# Total positivity in multivariate extremes

arXiv: https://arxiv.org/abs/2112.14727 · HTML: https://mlsanity.com/paper/2112.14727/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2021.

## Built on by (most cited first, 4 of 4)

- [Learning Large-Scale MTP2 Gaussian Graphical Models via Bridge-Block Decomposition](https://mlsanity.com/paper/2309.13405/) (2023)
- [Adaptive Estimation of Graphical Models under Total Positivity](https://mlsanity.com/paper/2210.15471/) (2022)
- [Extrapolation in Statistical Learning with Extreme Value Theory](https://mlsanity.com/paper/2605.01909/) (2026)
- [Spectral Sparsification of Laplacian-Constrained Gaussian and H"usler-Reiss Graphical Models](https://mlsanity.com/paper/2606.16681/) (2026)

## Abstract (arXiv)

Positive dependence is present in many real world data sets and has appealing stochastic properties that can be exploited in statistical modeling and in estimation. In particular, the notion of multivariate total positivity of order 2 ( MTP2 ) is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce  EMTP2 , an extremal version of  MTP2 . This notion turns out to appear prominently in extremes, and in fact, it is satisfied by many classical models. For a H"usler--Reiss distribution, the analogue of a Gaussian distribution in extremes, we show that it is  EMTP2  if and only if its precision matrix is a Laplacian of a connected graph. We propose an estimator for the parameters of the H"usler--Reiss distribution under  EMTP2  as the solution of a convex optimization problem with Laplacian constraint. We prove that this estimator is consistent and typically yields a sparse model with possibly nondecomposable extremal graphical structure. Applying our methods to a data set of Danube River flows, we illustrate this regularization and the superior performance compared to existing methods.