Identication and Estimation of Nonsymmetric Determinantal Point Processes

Christian Gouriéroux; Yang Lu

stat.MLOct 10, 2026

Identication and Estimation of Nonsymmetric Determinantal Point Processes

Authors: Christian Gouriéroux, Yang Lu

Organizations: University of Toronto, Toulouse School of Economics and CREST · Department of Mathematics and Statistics, Concordia University, Montreal, Canada

Abstract

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.

Figures & tables

$T$	Mean $N_{T}$	Median $e_{T}$	$\operatorname{RMSE}(\widehat{\beta}_{T})$	$\chi^{2}_{3}$ size	$\chi^{2}_{6}$ size
500	306.9	0.327	0.753	0.050	0.007
2000	1226.9	0.172	0.390	0.051	0.004
8000	4906.5	0.079	0.126	0.054	0.007

Table 1: Estimation of identified coordinates and rank-score inference. The table uses 2,000 MLE replications and 5,000 independent score replications per row. The penultimate column uses the critical value with three degrees of freedom, equal to the information rank; the final column uses six degrees of freedom, equal to the ambient parameter dimension.

Design	$\alpha_{0,0}$	$\rho_{0}$	$\mathbb{E}\|S\|$	$\pi_{4}$	$\pi_{5}$
Baseline	$-0.863$	$0.60$	$2.5$	$0.933$	$0.986$
Dense	$0.186$	$0.60$	$4.5$	$0.494$	$0.759$
Weak	$-0.821$	$0.15$	$2.5$	$0.935$	$0.987$

Table 2: Data-generating processes and exact retention probabilities

Design	$\bar{n}$	Mean $N_{T}$	RMSE $(\widehat{\rho})$	Trace efficiency	Regular share	Score size at $5\%$
Baseline	4	3730.0	0.180	0.745	0.985	0.060
	5	3945.5	0.159	0.902	0.989	0.058
	8	4000.0	0.150	1.000	0.993	0.058
Dense	4	1976.0	0.216	0.417	0.959	0.044
	5	3037.9	0.163	0.690	0.988	0.045
	8	4000.0	0.133	1.000	0.997	0.040

Table 3: MLE accuracy and score size at

T=4{,}000

. There are 2,000 MLE replications and 2,000 independent score replications per cell. Regular share is the proportion of MLE replications with an interior estimate and invertible plug-in information. The score statistic uses exact null information.

$\bar{n}$	Mean $N_{T}$	Size of $Q_{N_{T}}$	Size of $Q_{T}^{\mathrm{cal}}$
4	1975.2	0.044	0.267
5	3037.5	0.045	0.107
8	4000.0	0.040	0.040

Table 4: Effective-sample-size normalization in the dense design at

T=4{,}000

. Each row uses 2,000 independent replications and the

5\%

\chi^{2}_{3}

critical value; all replications enter.

Design	$\bar{n}$	Mean $N_{T}$	MLE RMSE	ASGD RMSE	ASGD/MLE
Baseline	4	238717	0.0221	0.2152	9.75
	5	252499	0.0200	0.2222	11.10
	8	256000	0.0186	0.2712	14.60
Dense	4	126386	0.0231	0.4725	20.42
	5	194360	0.0185	0.2546	13.75
	8	256000	0.0158	0.5522	34.87

Table 5: Calendar-time-indexed ASGD versus paired MLE at

T=256{,}000

, based on 200 paired replications. RMSE refers to

\widehat{\rho}

Specification	Dependence parameters	Log likelihood	$\Delta$ AIC	$\Delta$ BIC	Maximum absolute score
Time-adjusted independence	0	-6710.677	0.0	0.0	–
Symmetric Kronecker DPP	16	-6710.677	32.0	106.4	$1.22{\times}10^{-6}$
Axis	16	-6660.681	-68.0	6.4	$5.82{\times}10^{-6}$
Sector-led	22	-6637.049	-103.3	-1.0	$8.81{\times}10^{-6}$
Breach-led	26	-6619.405	-130.5	-9.7	$5.39{\times}10^{-6}$
Two-way	32	-6609.976	-137.4	11.4	$9.89{\times}10^{-6}$

Table 6: Fit of the symmetric and nonsymmetric DPP specifications. Information-criterion differences are relative to time-adjusted independence.

Rank	Pair	Correlation	Pair type
1	MED–HACK and MED–DISC	0.44799	Breach type
2	BUS–HACK and MED–DISC	0.29953	Cross
3	BUS–HACK and MED–HACK	0.29064	Sector
4	BUS–HACK and MED–ELET	0.29055	Cross
5	EDU–ELET and PUBLIC–ELET	0.28169	Sector

Table 7: Five largest fitted correlations after pooling the 772 weeks.

Pair class	Pairs	Mean $\bar{\rho}_{ij}$	Mean $\|\rho_{ij,t}\|$	Largest $\bar{\rho}_{ij}$
Same sector, different breach types	30	0.0151	0.0158	MED–DISC / MED–INSD (0.0354)
Different sectors, same breach type	40	0.0285	0.0300	BUS–ELET / MED–ELET (0.0733)
Different sectors, different breach types	120	0.0114	0.0136	BSF–INSD / EDU–HACK (0.0555)

Table 8: Contemporaneous dependence under the selected breach-led model. The mean absolute correlation averages

|\rho_{ij,t}|

over the relevant pairs and weeks.

Initial value	Initial $\ell_{T}$	Raw endpoint $\ell_{T}$	Averaged $\ell_{T}$	$\Delta_{\ell}$	$\\|\nabla\ell_{T}\\|_{\infty}/T$
Axis warm start	-6660.681	-6919.064	-6748.920	129.515	0.0777
Cellwise start	-6708.017	-6931.639	-6770.810	151.405	0.0559

Table 9: Chronological one-pass ASGD results for

\gamma_{0}=5

. The last two columns refer to the Polyak–Ruppert average; the deterministic full MLE has

\ell_{T}=-6619.405

Panel A: how the data days are used
Data days	Use
1	Select 20 high-activity candidate source computers
2–8	Choose the ten-computer panel; set starting values and coordinate scales
9–15	Not used for panel selection, tuning, or main estimation
16–19	Run candidate ASGD settings
20–22	Evaluate candidate estimates with coefficients held fixed

Table 10: Chronological design, computation, and predictive performance. The observed mean number of active sources on Days 52–58 is 2.88722. Over the 28 six-hour evaluation blocks, the NDPP has a higher log score than independence in 18 blocks under full-history averaging and 20 under common last-half averaging. Main-fit runtimes were measured on an Apple M1 iMac with 16GB of memory and exclude raw-data aggregation, calibration, ASGD-setting selection, and later-period evaluation.

Figure 1: Running full-history and last-half ASGD averages for the dynamic NDPP. The averages restart on Day 23, while the step-size clock includes the 604,800 earlier updates. At the end of each main-fit day, the two averages are frozen and scored on the first six hours of the next day. Panel (a) reports their mean joint log scores; Panel (b) reports the difference, with positive values favoring the last-half average. Last-half averaging has a higher log score in 20 of the 29 forecast windows; its mean and median gains are

0.0341

and

0.0103

per second, respectively. The diagnostic does not separate initialization effects from temporal change.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Panel A: normalized Fisher-risk ratio
Design	$\bar{n}$	$N=16{,}000$	$N=64{,}000$
Baseline	4	$734.017\ [564.026,920.869]$	$3047.295\ [2209.686,3951.232]$
	5	$1266.541\ [965.432,1601.564]$	$3005.759\ [2164.202,3960.102]$
	8	$1199.212\ [836.908,1592.544]$	$3521.345\ [2406.098,4776.091]$
Dense	4	$952.716\ [702.251,1251.617]$	$3152.242\ [2173.469,4215.633]$
	5	$1400.476\ [1002.209,1849.842]$	$3181.502\ [2030.623,4558.362]$

Appendix

Table B.1: Stabilized ASGD relative to paired MLE in the regular designs

Panel A: estimator accuracy
Design	$T$	MLE RMSE	ASGD RMSE	RMSE ratio	Risk ratio
Static	64,000	0.02369	0.07828	3.30	4.12
	256,000	0.01255	0.02569	2.05	2.02
	1,000,000	0.00706	0.01006	1.43	1.39
Contextual	64,000	0.02450	0.07283	2.97	4.08
	256,000	0.01216	0.02544	2.09	2.14

Appendix

Table B.2: Accuracy and computation with static and observation-specific kernels. Panel A reports RMSE for

\widehat{\rho}

; both ratios are ASGD divided by MLE within the same design. At

T=1{,}000{,}000

, paired-bootstrap

95\%

intervals for the contextual ASGD-to-MLE

\rho

-RMSE and Fisher-weighted risk ratios are

[1.25,1.53]

and

[1.22,1.49]

, respectively. All MLE optimizations satisfy the numerical convergence criterion, every ASGD path completes, and no ASGD path activates the projection. In Panel B, observation visits, state-weight evaluations, and elapsed seconds include both initialization and main estimation. MLE evaluations count only main-stage criterion evaluations. Observation visits and state-weight evaluations are in millions. One state-weight evaluation is one of the 256 terms used to form a normalizer; static evaluations use compressed counts. Elapsed seconds are median estimation times from ten single-threaded replications on an Apple M1 iMac; data generation is excluded.

\widehat{S}=\begin{array}[]{c|rrrrr}&\mathrm{BSF}&\mathrm{BUS}&\mathrm{MED}&\mathrm{EDU}&\mathrm{PUBLIC}\\ \hline\cr\mathrm{BSF}&0&1.433&1.168&1.073&0.924\\ \mathrm{BUS}&-1.433&0&1.978&-1.603&-1.537\\ \mathrm{MED}&-1.168&-1.978&0&1.344&-0.631\\ \mathrm{EDU}&-1.073&1.603&-1.344&0&2.068\\ \mathrm{PUBLIC}&-0.924&1.537&0.631&-2.068&0\end{array}

Appendix

Table B.3: Normalized point estimates for the selected time-adjusted breach-led NDPP. Rows and columns of

\widehat{S}

and

\widehat{C}_{S}

follow BSF, BUS, MED, EDU, PUBLIC; those of

\widehat{B}

follow HACK, ELET, DISC, INSD. Together, the signed matrices constitute one representative of the diagonal-sign and transposition equivalence class.

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

Hideitsu Hino, Keisuke Yano

The Institute of Statistical Mathematics, 10-3 Midori cho, Tachikawa City, 190-8562, Tokyo, Japan.

Parameter Identifiability Determinantal Point Process

Open paper AskPDF arXiv

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.

Abstract

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