# The Paulsen Problem Made Simple

arXiv: https://arxiv.org/abs/1809.04726 · HTML: https://mlsanity.com/paper/1809.04726/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2018.

## Built on by (most cited first, 2 of 2)

- [A Strongly Polynomial Algorithm for Approximate Forster Transforms and its Application to Halfspace Learning](https://mlsanity.com/paper/2212.03008/) (2022)
- [Optimal Bounds for Tyler's M-Estimator for Elliptical Distributions](https://mlsanity.com/paper/2510.13751/) (2025)

## Abstract (arXiv)

The Paulsen problem is a basic problem in operator theory that was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every ε-nearly equal norm Parseval frame in d dimensions is within squared distance O(ε d13/2) of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of O(ε d2).