Scalable Learning of Determinantal Point Processes with Kernel-Structured Interactions

Takahiro Kawashima; Hideitsu Hino

stat.MLOct 9, 2026

Scalable Learning of Determinantal Point Processes with Kernel-Structured Interactions

Authors: Takahiro Kawashima, Hideitsu Hino

Organizations: ZOZO Research · Institute of Statistical Mathematics · Waseda University

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Results on the MovieLens dataset. Each marker shows the median and each bar corresponds to minimum/maximum values of $3$ trials.

Panel of Figure 2 — Figure 2: Results on the Instacart dataset. Each marker shows the median and each bar corresponds to minimum/maximum values of $3$ trials.

Method	N=200	N=500	N=1000	N=2000	N=3000
MM	$-98.06^{\scriptscriptstyle-95.57}_{\scriptscriptstyle-98.37}$	$-191.33^{\scriptscriptstyle-185.40}_{\scriptscriptstyle-192.71}$	$-297.04^{\scriptscriptstyle-285.09}_{\scriptscriptstyle-297.94}$	$-420.40^{\scriptscriptstyle-401.43}_{\scriptscriptstyle-421.32}$	$-490.63^{\scriptscriptstyle-467.06}_{\scriptscriptstyle-491.55}$
LR+D r=32	$-134.95^{\scriptscriptstyle-128.61}_{\scriptscriptstyle-135.55}$	$-264.13^{\scriptscriptstyle-250.62}_{\scriptscriptstyle-266.96}$	$-394.02^{\scriptscriptstyle-370.97}_{\scriptscriptstyle-396.16}$	$-528.06^{\scriptscriptstyle-495.67}_{\scriptscriptstyle-530.21}$	$-598.72^{\scriptscriptstyle-561.71}_{\scriptscriptstyle-601.60}$
LR+D r=r max	$-98.24^{\scriptscriptstyle-95.97}_{\scriptscriptstyle-98.54}$	$-192.29^{\scriptscriptstyle-186.56}_{\scriptscriptstyle-193.32}$	$-299.63^{\scriptscriptstyle-287.62}_{\scriptscriptstyle-301.14}$	$-425.11^{\scriptscriptstyle-406.19}_{\scriptscriptstyle-426.03}$	$-497.28^{\scriptscriptstyle-474.39}_{\scriptscriptstyle-499.02}$
KS-DPP d=32	$-99.12^{\scriptscriptstyle-96.17}_{\scriptscriptstyle-99.40}$	$-193.79^{\scriptscriptstyle-186.30}_{\scriptscriptstyle-194.33}$	$-299.17^{\scriptscriptstyle-286.61}_{\scriptscriptstyle-300.81}$	$-423.00^{\scriptscriptstyle-402.65}_{\scriptscriptstyle-424.42}$	$-493.22^{\scriptscriptstyle-468.52}_{\scriptscriptstyle-494.52}$

Table 1: Comparison of the test mean log-likelihoods with the exact MM method on MovieLens . The values denote

\mbox{med}^{\mbox{max}}_{\mbox{min}}

of 3 trials.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Method	N
	200	500	1000	2000	3000
MM	$-98.06^{\scriptscriptstyle-95.57}_{\scriptscriptstyle-98.37}$	$-191.33^{\scriptscriptstyle-185.40}_{\scriptscriptstyle-192.71}$	$-297.04^{\scriptscriptstyle-285.09}_{\scriptscriptstyle-297.94}$	$-420.40^{\scriptscriptstyle-401.43}_{\scriptscriptstyle-421.32}$	$-490.63^{\scriptscriptstyle-467.06}_{\scriptscriptstyle-491.55}$
LR+D r=4	$-178.58^{\scriptscriptstyle-171.02}_{\scriptscriptstyle-179.68}$	$-311.40^{\scriptscriptstyle-296.57}_{\scriptscriptstyle-314.58}$	$-442.03^{\scriptscriptstyle-417.56}_{\scriptscriptstyle-444.55}$	$-575.97^{\scriptscriptstyle-542.57}_{\scriptscriptstyle-578.60}$	$-646.88^{\scriptscriptstyle-608.39}_{\scriptscriptstyle-649.53}$
LR+D r=8	$-169.19^{\scriptscriptstyle-161.78}_{\scriptscriptstyle-170.39}$	$-301.91^{\scriptscriptstyle-287.26}_{\scriptscriptstyle-305.11}$	$-432.51^{\scriptscriptstyle-408.20}_{\scriptscriptstyle-435.13}$	$-566.57^{\scriptscriptstyle-533.25}_{\scriptscriptstyle-569.30}$	$-637.41^{\scriptscriptstyle-599.12}_{\scriptscriptstyle-640.18}$
LR+D r=16	$-154.29^{\scriptscriptstyle-147.25}_{\scriptscriptstyle-155.49}$	$-286.06^{\scriptscriptstyle-271.78}_{\scriptscriptstyle-289.27}$	$-416.61^{\scriptscriptstyle-392.61}_{\scriptscriptstyle-419.19}$	$-550.56^{\scriptscriptstyle-517.61}_{\scriptscriptstyle-553.24}$	$-621.30^{\scriptscriptstyle-583.51}_{\scriptscriptstyle-624.31}$
LR+D r=32	$-134.95^{\scriptscriptstyle-128.61}_{\scriptscriptstyle-135.55}$	$-264.13^{\scriptscriptstyle-250.62}_{\scriptscriptstyle-266.96}$	$-394.02^{\scriptscriptstyle-370.97}_{\scriptscriptstyle-396.16}$	$-528.06^{\scriptscriptstyle-495.67}_{\scriptscriptstyle-530.21}$	$-598.72^{\scriptscriptstyle-561.71}_{\scriptscriptstyle-601.60}$

Appendix

Table C.1: Comparison of the test mean log-likelihoods with the exact MM method on MovieLens . The values denote

\mbox{med}^{\mbox{max}}_{\mbox{min}}

of 3 trials.

Determinantal Point Process

Determinantal point processes model and sample diverse subsets using determinant-based probabilities.

See all 18 papers

Generalized Determinantal Point Processes for Attraction and Repulsion with Distinct Interaction Kernels
Oct 9, 2026Hideaki Ishibashi, Takahiro Kawashima, Hideitsu HinoDeterminantal Point Process
Kyushu Institute of Technology · ZOZO Research · Institute of Statistical Mathematics +1
Spectral DPPs via NEPv: A Scalable Continuous Relaxation of Determinantal MAP for Diversity-Aware Data Selection
Jun 17, 2026Richard Yi Da XuDeterminantal Point Process Training Data Selection
Hong Kong Baptist University · TadReamk Limited
Fast determinantal sampling on general spaces and diffusion geometry
Jul 7, 2026Hoang-Son Tran, Pranav Gupta, Subhroshekhar GhoshDeterminantal Point Process
Department of Mathematics National University of Singapore

Oct 9, 2026J/K move · Enter open · S save

Generalized Determinantal Point Processes for Attraction and Repulsion with Distinct Interaction Kernels

Hideaki Ishibashi, Takahiro Kawashima, Hideitsu Hino

Kyushu Institute of Technology · ZOZO Research · Institute of Statistical Mathematics+1

Determinantal Point Process

Open paper AskPDF arXiv

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.

Abstract

Determinantal Point Process

Explore similar work

Generalized Determinantal Point Processes for Attraction and Repulsion with Distinct Interaction Kernels

Spectral DPPs via NEPv: A Scalable Continuous Relaxation of Determinantal MAP for Diversity-Aware Data Selection

Fast determinantal sampling on general spaces and diffusion geometry