Total variation multiscale estimators for linear inverse problems

Even though the statistical theory of linear inverse problems is a well-studied topic, certain relevant cases remain open. Among these is the estimation of functions of bounded variation (BV), meaning L-1 functions on a d-dimensional domain whose weak first derivatives are finite Radon measures. The estimation of BV functions is relevant in many applications, since it involves minimal smoothness assumptions and gives simplified, interpretable cartoonized reconstructions. In this paper, we propose a novel technique for estimating BV functions in an inverse problem setting and provide theoretical guaranties by showing that the proposed estimator is minimax optimal up to logarithms with respect to the L-q-risk, for any q is an element of [1,infinity). This is to the best of our knowledge the first convergence result for BV functions in inverse problems in dimension d >= 2, and it extends the results of Donoho (1995, Appl. Comput. Harmon. Anal., 2, 101-126) in d = 1. Furthermore, our analysis unravels a novel regime for large q in which the minimax rate is slower than n(-1/(d+2 beta+2)), where beta is the degree of ill-posedness: our analysis shows that this slower rate arises from the low smoothness of BV functions. The proposed estimator combines variational regularization techniques with the wavelet-vaguelette decomposition of operators.