论文标题
总变异正规化的界限和无限度
Boundedness and unboundedness in total variation regularization
论文作者
论文摘要
我们考虑，即使测得的数据不属于$ l^\ infty $，对于线性反问题的总变化正则化的最小化是否属于$ l^\ infty $。我们提供了最小化参数的最小化器界限的简单证明，并在源条件下得出了足够小的噪声的均匀边界，并提供了足够的先验参数选择。为了表明，对于每个保真度项和维度，我们都无法预期这样的结果，我们计算出明确的径向无限的最小化器，这是通过证明与加权的一维deno deo denoing的等效性来实现的。最后，我们讨论了将这种结果扩展到相关的高阶正规化功能的可能性，从而获得了对第一阶和二阶总变化的虚拟卷积的积极答案。
We consider whether minimizers for total variation regularization of linear inverse problems belong to $L^\infty$ even if the measured data does not. We present a simple proof of boundedness of the minimizer for fixed regularization parameter, and derive the existence of uniform bounds for sufficiently small noise under a source condition and adequate a priori parameter choices. To show that such a result cannot be expected for every fidelity term and dimension we compute an explicit radial unbounded minimizer, which is accomplished by proving the equivalence of weighted one-dimensional denoising with a generalized taut string problem. Finally, we discuss the possibility of extending such results to related higher-order regularization functionals, obtaining a positive answer for the infimal convolution of first and second order total variation.
