# Geometry and Symmetry in Short-and-Sparse Deconvolution

arXiv: https://arxiv.org/abs/1901.00256 · HTML: https://mlsanity.com/paper/1901.00256/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2019.

## Built on by (most cited first, 11 of 11)

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- [Resource-Efficient Invariant Networks: Exponential Gains by Unrolled Optimization](https://mlsanity.com/paper/2203.05006/) (2022)
- [Straight-Through meets Sparse Recovery: the Support Exploration Algorithm](https://mlsanity.com/paper/2301.13584/) (2023)
- [Blind Time-of-Flight Imaging: Sparse Deconvolution on the Continuum with Unknown Kernels](https://mlsanity.com/paper/2411.00893/) (2024)
- [Sharper Convergence Rates for Nonconvex Optimisation via Reduction Mappings](https://mlsanity.com/paper/2506.08428/) (2025)
- [Unlocking Off-the-Grid Sparse Recovery with Unlimited Sensing: Simultaneous Super-Resolution in Time and Amplitude](https://mlsanity.com/paper/2510.16948/) (2025)

## Abstract (arXiv)

We study the textitShort-and-Sparse (SaS) deconvolution problem of recovering a short signal mathbf a0 and a sparse signal mathbf x0 from their convolution. We propose a method based on nonconvex optimization, which under certain conditions recovers the target short and sparse signals, up to a signed shift symmetry which is intrinsic to this model. This symmetry plays a central role in shaping the optimization landscape for deconvolution. We give a textitregional analysis, which characterizes this landscape geometrically, on a union of subspaces. Our geometric characterization holds when the length-p0 short signal mathbf a0 has shift coherence μ, and mathbf x0 follows a random sparsity model with sparsity rate θ ∈ Bigl[fracc1p0, fracc2p0√μ + √p0Bigr]·frac1log2p0. Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.