# Algorithmic Foundations for the Diffraction Limit

arXiv: https://arxiv.org/abs/2004.07659 · HTML: https://mlsanity.com/paper/2004.07659/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Apr 2020.

## Built on by (most cited first, 6 of 6)

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- [A Fourier Approach to Mixture Learning](https://mlsanity.com/paper/2210.02415/) (2022)
- [Sample Complexity Bounds for Learning High-dimensional Simplices in Noisy Regimes](https://mlsanity.com/paper/2209.05953/) (2022)
- [Algorithms from Invariants: Smoothed Analysis of Orbit Recovery over SO(3)](https://mlsanity.com/paper/2106.02680/) (2021)
- [Learning Mixture Models via Efficient High-dimensional Sparse Fourier Transforms](https://mlsanity.com/paper/2601.05157/) (2026)
- [Dynamic super-resolution in particle tracking problems](https://mlsanity.com/paper/2204.04092/) (2022)

## Abstract (arXiv)

For more than a century and a half it has been widely-believed (but was never rigorously shown) that the physics of diffraction imposes certain fundamental limits on the resolution of an optical system. However our understanding of what exactly can and cannot be resolved has never risen above heuristic arguments which, even worse, appear contradictory. In this work we remedy this gap by studying the diffraction limit as a statistical inverse problem and, based on connections to provable algorithms for learning mixture models, we rigorously prove upper and lower bounds on the statistical and algorithmic complexity needed to resolve closely spaced point sources. In particular we show that there is a phase transition where the sample complexity goes from polynomial to exponential. Surprisingly, we show that this does not occur at the Abbe limit, which has long been presumed to be the true diffraction limit.