## Bernoulli

• Bernoulli
• Volume 25, Number 4B (2019), 3555-3589.

### Asymptotic equivalence of fixed-size and varying-size determinantal point processes

#### Abstract

Determinantal Point Processes (DPPs) are popular models for point processes with repulsion. They appear in numerous contexts, from physics to graph theory, and display appealing theoretical properties. On the more practical side of things, since DPPs tend to select sets of points that are some distance apart (repulsion), they have been advocated as a way of producing random subsets with high diversity. DPPs come in two variants: fixed-size and varying-size. A sample from a varying-size DPP is a subset of random cardinality, while in fixed-size “$k$-DPPs” the cardinality is fixed. The latter makes more sense in many applications, but unfortunately their computational properties are less attractive, since, among other things, inclusion probabilities are harder to compute. In this work, we show that as the size of the ground set grows, $k$-DPPs and DPPs become equivalent, in the sense that fixed-order inclusion probabilities converge. As a by-product, we obtain saddlepoint formulas for inclusion probabilities in $k$-DPPs. These turn out to be extremely accurate, and suffer less from numerical difficulties than exact methods do. Our results also suggest that $k$-DPPs and DPPs also have equivalent maximum likelihood estimators. Finally, we obtain results on asymptotic approximations of elementary symmetric polynomials which may be of independent interest.

#### Article information

Source
Bernoulli, Volume 25, Number 4B (2019), 3555-3589.

Dates
Revised: August 2018
First available in Project Euclid: 25 September 2019

https://projecteuclid.org/euclid.bj/1569398777

Digital Object Identifier
doi:10.3150/18-BEJ1102

Mathematical Reviews number (MathSciNet)
MR4010965

Zentralblatt MATH identifier
07110148

#### Citation

Barthelmé, Simon; Amblard, Pierre-Olivier; Tremblay, Nicolas. Asymptotic equivalence of fixed-size and varying-size determinantal point processes. Bernoulli 25 (2019), no. 4B, 3555--3589. doi:10.3150/18-BEJ1102. https://projecteuclid.org/euclid.bj/1569398777

#### References

• [1] Billingsley, P. (2008). Probability and Measure. John Wiley & Sons.
• [2] Chen, X.-H., Dempster, A.P. and Liu, J.S. (1994). Weighted finite population sampling to maximize entropy. Biometrika 81 457–469.
• [3] Couillet, R. and Debbah, M. (2011). Random Matrix Methods for Wireless Communications. Cambridge: Cambridge Univ. Press.
• [4] Daniels, H.E. (1954). Saddlepoint approximations in statistics. Ann. Math. Stat. 25 631–650.
• [5] DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. New York: Springer.
• [6] Deshpande, A. and Rademacher, L. (2010). Efficient volume sampling for row/column subset selection. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science—FOCS 2010 329–338. Los Alamitos, CA: IEEE Computer Soc.
• [7] Deshpande, A., Rademacher, L., Vempala, S. and Wang, G. (2006). Matrix approximation and projective clustering via volume sampling. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms 1117–1126. New York: ACM.
• [8] Gautier, G., Bardenet, R. and Valko, M. (2017). Zonotope hit-and-run for efficient sampling from projection DPPs. ArXiv Preprint arXiv:1705.10498.
• [9] Jozsa, R. and Mitchison, G. (2015). Symmetric polynomials in information theory: Entropy and subentropy. J. Math. Phys. 56 062201, 17.
• [10] Kulesza, A. and Taskar, B. (2011). k-DPPs: Fixed-size determinantal point processes. In Proceedings of the 28th International Conference on Machine Learning (ICML-11) 1193–1200.
• [11] Kulesza, A. and Taskar, B. (2012). Determinantal point processes for machine learning. Found. Trends Mach. Learn. 5 123–286.
• [12] Li, C., Jegelka, S. and Sra, S. (2016). Efficient sampling for k-determinantal point processes.
• [13] Macchi, O. (1975). The coincidence approach to stochastic point processes. Adv. in Appl. Probab. 7 83–122.
• [14] Mariet, Z. and Sra, S. (2017). Elementary symmetric polynomials for optimal experimental design. ArXiv Eprint arXiv:1705.09677v1.
• [15] Rue, H. and Held, L. (2005). Gaussian Markov Random Fields: Theory and Applications. Monographs on Statistics and Applied Probability 104. Boca Raton, FL: CRC Press/CRC.
• [16] Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55 923–975.
• [17] Sra, S. (2019). Logarithmic inequalities under a symmetric polynomial dominance order. Proc. Amer. Math. Soc. 147 481–486.
• [18] Touchette, H. (2015). Equivalence and nonequivalence of ensembles: Thermodynamic, macrostate, and measure levels. J. Stat. Phys. 159 987–1016.
• [19] Tremblay, N., Barthelmé, S. and Amblard, P.-O. (2018). Determinantal point processes for coresets. ArXiv Preprint arXiv:1803.08700.
• [20] Tremblay, N., Barthelme, S. and Amblard, P.O. (2018). Optimized algorithms to sample determinantal point processes. ArXiv E-print arXiv:1802.08471.