On Ambiguity in Linear Inverse Problems: Entrywise Bounds on Nearly Data-Consistent Solutions and Entrywise Condition Numbers

Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree of ambiguity that may exist about its solution. Traditional measures of ill-posedness, such as the condition number of a matrix, provide characterizations that are global in nature. While such characterizations can be powerful, they can also fail to provide full insight into situations where certain entries of the solution vector are more or less ambiguous than others. In this work, we derive novel theoretical lower- and upper-bounds that apply to individual entries of the solution vector, and are valid for all potential solution vectors that are nearly data-consistent. These bounds are agnostic to the noise statistics and the specific method used to solve the inverse problem, and are also shown to be tight. In addition, our results also lead us to introduce an entrywise version of the traditional condition number, which provides a substantially more nuanced characterization of scenarios where certain elements of the solution vector are less sensitive to perturbations than others. Our results are illustrated in an application to magnetic resonance imaging reconstruction, and we include discussions of practical computation methods for large-scale inverse problems, connections between our new theory and the traditional Cram\'{e}r-Rao bound under statistical modeling assumptions, and potential extensions to cases involving constraints beyond just data-consistency

On Optimality in ROVir

We recently published an approach named ROVir (Region-Optimized Virtual coils) that uses the beamforming capabilities of a multichannel magnetic resonance imaging (MRI) receiver array to achieve coil compression (reducing an original set of receiver channels into a much smaller number of virtual channels for the purposes of dimensionality reduction), while simultaneously preserving the MRI signal from desired spatial regions and suppressing the MRI signal arising from unwanted spatial regions. The original ROVir procedure is computationally-simple to implement (involving just a single small generalized eigendecomposition), and its signal-suppression capabilities have proven useful in an increasingly wide range of MRI applications. Our original paper made claims about the theoretical optimality of this generalized eigendecomposition procedure, but did not present the details. The purpose of this write-up is to elaborate on these mathematical details, and to introduce a new greedy iterative ROVir algorithm that enjoys certain advantages over the original ROVir calculation approach. This discussion is largely academic, with implications that we suspect will be minor for practical applications -- we have only observed small improvements to ROVir performance in the cases we have tried, and it would have been safe in these cases to still use the simpler original calculation procedure with negligible practical impact on the final imaging results.Comment: 7 pages, 4 figure