Estimating processes in adapted Wasserstein distance

Backhoff, Julio; Bartl, Daniel; Beiglböck, Mathias; Wiesel, Johannes

doi:10.48550/arxiv.2002.07261

Cited by 5 publications

(8 citation statements)

References 28 publications

(44 reference statements)

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“…On an intuitive level, the nested Wasserstein distance only considers those couplings γ ∈ Π(π, π), which respect the information flow formalised by the canonical (i.e. coordinate) filtration (F t ) t∈{1,2} : in (1) this is achieved by first taking an infimum over couplings of π 1 , π1 (i.e. "couplings at time one") and then a second (nested) infimum with respect to the respective disintegrations (i.e.…”

Section: Notation and Main Resultsmentioning

confidence: 99%

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Measuring association with Wasserstein distances

Wiesel¹

2021

Preprint

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Let π ∈ Π(µ, ν) be a coupling between two probability measures µ and ν on a Polish space. In this article we propose and study a class of nonparametric measures of association between µ and ν. The analysis is based on the Wasserstein distance between ν and the disintegration πx 1 of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate. Throughout our analysis we make use of the so-called adapted/causal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our class of measures offers on alternative to the correlation coefficient proposed by [Dette, Siburg and Stoimenov (2013). A copula-based non-parametric measure of regression dependence. Scandinavian Journal of Statistics 40(1), 21-41] and [Chatterjee (2020). A new coefficient of correlation. Journal of the American Statistical Association, 1-21]. In contrast to these works, our approach also applies to probability laws on general Polish spaces.

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Section: Notation and Main Resultsmentioning

confidence: 99%

“…One such AW-consistent estimator of π has recently been constructed in [1] and throughout this article, we will make use of results established there. In particular continuity of T in AW will directly enable us to establish convergence rates.…”

Section: Notation and Main Resultsmentioning

confidence: 99%

Measuring association with Wasserstein distances

Wiesel¹

2021

Preprint

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“…Thus, to quantize (P p ([0, 1] dT ), AW p ), we can not take functions as in (3.1). Instead, [10] suggests the use of an adapted empirical distribution. Let r = (T + 1) −1 for d = 1, and r = (dT ) −1 for d ≥ 2.…”

Section: Definition 34 (Metric Attention Mechanism)mentioning

confidence: 99%

Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

Acciaio¹,

Kratsios²,

Pammer³

2022

Preprint

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A. We introduce a universal class of geometric deep learning models, called metric hypertransformers (MHTs), capable of approximating any adapted map F : X Z → Y Z with approximable complexity, where X ⊆ R d and Y is any suitable metric space, and X Z (resp. Y Z ) capture all discrete-time paths on X (resp. Y ). Suitable spaces Y include various (adapted) Wasserstein spaces, all Fréchet spaces admitting a Schauder basis, and a variety of Riemannian manifolds arising from information geometry. Even in the static case, where f : X → Y is a Hölder map, our results provide the first (quantitative) universal approximation theorem compatible with any such X and Y . Our universal approximation theorems are quantitative, and they depend on the regularity of F, the choice of activation function, the metric entropy and diameter of X , and on the regularity of the compact set of paths whereon the approximation is performed. Our guiding examples originate from mathematical finance. Notably, the MHT models introduced here are able to approximate a broad range of stochastic processes' kernels, including solutions to SDEs, many processes with arbitrarily long memory, and functions mapping sequential data to sequences of forward rate curves.

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“…He derives convergence rates if 𝜏 𝑌 (𝛾) is estimated by 𝜏 𝑌 ( γ𝑛 ) for a so-called adapted empirical measure γ𝑛 . One example for such an adaption is the projection of the individual observations 𝜉 𝑖 and 𝜁 𝑖 to a grid in the unit cube (see Backho et al 2020).…”

Section: 𝛼mentioning

confidence: 99%

Transport Dependency: Optimal Transport Based Dependency Measures

Nies¹,

Staudt²,

Munk³

2021

Preprint

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Finding meaningful ways to determine the dependency between two random variables 𝜉 and 𝜁 is a timeless statistical endeavor with vast practical relevance. In recent years, several concepts that aim to extend classical means (such as the Pearson correlation or rank-based coe cients like Spearman's 𝜌) to more general spaces have been introduced and popularized, a well-known example being the distance correlation. In this article, we propose and study an alternative framework for measuring statistical dependency, the transport dependency 𝜏 ≥ 0, which relies on the notion of optimal transport and is applicable in general Polish spaces. It can be estimated consistently via the corresponding empirical measure, is versatile and adaptable to various scenarios by proper choices of the cost function, and intrinsically respects metric properties of the ground spaces. Notably, statistical independence is characterized by 𝜏 = 0, while large values of 𝜏 indicate highly regular relations between 𝜉 and 𝜁 . Indeed, for suitable base costs, 𝜏 is maximized if and only if 𝜁 can be expressed as 1-Lipschitz function of 𝜉 or vice versa. Based on sharp upper bounds, we exploit this characterization and de ne three distinct dependency coe cients (a-c) with values in [0, 1], each of which emphasizes di erent functional relations. ese transport correlations a ain the value 1 if and only if 𝜁 = 𝜑 (𝜉), where 𝜑 is a) a Lipschitz function, b) a measurable function, c) a multiple of an isometry. e properties of coe cient c) make it comparable to the distance correlation, while coe cient b) is a limit case of a) that was recently studied independently by Wiesel (2021). Numerical results suggest that the transport dependency is a robust quantity that e ciently discerns structure from noise in simple se ings, o en out-performing other commonly applied coe cients of dependency.

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Estimating processes in adapted Wasserstein distance

Cited by 5 publications

References 28 publications

Measuring association with Wasserstein distances

Measuring association with Wasserstein distances

Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

Transport Dependency: Optimal Transport Based Dependency Measures

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