Poisson intensity estimation with reproducing kernels

Seth Flaxman, Yee Whye Teh, Dino Sejdinovic
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:270-279, 2017.

Abstract

Despite the fundamental nature of the Poisson process in the theory and application of stochastic processes, and its attractive generalizations (e.g. Cox process), few tractable nonparametric modeling approaches exist, especially in high dimensional settings. In this paper we develop a new, computationally tractable Reproducing Kernel Hilbert Space (RKHS) formulation for the inhomogeneous Poisson process. We model the square root of the intensity as an RKHS function. The modeling challenge is that the usual representer theorem arguments no longer apply due to the form of the inhomogeneous Poisson process likelihood. However, we prove that the representer theorem does hold in an appropriately transformed RKHS, guaranteeing that the optimization of the penalized likelihood can be cast as a tractable finite-dimensional problem. The resulting approach is simple to implement, and readily scales to high dimensions and large-scale datasets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v54-flaxman17a, title = {{Poisson intensity estimation with reproducing kernels}}, author = {Flaxman, Seth and Teh, Yee Whye and Sejdinovic, Dino}, booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics}, pages = {270--279}, year = {2017}, editor = {Singh, Aarti and Zhu, Jerry}, volume = {54}, series = {Proceedings of Machine Learning Research}, month = {20--22 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v54/flaxman17a/flaxman17a.pdf}, url = {https://proceedings.mlr.press/v54/flaxman17a.html}, abstract = {Despite the fundamental nature of the Poisson process in the theory and application of stochastic processes, and its attractive generalizations (e.g. Cox process), few tractable nonparametric modeling approaches exist, especially in high dimensional settings. In this paper we develop a new, computationally tractable Reproducing Kernel Hilbert Space (RKHS) formulation for the inhomogeneous Poisson process. We model the square root of the intensity as an RKHS function. The modeling challenge is that the usual representer theorem arguments no longer apply due to the form of the inhomogeneous Poisson process likelihood. However, we prove that the representer theorem does hold in an appropriately transformed RKHS, guaranteeing that the optimization of the penalized likelihood can be cast as a tractable finite-dimensional problem. The resulting approach is simple to implement, and readily scales to high dimensions and large-scale datasets. } }
Endnote
%0 Conference Paper %T Poisson intensity estimation with reproducing kernels %A Seth Flaxman %A Yee Whye Teh %A Dino Sejdinovic %B Proceedings of the 20th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2017 %E Aarti Singh %E Jerry Zhu %F pmlr-v54-flaxman17a %I PMLR %P 270--279 %U https://proceedings.mlr.press/v54/flaxman17a.html %V 54 %X Despite the fundamental nature of the Poisson process in the theory and application of stochastic processes, and its attractive generalizations (e.g. Cox process), few tractable nonparametric modeling approaches exist, especially in high dimensional settings. In this paper we develop a new, computationally tractable Reproducing Kernel Hilbert Space (RKHS) formulation for the inhomogeneous Poisson process. We model the square root of the intensity as an RKHS function. The modeling challenge is that the usual representer theorem arguments no longer apply due to the form of the inhomogeneous Poisson process likelihood. However, we prove that the representer theorem does hold in an appropriately transformed RKHS, guaranteeing that the optimization of the penalized likelihood can be cast as a tractable finite-dimensional problem. The resulting approach is simple to implement, and readily scales to high dimensions and large-scale datasets.
APA
Flaxman, S., Teh, Y.W. & Sejdinovic, D.. (2017). Poisson intensity estimation with reproducing kernels. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 54:270-279 Available from https://proceedings.mlr.press/v54/flaxman17a.html.

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