A Stability Result for Sparse Convolutions

TL;DR

This paper proves a stability property for sparse convolutions on torsion-free abelian groups, showing that injectivity is guaranteed with a universal lower bound depending only on support sizes.

Contribution

It establishes a dimension-independent stability result for sparse convolutions on torsion-free groups, using a novel reduction and compression technique.

Findings

Sparse convolutions are free of cancellations on torsion-free groups.

Injectivity with a universal lower bound depends only on support sizes.

The result is dimension-independent and relies on a reduction argument.

Abstract

We will establish in this note a stability result for sparse convolutions on torsion-free additive (discrete) abelian groups. Sparse convolutions on torsion-free groups are free of cancellations and hence admit stability, i.e. injectivity with a universal lower bound $\alpha=\alpha(s,f)$, only depending on the cardinality $s$ and $f$ of the supports of both input sequences. More precisely, we show that $\alpha$ depends only on $s$ and $f$ and not on the ambient dimension. This statement follows from a reduction argument which involves a compression into a small set preserving the additive structure of the supports.