Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
156 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Extremal graphical modeling with latent variables via convex optimization (2403.09604v3)

Published 14 Mar 2024 in stat.ME, math.ST, stat.ML, and stat.TH

Abstract: Extremal graphical models encode the conditional independence structure of multivariate extremes and provide a powerful tool for quantifying the risk of rare events. Prior work on learning these graphs from data has focused on the setting where all relevant variables are observed. For the popular class of H\"usler-Reiss models, we propose the \texttt{eglatent} method, a tractable convex program for learning extremal graphical models in the presence of latent variables. Our approach decomposes the H\"usler-Reiss precision matrix into a sparse component encoding the graphical structure among the observed variables after conditioning on the latent variables, and a low-rank component encoding the effect of a few latent variables on the observed variables. We provide finite-sample guarantees of \texttt{eglatent} and show that it consistently recovers the conditional graph as well as the number of latent variables. We highlight the improved performances of our approach on synthetic and real data.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (44)
  1. Statistical mechanics of complex networks. ArXiv cond-mat/0106096. URL: https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.74.47.
  2. Extremes on river networks. Annals of Applied Statistics 9, 2023–2050. URL: https://www.jstor.org/stable/43826454.
  3. Inference on extremal dependence in the domain of attraction of a structured Hüsler–Reiss distribution motivated by a Markov tree with latent variables. Extremes 24, 461–500. arXiv:2001.09510.
  4. Extremes of Markov random fields on block graphs: max-stable limits and structured Hüsler–Reiss distributions. Extremes 26, 433–468. URL: https://link.springer.com/article/10.1007/s10687-023-00467-9.
  5. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundational Trends of Machine Learning 3, 1–122. URL: https://dl.acm.org/doi/10.1561/2200000016.
  6. Robust principal component analysis? Journal of the ACM 58, 1–37. URL: https://dl.acm.org/doi/10.1145/1970392.1970395.
  7. Exact matrix completion via convex optimization. Foundations of Computational Mathematics 55, 111–119. URL: https://link.springer.com/article/10.1007/s10208-009-9045-5.
  8. Latent variable graphical model selection via convex optimization. Annals of Statistics 40, 1935–1967. URL: https://www.jstor.org/stable/41806519.
  9. Rank-sparsity incoherence for matrix decomposition. SIAM Journal of Optimization 21, 572–596. URL: https://epubs.siam.org/doi/10.1137/090761793.
  10. Exact simulation of max-stable processes. Biometrika 103, 303–317. arXiv:1506.04430.
  11. Graphical models for multivariate extremes. arXiv:2402.02187.
  12. Graphical models for extremes (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol 82, 871–932. URL: https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12355.
  13. graphicalExtremes: Statistical Methodology for Graphical Extreme Value Models. URL: https://github.com/sebastian-engelke/graphicalExtremes.
  14. Sparse structures for multivariate extremes. Annu. Rev. Stat. Appl. 8, 241–270. URL: https://www.annualreviews.org/doi/abs/10.1146/annurev-statistics-040620-041554.
  15. Graphical models for infinite measures with applications to extremes and Lévy processes. URL: https://arxiv.org/abs/2211.15769, doi:10.48550/ARXIV.2211.15769.
  16. Learning extremal graphical structures in high dimensions. URL: https://arxiv.org/abs/2111.00840.
  17. Extremal dependence of random scale constructions. Extremes 22, 623–666. URL: https://link.springer.com/article/10.1007/s10687-019-00353-3.
  18. Structure learning for extremal tree models. Journal of the Royal Statistical Society Series B: Statistical Methodology 84, 2055–2087. URL: https://doi.org/10.1111/rssb.12556, doi:10.1111/rssb.12556.
  19. Rank minimization and applications in system theory. Proceedings of the 2004 American Control Conference 4, 3273–3278 vol.4. URL: https://ieeexplore.ieee.org/document/1384521.
  20. Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9, 432–441. URL: https://doi.org/10.1093/biostatistics/kxm045.
  21. Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 40, 317–337. URL: https://link.springer.com/article/10.1007/BF00533086.
  22. Statistical inference for Hüsler–Reiss graphical models through matrix completions. URL: https://arxiv.org/abs/2210.14292.
  23. Modelling multivariate extreme value distributions via markov trees. URL: https://arxiv.org/abs/2208.02627, arXiv:2208.02627.
  24. Maxima of normal random vectors: Between independence and complete dependence. Statist. Prob. Letters 7, 283–286. URL: https://ideas.repec.org/a/eee/stapro/v7y1989i4p283-286.html.
  25. Graphical models. volume 17 of Oxford statistical science series. Clarendon Press, Oxford. URL: https://www.tib.eu/de/suchen/id/TIBKAT%3A197598226.
  26. Extremes in high dimensions: Methods and scalable algorithms. arXiv:2303.04258. available from https://arxiv.org/abs/2303.04258.
  27. Alternating direction methods for latent variable gaussian graphical model selection. Neural Computation 25, 2172–2198. URL: https://direct.mit.edu/neco/article/25/8/2172/7900/Alternating-Direction-Methods-for-Latent-Variable.
  28. High-dimensional graphs and variable selection with the lasso. Annals of Statistics 34, 1436–1462. URL: https://projecteuclid.org/journals/annals-of-statistics/volume-34/issue-3/High-dimensional-graphs-and-variable-selection-with-the-Lasso/10.1214/009053606000000281.full.
  29. Conditional independence among max-stable laws. Statistics & Probability Letters 108, 9–15. URL: https://www.sciencedirect.com/science/article/pii/S0167715215002874.
  30. High-dimensional covariance estimation by minimizing ℓ1subscriptℓ1\ell_{1}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-penalized log-determinant divergence. Electronic Journal of Statistics 5, 935–980. URL: https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-5/issue-none/High-dimensional-covariance-estimation-by-minimizing-%E2%84%931-penalized-log-determinant/10.1214/11-EJS631.full.
  31. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Review 52, 471–501. URL: https://epubs.siam.org/doi/10.1137/070697835.
  32. Extreme Values, Regular Variation and Point Processes. Springer, New York.
  33. Multivariate generalized Pareto distributions: Parametrizations, representations, and properties. J. Multivariate Anal. 165, 117–131. URL: https://www.sciencedirect.com/science/article/pii/S0047259X17303147.
  34. Multivariate generalized Pareto distributions. Bernoulli 12, 917–930. URL: https://projecteuclid.org/journals/bernoulli/volume-12/issue-5/Multivariate-generalized-Pareto-distributions/10.3150/bj/1161614952.full.
  35. Parametric and nonparametric symmetries in graphical models for extremes. arXiv:2306.00703. available from https://arxiv.org/ags/2306.00703.
  36. Total positivity in multivariate extremes. The Annals of Statistics 51, 962 – 1004. URL: https://doi.org/10.1214/23-AOS2272.
  37. One- versus multi-component regular variation and extremes of Markov trees. Adv. in Appl. Probab. 52, 855–878. arXiv:1902.02226.
  38. Interpreting latent variables in factor models via convex optimization. Mathematical Programming 167, 129–154. URL: https://link.springer.com/article/10.1007/s10107-017-1187-7.
  39. A statistical graphical model of the California reservoir system. Water Resources Research 53, 9721 – 9739. URL: https://agupubs.onlinelibrary.wiley.com/doi/10.1002/2017WR020412.
  40. Sharp thresholds for high-dimensional and noisy sparsity recovery using ℓ1subscriptℓ1\ell_{1}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-constrained quadratic programming (lasso). IEEE Transactions on Information Theory 55, 2183–2202. URL: https://ieeexplore.ieee.org/document/4839045.
  41. Graphical lasso for extremes. arXiv:2307.15004.
  42. On model selection consistency of lasso. Journal of Machine Learning Research 7, 2541–2563. URL: https://www.jmlr.org/papers/volume7/zhao06a/zhao06a.pdf.
  43. Dependence structure of risk factors and diversification effects. Risk and Insurance / Measures and Control 2 URL: https://api.semanticscholar.org/CorpusID:15682170.
  44. Dependence of drivers affects risks associated with compound events. Science Advances 3. URL: https://www.science.org/doi/10.1126/sciadv.1700263.
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now