Placeholder notes for a type of point process, with which I am unfamiliar, but about which I am incidentally curious.

Wikipedia says:

Let \(\Lambda\) be a locally compact Polish space and \(\mu\) be a Radon measure on \(\Lambda\). Also, consider a measurable function \(K:\Lambda^2\rightarrow \mathbb{C}\).

We say that \(X\) is a

determinantal point processon \(\Lambda\) with kernel \(K\) if it is a simple point process on \(\Lambda\) with a joint intensity/Factorial_moment_densityorcorrelation function (which is the density of its factorial moment measure) given by\[ \rho_n(x_1,\ldots,x_n) = \det[K(x_i,x_j)]_{1 \le i,j \le n} \]

for every \(n\ geq 1\) and \(x_1,\dots, x_n\in \Lambda.\)

The most popular tutorial introduction to this topic seems to be (Kulesza and Taskar 2012). I found it unhelpful as it is rooted in discrete-space problems which is precisely where I do not work. For continuous state space, (Møller and Waagepetersen 2007, 2017) and Terry Tao’s summary of (Ben Hough et al. 2006) are good.

Interesting property: The zeros random polynomials with Gaussian coefficients are apparently to be distributed as DPPs. (Ben Hough et al. 2009; Krishnapur 2006)

Ben Hough, J., Manjunath Krishnapur, Yuval Peres, and Bálint Virág. 2006. “Determinantal Processes and Independence.” *Probability Surveys* 3 (0): 206–29. https://doi.org/10.1214/154957806000000078.

Ben Hough, John, Manjunath Krishnapur, Yuval Peres, and Bálint Virág. 2009. *Zeros of Gaussian Analytic Functions and Determinantal Point Processes*. University Lecture Series, v. 51. Providence, R.I: American Mathematical Soc. http://math.iisc.ernet.in/~manju/GAF_book.pdf.

Borodin, Alexei. 2009. “Determinantal Point Processes.” In *Oxford Handbook of Random Matrix Theory*. http://arxiv.org/abs/0911.1153.

Gillenwater, Jennifer A, Alex Kulesza, Emily Fox, and Ben Taskar. 2014. “Expectation-Maximization for Learning Determinantal Point Processes.” In *Advances in Neural Information Processing Systems 27*, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, 3149–57. Curran Associates, Inc. http://papers.nips.cc/paper/5564-expectation-maximization-for-learning-determinantal-point-processes.pdf.

Iyer, Rishabh, and Jeffrey Bilmes. 2015. “Submodular Point Processes with Applications to Machine Learning.” In *Artificial Intelligence and Statistics*, 388–97. http://proceedings.mlr.press/v38/iyer15.html.

Krishnapur, Manjunath. 2006. “Zeros of Random Analytic Functions,” July. http://arxiv.org/abs/math/0607504.

Kulesza, Alex, and Ben Taskar. 2011. “Learning Determinantal Point Processes.” In *Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence*, 419–27. UAI’11. Arlington, Virginia, United States: AUAI Press. http://arxiv.org/abs/1202.3738.

———. 2012. *Determinantal Point Processes for Machine Learning*. Vol. 5. Foundations and Trends® in Machine Learning 5,2-3. Boston, Mass.: Now Foundations and Trends. http://www.alexkulesza.com/pubs/dpps_fnt12.pdf.

Lavancier, Frédéric, Jesper Møller, and Ege Rubak. 2015. “Determinantal Point Process Models and Statistical Inference.” *Journal of the Royal Statistical Society: Series B (Statistical Methodology)* 77 (4): 853–77. https://doi.org/10.1111/rssb.12096.

Lyons, Russell. 2003. “Determinantal Probability Measures.” *Publications Mathématiques de L’Institut Des Hautes Études Scientifiques* 98 (1): 167–212. https://doi.org/10.1007/s10240-003-0016-0.

Lyons, Russell, and Y. Peres. 2016. *Probability on Trees and Networks*. Cambridge Series in Statistical and Probabilistic Mathematics. New York NY: Cambridge University Press.

McCullagh, Peter, and Jesper Møller. 2006. “The Permanental Process.” *Advances in Applied Probability* 38 (4): 873–88. https://doi.org/10.1017/S0001867800001361.

Møller, Jesper, and Rasmus Waagepetersen. 2017. “Some Recent Developments in Statistics for Spatial Point Patterns.” *Annual Review of Statistics and Its Application* 4 (1): 317–42. https://doi.org/10.1146/annurev-statistics-060116-054055.

Møller, Jesper, and Rasmus Plenge Waagepetersen. 2007. “Modern Statistics for Spatial Point Processes.” *Scandinavian Journal of Statistics* 34 (4): 643–84. https://doi.org/10.1111/j.1467-9469.2007.00569.x.

Osogami, Takayuki, Rudy Raymond, Akshay Goel, Tomoyuki Shirai, and Takanori Maehara. 2018. “Dynamic Determinantal Point Processes.” In *Thirty-Second AAAI Conference on Artificial Intelligence*. https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16081.

Pemantle, Robin, and Igor Rivin. 2013. “The Distribution of Zeros of the Derivative of a Random Polynomial.” In *Advances in Combinatorics*, edited by Ilias S. Kotsireas and Eugene V. Zima, 259–73. Springer Berlin Heidelberg.

Soshnikov, A. 2000. “Determinantal Random Point Fields.” *Russian Mathematical Surveys* 55 (5): 923. https://doi.org/10.1070/RM2000v055n05ABEH000321.

Torrisi, Giovanni Luca, and Emilio Leonardi. 2014. “Large Deviations of the Interference in the Ginibre Network Model.” *Stochastic Systems* 4 (1): 173–205. https://doi.org/10.1287/13-SSY109.

Xie, Pengtao, Ruslan Salakhutdinov, Luntian Mou, and Eric P. Xing. 2017. “Deep Determinantal Point Process for Large-Scale Multi-Label Classification.” In, 473–82. IEEE. https://doi.org/10.1109/ICCV.2017.59.