Jeffrey Hoch

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 Abstract   Theoretical results from the field of compressed sensing (CS), principally theorems specifying the minimum number of samples need to reconstruct a sparse spectrum, led to an explosion of activity in NMR on efficient sampling strategies and spectral reconstruction methods. While these results firmly established the notion that sparse multidimensional NMR spectra can be accurately recovered using far fewer samples and with higher resolution than is possible using conventional uniform sampling, quantitative agreement between the theorems and empirical observations remains elusive. Contributing to this discordance, NMR spectra do not satisfy the strict definitions of sparseness assumed by CS, and practical algorithms may not achieve the limitning.limit behav.Inbehaviorprescribed by the .theorems. addition, Monajemi and Donoho \cite{MoDo17} recently showed that certain specifics of NMR experiment design require modification of CS theories. For example, when uniform sampling is conducted in a dimension orthogonal to dimensions that are sampled non-uniformly,  an excess coherence is introduced that increases the minimum number of samples required to recover the spectrum. Similarly, the simple quantification of coherence used in CS requires modification when phase  subdimensions are sampled nonuniformly (as in random phase detection or partial component sampling \cite{21949370,23246651}). Beyond the minimum bounds on the number of samples needed to recover a sparse multidimensional NMR spectrum, the notion of phase transitions in CS, the sharp transition from successful recovery to failure, holds important implications for the design of NMR experiments, including the impact of higher magnetic fields. Here we consider the broader implications of CS theorems for nonuniform sampling in multidimensional NMR via a dialog between mathematicians and NMR spectroscopists. Promising avenues for further investigation emerge from the dialog. IntroductionNMR spectroscopists have long intuited, with ample empirical evidence, that sparse multidimensional NMR spectra can be faithfully reconstructed from far fewer measurements than are required in the Jeener paradigm of parametric uniform sampling of indirect time dimensions. The field of compressed sensing (CS) didn’t so much emerge as coalesce beginning in 2006, around rigorous theorems placing bounds on the minimum number of samples needed to reconstruct a spectrum. CS theorems have continued to evolve, decreasing the lower bounds below the initial rather conservative limits. Beyond the quantitative lower bounds provided by CS theorems, the concepts of a phase diagram linking sampling coverage and spectral sparsity and the phase transition separating the regimes corresponding to successful and unsuccessful recovery have important implications for experiment design in multidimensional NMR, including the effect of higher magnetic fields. In addition to discussing these implications, we consider some recent results from CS that Reconstruction guaranteesResearchers in the field of compresses sensing have carefully studied the number of required samples for successful reconstruction of unknown signals from undersampled data.  The most well-known theoretical arguments include coherence\cite{Donoho_2006,Tropp_2004,Monajemi2016ACHA} , restricted isometry property (RIP) \cite{CandesStable} and phase transition \cite{DoTa10,MoDo17,DoTa05}  theories. Coherence and RIP arguments provide sufficient conditions for a successful reconstruction, which often lead to pessimistic lower bounds on the number required samples. Phase transition theories, on the other hand, measure the exact probability of successful reconstruction and lead to accurate theoretical lower bounds that match the empirical results exactly \cite{Monajemi_2012}. New results for multidimensional NMR Traditional CS theories are not directly applicable for reconstruction of  NMR spectra due to anisotropy of sampling and  phase-sensitive nature of the experiments. Accommodating these additional considerations requires a new new notion of coherence and further theoretical investigation of phase transition phenomena, which has been recently examined in detail [cite Hatef's theiss]Hatef']Stanford.   thesis]. Here, we discuss their implications  for  the field of NMR spectroscopy.  Coherence guarantee