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Total variation (TV) minimization is one of the most important techniques inmodern signal/image processing, and has wide range of applications. While thereare numerous recent works on the restoration guarantee of the TV minimizationin the framework of compressed sensing, there are few works on the restorationguarantee of the restoration from partial observations. This paper is toanalyze the error of TV based restoration from random entrywise samples. Inparticular, we estimate the error between the underlying original data and theapproximate solution that interpolates (or approximates with an error bounddepending on the noise level) the given data that has the minimal TV seminormamong all possible solutions. Finally, we further connect the error estimatefor the discrete model to the sparse gradient restoration problem and to theapproximation to the underlying function from which the underlying true datacomes.

Approximation Theory of Total Variation Minimization for Data Completion