Of particular interest for super-resolution is the line spectrum estimation problem, which consists in recovering a stream of spikes (point sources) from the noisy observation of a few number of its first trigonometric moments weighted by the ones of the point-spread function (PSF). The empirical feasibility of this problem has been known since the work of Rayleigh on diffraction to be essentially driven by the separation between the spikes to recover.

We present a novel statistical framework based on the spectrum of the Fisher information matrix (FIM) to quantify the stability limit of super-resolution as a function of the PSF. In the regime where the minimal separation is inversely proportional to the number of acquired moments, we show the existence of a separation constant above which the minimal eigenvalue of the FIM is not asymptotically vanishing—defining a statistical resolution limit. Notably, a relationship between the total variation of the autocorrelation function of the PSF and its association resolution limit is highlighted. Those novel bounds are derived by relating the extremal eigenvalues of the FIM with a higher-order Beurling–Selberg type extremal approximation problem over the functions of bounded variation, for which we provide solutions.

In this talk, we consider the problem of learning the parameters from the Fourier measurements of the one-dimensional Gaussian mixture models(GMM). Unlike most algorithms requires the number of Gaussians a prior, our method only need to know the number of different variances as prior information. We also illustrate that for stably recovering all the components under certain noise level, a separation condition for the variances is necessary. Our method can be generalized into high dimensional cases.

In this paper, we show that it is possible to overcome one of the fundamental limitations of super-resolution microscopy techniques: the necessity to be in an $\text{optically homogeneous}$ environment. Using recent modal approximation results we show as a proof of concept that it is possible to recover the position of a single point-like emitter in a $\text{known resonant environment}$ from far-field measurements with a precision two orders of magnitude below the classical Rayleigh limit. The procedure does not involve solving any partial differential equation, is computationally light (optimisation in $\mathbb{R}^d$ with $d$ of the order of 10) and therefore suited for the recovery of a very large number of single emitters.

We consider the problem of recovering a linear combination of Dirac masses from noisy Fourier samples, also known as the problem of super-resolution. Following recent derivation of min-max bounds for this problem when some of the sources collide, we develop an optimal algorithm which provably achieves these bounds in such a challenging scenario. Our method is based on the well-known Prony's method for exponential fitting, and a novel analysis of its stability in the near-colliding regime, combined with the decimation technique for improving the conditioning of the problem.

Based on joint works with N.Diab and R.Katz:

[1] R. Katz, N. Diab, D. Batenkov. Decimated Prony's Method for Stable Super-resolution. 2022. http://arxiv.org/abs/2210.13329

[2] R. Katz, N. Diab, D. Batenkov. On the accuracy of Prony's method for recovery of exponential sums with closely spaced exponents. 2023. http://arxiv.org/abs/2302.05883