# On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data

arXiv: https://arxiv.org/abs/1407.2490 · HTML: https://mlsanity.com/paper/1407.2490/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jul 2014.

## Built on by (most cited first, 1 of 1)

- [The Optimization Landscape of Carath'eodory Decomposition of Toeplitz Covariances](https://mlsanity.com/paper/2511.01605/) (2025)

## Abstract (arXiv)

This paper is concerned about sparse, continuous frequency estimation in line spectral estimation, and focused on developing gridless sparse methods which overcome grid mismatches and correspond to limiting scenarios of existing grid-based approaches, e.g., ℓ1 optimization and SPICE, with an infinitely dense grid. We generalize AST (atomic-norm soft thresholding) to the case of nonconsecutively sampled data (incomplete data) inspired by recent atomic norm based techniques. We present a gridless version of SPICE (gridless SPICE, or GLS), which is applicable to both complete and incomplete data without the knowledge of noise level. We further prove the equivalence between GLS and atomic norm-based techniques under different assumptions of noise. Moreover, we extend GLS to a systematic framework consisting of model order selection and robust frequency estimation, and present feasible algorithms for AST and GLS. Numerical simulations are provided to validate our theoretical analysis and demonstrate performance of our methods compared to existing ones.