# Inducing Riesz and orthonormal bases in L2 via composition operators

arXiv: https://arxiv.org/abs/2406.18613 · HTML: https://mlsanity.com/paper/2406.18613/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 27 references that are in the map.
- First posted on arXiv 25 Jun 2024.

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

- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Ab-initio study of interacting fermions at finite temperature with neural canonical transformation](https://mlsanity.com/paper/2105.08644/) (2021)
- [Inferring the quantum density matrix with machine learning](https://mlsanity.com/paper/1904.05903/) (2019)
- [Computing excited states of molecules using normalizing flows](https://mlsanity.com/paper/2308.16468/) (2023)

## Abstract (arXiv)

Let Ch be a composition operator mapping L2(Omega1) into L2(Omega2) for some open sets Omega1, Omega2 subseteq ℝn. We characterize the mappings h that transform Riesz bases of L2(Omega1) into Riesz bases of L2(Omega2). Restricting our analysis to differentiable mappings, we demonstrate that mappings h that preserve Riesz bases have Jacobian determinants that are bounded away from zero and infinity. We discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct Riesz bases with favorable approximation properties.