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Techniques for finding regularized solutions to underdetermined linearsystems can be viewed as imposing prior knowledge on the unknown vector. Thesuccess of modern techniques, which can impose priors such as sparsity andnon-negativity, is the result of advances in optimization algorithms to solveproblems which lack closed-form solutions. Techniques for characterization andanalysis of the system to determined when information is recoverable, however,still typically rely on closed-form solution techniques such as singular valuedecomposition or a filter cutoff, for example. In this letter we poseoptimization approaches to broaden the approach to system characterization. Westart by deriving conditions for when each unknown element of a system admits aunique solution, subject to a broad class of types of prior knowledge. Withthis approach we can pose a convex optimization problem to find "how unique"each element of the solution is, which may be viewed as a generalization ofresolution to incorporate prior knowledge. We find that the result varies withthe unknown vector itself, i.e. is data-dependent, such as when the sparsity ofthe solution improves the chance it can be uniquely reconstructed. The approachcan be used to analyze systems on a case-by-case basis, estimate the amount ofimportant information present in the data, and quantitatively understand thedegree to which the regularized solution may be trusted.

Element-wise uniqueness, prior knowledge, and data-dependent resolution