# Sparse approximation of triangular transports. Part II: the infinite dimensional case

@article{Zech2021SparseAO, title={Sparse approximation of triangular transports. Part II: the infinite dimensional case}, author={Jakob Zech and Youssef M. Marzouk}, journal={ArXiv}, year={2021}, volume={abs/2107.13422} }

For two probability measures ρ and π on [−1, 1] we investigate the approximation of the triangular Knothe–Rosenblatt transport T : [−1, 1] → [−1, 1] that pushes forward ρ to π. Under suitable assumptions, we show that T can be approximated by rational functions without suffering from the curse of dimension. Our results are applicable to posterior measures arising in certain inference problems where the unknown belongs to an (infinite dimensional) Banach space. In particular, we show that it is… Expand

#### References

SHOWING 1-10 OF 113 REFERENCES

On the geometry of Stein variational gradient descent

- Mathematics, Computer Science
- ArXiv
- 2019

This paper focuses on the recently introduced Stein variational gradient descent methodology, a class of algorithms that rely on iterated steepest descent steps with respect to a reproducing kernel Hilbert space norm, and considers certain nondifferentiable kernels with adjusted tails. Expand

Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs

- Mathematics
- 2015

Abstract The numerical approximation of parametric partial differential equations D ( u , y ) = 0 is a computational challenge when the dimension d of the parameter vector y is large, due to the… Expand

Well-Posed Bayesian Inverse Problems: Priors with Exponential Tails

- Mathematics, Computer Science
- SIAM/ASA J. Uncertain. Quantification
- 2017

A general recipe for construction of convex priors on Banach spaces is presented which will be of interest in practical applications where one often works with spaces such as $L^2$ or the continuous functions. Expand

Sparse deterministic approximation of Bayesian inverse problems

- Mathematics
- 2012

We present a parametric deterministic formulation of Bayesian inverse problems with an input parameter from infinite-dimensional, separable Banach spaces. In this formulation, the forward problems… Expand

Convergence rates of high dimensional Smolyak quadrature

- Mathematics
- 2020

We analyse convergence rates of Smolyak integration for parametric maps u : U → X taking values in a Banach space X , defined on the parameter domain U = [−1,1]N . For parametric maps which are… Expand

Inference via Low-Dimensional Couplings

- Mathematics, Computer Science
- J. Mach. Learn. Res.
- 2018

This paper establishes a link between the Markov properties of the target measure and the existence of low-dimensional couplings, induced by transport maps that are sparse and/or decomposable, and suggests new inference methodologies for continuous non-Gaussian graphical models. Expand

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

- Mathematics, Computer Science
- Found. Comput. Math.
- 2014

An interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving, which is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. Expand

Higher Order Quasi-Monte Carlo Integration for Holomorphic, Parametric Operator Equations

- Mathematics, Computer Science
- SIAM/ASA J. Uncertain. Quantification
- 2016

We analyze the convergence of higher order quasi--Monte Carlo (QMC) quadratures of solution functionals to countably parametric, nonlinear operator equations with distributed uncertain parameters… Expand

Triangular transformations of measures

- Mathematics
- 2004

A new identity for the entropy of a non-linear image of a measure on is obtained, which yields the well-known Talagrand's inequality. Triangular mappings on and are studied, that is, mappings such… Expand

Approximation and sampling of multivariate probability distributions in the tensor train decomposition

- Mathematics, Computer Science
- Stat. Comput.
- 2020

A sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor train format, a methodology that has been exploited for many years for scalable, high-dimensional density function approximation in quantum physics and chemistry. Expand