# Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction

arXiv: https://arxiv.org/abs/1503.02893 · HTML: https://mlsanity.com/paper/1503.02893/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Mar 2015.

## Built on by (most cited first, 5 of 5)

- [Efficient learning of hidden state LTI state space models of unknown order](https://mlsanity.com/paper/2202.01625/) (2022)
- [Network Topology Inference with Sparsity and Laplacian Constraints](https://mlsanity.com/paper/2309.00960/) (2023)
- [System Identification via Nuclear Norm Regularization](https://mlsanity.com/paper/2203.16673/) (2022)
- [Outlier Detection Using Generative Models with Theoretical Performance Guarantees](https://mlsanity.com/paper/2310.09999/) (2023)
- [Frequency Response Identification of Low-Order Systems: Finite-Sample Analysis](https://mlsanity.com/paper/2508.17142/) (2025)

## Abstract (arXiv)

This paper explores robust recovery of a superposition of R distinct complex exponential functions from a few random Gaussian projections. We assume that the signal of interest is of 2N-1 dimensional and R<<2N-1. This framework covers a large class of signals arising from real applications in biology, automation, imaging science, etc. To reconstruct such a signal, our algorithm is to seek a low-rank Hankel matrix of the signal by minimizing its nuclear norm subject to the consistency on the sampled data. Our theoretical results show that a robust recovery is possible as long as the number of projections exceeds O(Rln2N). No incoherence or separation condition is required in our proof. Our method can be applied to spectral compressed sensing where the signal of interest is a superposition of R complex sinusoids. Compared to existing results, our result here does not need any separation condition on the frequencies, while achieving better or comparable bounds on the number of measurements. Furthermore, our method provides theoretical guidance on how many samples are required in the state-of-the-art non-uniform sampling in NMR spectroscopy. The performance of our algorithm is further demonstrated by numerical experiments.