## Abstract (arXiv)

We study the geometry of determinantal point processes (DPPs) through the spectral decomposition L=ULambda Utop. The spectrum Lambda 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 binomNk<N(N+1)/2. In contrast, for the full DPP the non-identifiability comes only from the discrete sign similarity.