# Reconstructing Rational Functions on Finite Abelian Groups with Higher Autocorrelations

arXiv: https://arxiv.org/abs/2503.21022 · HTML: https://mlsanity.com/paper/2503.21022/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 44 references that are in the map.
- First posted on arXiv 26 Mar 2025.

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

- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [Estimation under group actions: recovering orbits from invariants](https://mlsanity.com/paper/1712.10163/) (2017)
- [Low Autocorrelation Binary Sequences](https://mlsanity.com/paper/1512.02475/) (2015)
- [The sample complexity of multi-reference alignment](https://mlsanity.com/paper/1707.00943/) (2017)
- [Optimization Methods for Designing Sequences with Low Autocorrelation Sidelobes](https://mlsanity.com/paper/1501.02252/) (2014)

## Abstract (arXiv)

The higher-order autocorrelations of integer-valued or rational-valued functions on finite Abelian groups appear naturally in X-ray crystallography, and have applications in computer vision systems, correlation tomography, correlation spectroscopy, and pattern recognition. In this paper, we consider the problem of reconstructing a rational-valued function on finite Abelian groups from its higher-order autocorrelations. We describe an explicit reconstruction algorithm, and prove that the autocorrelations up to order 3r+3 are always sufficient to determine the data up to translation, where r is the rank of the group. We also provide examples of rational-valued functions on finite Abelian group which are not determined by their autocorrelations up to order 3r+2. In particular, we provide a sharp upper bound on the separating degree of the regular representation of a finite Abelian group in terms of its rank.