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Abstract We consider the problem of recovering the superposition of $$R$$ distinct complex exponential functions from compressed non-uniform time-domain samples. Total variation (TV) minimization or atomic norm minimization was proposed in the literature to recover the $$R$$ frequencies or the missing data. However, it is known that in order for TV minimization and atomic norm minimization to recover the missing data or the frequencies, the underlying $$R$$ frequencies are required to be well separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve the $$R$$ complex exponentials and their frequencies from compressed non-uniform measurements, regardless of how close their frequencies are to each other. We propose a new concept of orthonormal atomic norm minimization (OANM), and demonstrate that the success of Hankel matrix recovery in separation-free super-resolution comes from the fact that the nuclear norm of a Hankel matrix is an orthonormal atomic norm. More specifically, we show that, in traditional atomic norm minimization, the underlying parameter values must be well separated to achieve successful signal recovery, if the atoms are changing continuously with respect to the continuously valued parameter. In contrast, for the OANM, it is possible the OANM is successful even though the original atoms can be arbitrarily close. As a byproduct of this research, we provide one matrix-theoretic inequality of nuclear norm, and give its proof using the theory of compressed sensing. 

The rapid spread of SARS-CoV-2 has placed a significant burden on public health systems to provide swift and accurate diagnostic testing highlighting the critical need for innovative testing approaches for future pandemics. In this study, we present a novel sample pooling procedure based on compressed sensing theory to accurately identify virally infected patients at high prevalence rates utilizing an innovative viral RNA extraction process to minimize sample dilution. At prevalence rates ranging from 0–14.3%, the number of tests required to identify the infection status of all patients was reduced by 69.26% as compared to conventional testing in primary human SARS-CoV-2 nasopharyngeal swabs and a coronavirus model system. Our method provided quantification of individual sample viral load within a pool as well as a binary positive-negative result. Additionally, our modified pooling and RNA extraction process minimized sample dilution which remained constant as pool sizes increased. Compressed sensing can be adapted to a wide variety of diagnostic testing applications to increase throughput for routine laboratory testing as well as a means to increase testing capacity to combat future pandemics.