# Structured Approximations of Measures

arXiv: https://arxiv.org/abs/2310.09149 · HTML: https://mlsanity.com/paper/2310.09149/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 57 references that are in the map.
- First posted on arXiv 13 Oct 2023.

## Builds on (most cited first, 5 of 5)

- [Wasserstein Task Embedding for Measuring Task Similarities](https://mlsanity.com/paper/2208.11726/) (2022)
- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Supervised learning of sheared distributions using linearized optimal transport](https://mlsanity.com/paper/2201.10590/) (2022)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)
- [Applications of No-Collision Transportation Maps in Manifold Learning](https://mlsanity.com/paper/2304.00199/) (2023)

## Abstract (arXiv)

We study the approximation of probability measures in the Wasserstein-p distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints. We obtain three sets of results. First, for measures with densities bounded away from zero on a bounded Lipschitz domain Omega, we prove that any approximation scheme for functions in Lp(Omega) transfers, with linear rate, to a corresponding approximation scheme for measures in Wp(Omega). The argument applies a theorem of Bogovskii on regularity of solutions to the continuity equation in the Benamou-Brenier formulation of optimal transport. We exhibit concrete approximation schemes (polynomials, shift-invariant spaces, cardinal interpolation with radial basis functions, kernel density estimators, and piecewise approximations on nonuniform Voronoi partitions) that fit the framework. As a matter of independent interest, we prove a negative Sobolev lower bound that generalizes existing bounds from p=2 to all p∈(1,∞). We also consider deterministic bounds for discrete approximations to arbitrary measures in terms of the mesh norm of a quasi-uniform set of points. We specialize these bounds to show that compactly supported measures admit a deterministic N-term approximation μN such that Wp(μ,μN) = O(N^-frac1d) for all d≥ 1, which matches the asymptotic optimal quantizer rate. We also extend these results to non-compactly supported measures with appropriate tail decay.