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.
Figures & tables
Figure 1: Number of parameters vs. validation NLL (lower is better) on the synthetic datasets (a)–(c) and the real datasets (d)–(f). Each point corresponds to one hyperparameter setting.
Figure 2: Comparison between the rank of DRPPs and that of NDPPs on cluster dataset. Error bars indicate mean ± standard deviation over five independent runs.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Raw
∣A∣≥2
Dataset
N
Split
Train
Valid
Test
Train
Valid
Test
Texcess
Amazon
300
70/15/15
21,852
4,682
4,684
21,852
4,682
4,684
+0.582
COCO
80
70/15/15
82,800
17,743
17,744
65,155
13,962
13,963
+0.290
KDD
64
given
180,092
19,907
34,955
15,841
1,778
2,668
+0.208
Appendix
Table 1: Real datasets: numbers of subsets before (raw) and after removing subsets with fewer than two items, and transitivity.
Determinantal point processes (DPPs) on a finite set are probability models that exhibit repulsion where inter-item similarities are encoded by a positive definite kernel matrix. Scaling DPP learning to large datasets typically requires imposing structural constraints on the kernel matrix, such as a low-rank-plus-diagonal form. However, structural assumptions introduce a severe trade-off between expressive power and computational scalability. Our study addresses this challenge by (a) modeling the full-rank interactions via a nonlinear kernel function and latent features, (b) designing a surrogate function for the maximum likelihood estimator, and (c) adopting stochastic optimization using random features of the nonlinear kernel. Numerical experiments on real-world datasets demonstrate that our method significantly improves this trade-off, achieving better generalization performance with remarkable parameter efficiency.
Takahiro Kawashima, Hideitsu Hino
ZOZO Research · Institute of Statistical Mathematics · Waseda University
We introduce the distributional determinantal point process (dDPP) as a novel repulsive point process whose atoms are probability distributions rather than points in a real space. The dDPP is constructed via an L-ensemble with a sliced Wasserstein (SW) kernel between distributions. We show its validity as a well-defined point process. In the discrete setting, we derive concentration results for plug-in estimators of the L-ensemble, the correlation kernel, and their determinants given i.i.d. samples from the distributional atoms. Leveraging this framework, we propose a distribution-valued random partition model by way of a repulsive generalized Bayesian mixture model. The model places a dDPP prior over the atoms of the mixing measure and defines a generalized likelihood based on SW distance. To summarize posterior inference, we develop a decision-theoretic approach to report a point estimate of the mixing measure as a Bayes rule under a hierarchical optimal transport utility function. The latter is a natural choice given that the mixing measure is itself a distribution over distributions. We use the proposed framework for inference with single-cell gene expression data and human epilepsy data, producing interpretable and well-separated clusters that reflect meaningful structure in the data.
Khai Nguyen, Yang Ni, Elizabeth Juarez-Colunga +1
Department of Statistics and Data Sciences, University of Texas at Austin · Department of Biostatistics and Informatics, University of Colorado Anschutz · Department of Mathematics, University of Texas at Austin
Determinantal point processes (DPPs) provide tractable distributions for nite random sets, but the usual symmetric-kernel specication imposes nonpositive pairwise dependence. Statistical research on nonsymmetric DPPs, which can accommodate positive dependence, remains limited. We study identication and estimation for this class. First, we establish the equivalence of two sucient conditions for a nonsymmetric kernel to dene a valid DPP. Second, we develop the identication theory for nonsymmetric DPPs. Third, we develop cardinality-truncated maximum-likelihood estimation and estimation by averaged stochastic gradient descent. Under suitable identication and regularity conditions, we establish consistency and asymptotic normality for both estimators. Simulations examine identication, the eects of truncation, and the nite-sample performance of the estimators. We illustrate the methods with two cyber applications at markedly dierent sampling frequencies: weekly sector-by-breach data and second-by-second enterprise authentication data with more than ve million set observations.
Christian Gouriéroux, Yang Lu
University of Toronto, Toulouse School of Economics and CREST · Department of Mathematics and Statistics, Concordia University, Montreal, Canada