Super-resolution of near-colliding point sources

Abstract

<jats:title>Abstract</jats:title> <jats:p>We consider the problem of stable recovery of sparse signals of the form $$\begin{equation*}F(x)=\sum_{j=1}^d a_j\delta(x-x_j),\quad x_j\in\mathbb{R},\;a_j\in\mathbb{C}, \end{equation*}$$from their spectral measurements, known in a bandwidth $\varOmega $ with absolute error not exceeding $\epsilon&amp;gt;0$. We consider the case when at most $p\leqslant d$ nodes $\{x_j\}$ of $F$ form a cluster whose extent is smaller than the Rayleigh limit ${1\over \varOmega }$, while the rest of the nodes is well separated. Provided that $\epsilon \lessapprox \operatorname{SRF}^{-2p+1}$, where $\operatorname{SRF}=(\varOmega \varDelta )^{-1}$ and $\varDelta $ is the minimal separation between the nodes, we show that the minimax error rate for reconstruction of the cluster nodes is of order ${1\over \varOmega }\operatorname{SRF}^{2p-1}\epsilon $, while for recovering the corresponding amplitudes $\{a_j\}$ the rate is of the order $\operatorname{SRF}^{2p-1}\epsilon $. Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are ${\epsilon \over \varOmega }$ and $\epsilon $, respectively. These results suggest that stable super-resolution is possible in much more general situations than previously thought. Our numerical experiments show that the well-known matrix pencil method achieves the above accuracy bounds.</jats:p>

Sparse super resolution is Lipschitz continuous

“…3 Compared to the ESPRIT-map, the order in N in the stability result (3) might be improved for the Prony-map: Remark 2.7. Assume as in [4] that the node set of μ, Y μ = {t j } j , does only have one cluster Y ⊂ Y μ where the nodes can lie very closely together, whereas all nodes outside of the cluster, i.e.…”

“…for all t j ∈ Y μ, j = ℓ, and a constant c > 0. In contrast to our setting, [4] starts from a more information theory based approach of assuming continuous knowledge about the Fourier transform of the measure µ instead of discrete moments as in (2). The main result [4, Thm.…”

Sum-Rate Channel Capacity for Line-of-Sight Models

“…Wireless performance studies usually employ Vandermonde matrices as mathematical channel models in applications such as system identification, harmonic analysis, direction-finding and precoding [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 ]. Specifically, given a particular wireless environment, the sum-rate channel capacity is better modelled by the Vandermonde matrix approach as presented in [ 20 , 21 , 22 , 23 ].…”

On Overfitting in Discrete Super-Resolution Recovery

“…In [7][8][9][10], the target signal is assumed to be supported on a predetermined fine grid. In [11][12][13][14][15][16], gridless approaches are considered to allow continuously located sources and avoid the grid mismatch problem. Available gridless sparse super-resolution algorithms are based on atomic norm or TV-norm which are continuous surrogates for the 1norm in the discrete case.…”

“…The first non-asymptotic stability analysis of discrete super-resolution problem was developed in [8] with a novel interpolationbased proof technique. Similar ideas have been adopted and extended in many following studies that sought to establish the stability guarantees of different super-resolution algorithms [9,10,[15][16][17][18].…”