Fixed-Size Determinantal Point Processes Sampling For Species Phylogeny

Diala Wehbe; Nicolas Wicker; Baydaa Al-Ayoubi; Luc Moulinier

doi:10.5802/msia.13

Diala Wehbe ¹; Nicolas Wicker ²; Baydaa Al-Ayoubi ³; Luc Moulinier ⁴

¹ Paul Painlevé Laboratory, University of Lille, 59650 Villeneuve D’Ascq, France and EDST, Lebanese University, Tripoli, Lebanon
² Paul Painlevé Laboratory, University of Lille, 59650 Villeneuve D’Ascq, France
³ Faculty of Sciences, Lebanese University, Rafic Hariri University Campus - Hadas, Lebanon
⁴ ICube, CSTB (Complex Systems and Translational Bioinformatics), University of Strasbourg, 67085 Strasbourg, France

MathematicS In Action, Volume 10 (2021) no. 1, pp. 1-13.

Abstract

Determinantal point processes (DPPs) are popular tools that supply useful information for repulsiveness. They provide coherent probabilistic models when negative correlations arise and also represent new algorithms for inference problems like sampling, marginalization and conditioning. Recently, DPPs have played an increasingly important role in machine learning and statistics, since they are used for diverse subset selection problems. In this paper we use $k$ -DPP, a conditional DPP that models only sets of cardinality $k$ , to sample a diverse subset of species from a large phylogenetic tree. The tree sampling task is important in many studies in modern bioinformatics. The results show a fast mixing sampler for $k$ -DPP, for which a polynomial bound on the mixing time is given. This approach is applied to a real-world dataset of species, and we observe that leaves joined by a higher subtree are more likely to appear.

Published online: 2021-02-16

DOI: 10.5802/msia.13
Keywords: Determinantal point process, Kernel, Markov chain, Metropolis-Hasting, Mixing time, Phylogenetic tree
Author's affiliations:

Diala Wehbe ¹; Nicolas Wicker ²; Baydaa Al-Ayoubi ³; Luc Moulinier ⁴

@article{MSIA_2021__10_1_1_0,
     author = {Diala Wehbe and Nicolas Wicker and Baydaa Al-Ayoubi and Luc Moulinier},
     title = {Fixed-Size {Determinantal} {Point} {Processes} {Sampling} {For} {Species} {Phylogeny}},
     journal = {MathematicS In Action},
     pages = {1--13},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {10},
     number = {1},
     year = {2021},
     doi = {10.5802/msia.13},
     language = {en},
     url = {https://msia.centre-mersenne.org/articles/10.5802/msia.13/}
}

TY  - JOUR
TI  - Fixed-Size Determinantal Point Processes Sampling For Species Phylogeny
JO  - MathematicS In Action
PY  - 2021
DA  - 2021///
SP  - 1
EP  - 13
VL  - 10
IS  - 1
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://msia.centre-mersenne.org/articles/10.5802/msia.13/
UR  - https://doi.org/10.5802/msia.13
DO  - 10.5802/msia.13
LA  - en
ID  - MSIA_2021__10_1_1_0
ER  -

%0 Journal Article
%T Fixed-Size Determinantal Point Processes Sampling For Species Phylogeny
%J MathematicS In Action
%D 2021
%P 1-13
%V 10
%N 1
%I Société de Mathématiques Appliquées et Industrielles
%U https://doi.org/10.5802/msia.13
%R 10.5802/msia.13
%G en
%F MSIA_2021__10_1_1_0

Diala Wehbe; Nicolas Wicker; Baydaa Al-Ayoubi; Luc Moulinier. Fixed-Size Determinantal Point Processes Sampling For Species Phylogeny. MathematicS In Action, Volume 10 (2021) no. 1, pp. 1-13. doi : 10.5802/msia.13. https://msia.centre-mersenne.org/articles/10.5802/msia.13/

References
Cited by

[1] N. Anari; S. O. Gharan; A. Rezaei Monte Carlo Markov chain algorithms for sampling strongly Rayleigh distributions and determinantal point processes, In COLT (2016)

[2] J. Borcea; P. Branden; T. M. Liggett Negative dependence and the geometry of polynomials, J. Am. Math. Soc., Volume 22 (2009), pp. 521-567 | Article | MR: 2476782 | Zbl: 1206.62096

[3] A. Çivril; M. Magdon-Ismail On selecting a maximum volume sub-matrix of a matrix and related problems, Theor. Comput. Sci. (2009), pp. 4801-4811 | Article | MR: 2583677 | Zbl: 1181.15002

[4] D. M. Cvetkovic; M. Doob; H. Sachs Spectra of Graphs, Academic Press Inc., 1980 | Zbl: 0458.05042

[5] A. Deshpande; L. Rademacher Efficient volume sampling for row/column subset selection, FOCS (2010) no. 1-3-4, pp. 329-338

[6] P. G. Doyle; J. L. Snell Random Walks and Electric Networks, Mathematical Association of America, 1984 | Zbl: 0583.60065

[7] J. Ginibre Statistical ensembles of complex, quaternion,and real matrices, J. Math. Phys., Volume 6 (1965), pp. 440-449 | Article | MR: 173726 | Zbl: 0127.39304

[8] F. Gobel; A. Jagers Random walks on graphs., Stochastic Processes Appl., Volume 2 (1974), pp. 311-336 | Article | MR: 397887 | Zbl: 0296.60046

[9] B. Kang Fast determinantal point process sampling with application to clustering, NIPS (2013), pp. 2319-2327

[10] R. Kannan; S. Vempala Spectral algorithms, Foundations and Trends in Theoretical Computer Science, Volume 4 (2009), pp. 157-288 | Article | MR: 2558901

[11] R. I. Kondor; J. Lafferty Diffusion kernels on graphs and other discrete structures, Proceedings of the 19th International Conference On Machine Learning (2002), pp. 315-322

[12] A. Kulesza; B. Taskar kDPPs: fixed size determinantal point processes, Proceedings of the 28th International Conference on Machine Learning, Omnipress, 2011, pp. 1193-1200

[13] A. Kulesza; B. Taskar Determinantal point processes for machine learning, Foundations and Trends in Machine Learning, Volume 5 (2012) no. 2-3, pp. 123-286 | Article | Zbl: 1278.68240

[14] L. Lovász Random walks on graphs: A survey, Combinatorics, Volume 2 (1993), pp. 1-46

[15] O. Macchi The Coincidence approach to stochastic point processes, Adv. Appl. Probab., Volume 7 (1975) no. 1, pp. 83-122 | Article | MR: 380979 | Zbl: 0366.60081

[16] M. L. Mehta; M. Gaudin On the density of eigenvalues of a random matrix, Nuclear Physics, Volume 18 (1960), pp. 420-427 | Article | MR: 112895 | Zbl: 0107.35702

[17] R. Merris Laplacian matrices of graphs: A survey, Linear Algebra Appl., Volume 197-198 (1994), pp. 143-176 | Article | MR: 1275613 | Zbl: 0802.05053

[18] S. Pundir; M. J. Martin; C. O’Donovan UniProt Protein Knowledgebase, Methods Mol. Biol, Volume 1558 (2017), pp. 41-55 | Article

Cited by Sources: