  Techniques for finding regularized solutions to underdetermined linear
systems can be viewed as imposing prior knowledge on the unknown vector. The
success of modern techniques, which can impose priors such as sparsity and
non-negativity, is the result of advances in optimization algorithms to solve
problems which lack closed-form solutions. Techniques for characterization and
analysis of the system to determined when information is recoverable, however,
still typically rely on closed-form solution techniques such as singular value
decomposition or a filter cutoff, for example. In this letter we pose
optimization approaches to broaden the approach to system characterization. We
start by deriving conditions for when each unknown element of a system admits a
unique solution, subject to a broad class of types of prior knowledge. With
this 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 of
resolution to incorporate prior knowledge. We find that the result varies with
the unknown vector itself, i.e. is data-dependent, such as when the sparsity of
the solution improves the chance it can be uniquely reconstructed. The approach
can be used to analyze systems on a case-by-case basis, estimate the amount of
important information present in the data, and quantitatively understand the
degree to which the regularized solution may be trusted.
