# Determinantal point processes

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 process on $$\Lambda$$ 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.