Optimal transport vs. Fisher-Rao distance between copulas for clustering multivariate time series

Marti, Gautier; Andler, Sébastien; Nielsen, Frank; Donnat, Philippe

doi:10.1109/ssp.2016.7551770

Cited by 13 publications

(9 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also, there have been many works linking information geometry and optimal transport [4] [45]. In particular, the Wasserstein metric tensor for Gaussian distributions exhibits an explicit form [26][43] [10], which leads to extensive studies between the Wasserstein and Fisher-Rao metric for this model [31][34] [42]. In contrast to their works, our Wasserstein metric tensor can work for general parametric models.…”

Section: Introductionmentioning

confidence: 99%

Optimal transport natural gradient for statistical manifolds with continuous sample space

Chen¹,

Li²

2018

Preprint

View full text Add to dashboard Cite

We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the L 2 -Wasserstein metric tensor in the probability density space to a parameter space, equipping the latter with a positive definite metric tensor, under which it becomes a Riemannian manifold, named the Wasserstein statistical manifold. In general, it is not a totally geodesic sub-manifold of the density space, and therefore its geodesics will differ from the Wasserstein geodesics, except for the wellknown Gaussian distribution case, a fact which can also be validated under our framework. We use the sub-manifold geometry to derive a gradient flow and natural gradient descent method in the parameter space. When parametrized densities lie in R, the induced metric tensor establishes an explicit formula. In optimization problems, we observe that the natural gradient descent outperforms the standard gradient descent when the Wasserstein distance is the objective function. In such a case, we prove that the resulting algorithm behaves similarly to the Newton method in the asymptotic regime. The proof calculates the exact Hessian formula for the Wasserstein distance, which further motivates another preconditioner for the optimization process. To the end, we present examples to illustrate the effectiveness of the natural gradient in several parametric statistical models, including the Gaussian measure, Gaussian mixture, Gamma distribution, and Laplace distribution.

show abstract

Section: Introductionmentioning

confidence: 99%

Optimal transport natural gradient for statistical manifolds with continuous sample space

Chen¹,

Li²

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…The paper [8] shows that it is not always observed empirically. However, the Cramér-Rao lower bound (CRLB) for correlation [135] points out that the higher the correlation, the easier its estimation, i.e. less statistical uncertainty for high correlations.…”

Section: Moot Points and Controversiesmentioning

confidence: 99%

A review of two decades of correlations, hierarchies, networks and clustering in financial markets

Marti,

Nielsen,

Bińkowski

et al. 2017

Preprint

View full text Add to dashboard Cite

We review the state of the art of clustering financial time series and the study of their correlations alongside other interaction networks. The aim of the review is to gather in one place the relevant material from different fields, e.g. machine learning, econophysics, statistical physics, econometrics, behavioral finance. We hope it will help researchers to use more effectively this alternative modeling of the financial time series. Decision makers may also be able to leverage its insights. Finally, we also hope that this review will form the basis of an open toolbox to study correlations, hierarchies, networks and clustering in financial markets.

show abstract

“…Clustering with Optimal Transport Using the concept of Wasserstein barycenters [2,13,22], there exists many clustering algorithms based on Optimal Transport [23,41,42,44,45,55]. A recent generalization of KM++ with Optimal Transport cost has also been proposed in [72].…”

Section: Comparison Of Exponential Family Mixture Models By Optimal T...mentioning

confidence: 99%

Parameter Estimation in Finite Mixture Models by Regularized Optimal Transport: A Unified Framework for Hard and Soft Clustering

Dessein¹,

Papadakis²,

Deledalle³

2017

Preprint

View full text Add to dashboard Cite

In this short paper, we formulate parameter estimation for finite mixture models in the context of discrete optimal transportation with convex regularization. The proposed framework unifies hard and soft clustering methods for general mixture models. It also generalizes the celebrated k-means and expectation-maximization algorithms in relation to associated Bregman divergences when applied to exponential family mixture models.

show abstract

Optimal transport vs. Fisher-Rao distance between copulas for clustering multivariate time series

Cited by 13 publications

References 15 publications

Optimal transport natural gradient for statistical manifolds with continuous sample space

Optimal transport natural gradient for statistical manifolds with continuous sample space

A review of two decades of correlations, hierarchies, networks and clustering in financial markets

Parameter Estimation in Finite Mixture Models by Regularized Optimal Transport: A Unified Framework for Hard and Soft Clustering

Contact Info

Product

Resources

About