A Stability Result for Sparse Convolutions

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.

(i) minor revision of the text (ii) use definition of Freiman dimension as in [3] (iii) updated references

References in corpus (1)

• Compressed Sensing on the Image of Bilinear Maps

Cited by in corpus (2)

• Self-Calibration and Biconvex Compressive Sensing
• Sparse Signal Processing Concepts for Efficient 5G System Design