Determining the radial current density profile j(r) in a tokamak plasma from magnetic probe measurements taken outside the current column is an example of a poorly-determined inverse problem, since the measurements are generally responsive to integral rather than local properties of j(r). An exception occurs when the plasma boundary surface includes one or more magnetic X-points. The current flowing in radial annuli close to the boundary is increasingly concentrated around the X-point(s) and its influence is accordingly strongest on nearby probes, thus admitting the possibility of determining edge-localized moments of j(r) from probe data. Singular Value Decomposition provides an intuitively powerful method of identifying the spatial moments of j(r) that can be determined from probe data: The singular vectors give the spatial structure of such moments and the singular values can be interpreted as the signal strength of these moments. Judging the recoverability of such moments is a simple matter of comparing the singular values with the experimental noise level in the probes. Examples of this approach from the ASDEX Upgrade tokamak experiment will be provided.