Beyond convergence rates: exact recovery with the Tikhonov regularization with sparsity constraints
Beyond convergence rates: exact recovery with the Tikhonov regularization with sparsity constraints
复制标题
超越收敛率：通过具有稀疏性约束的吉洪诺夫正则化进行精确恢复
DOI：
10.1088/0266-5611/27/8/085009
复制
发表时间：
2010
期刊：
影响因子：
2.1
通讯作者：
D. Trede
中科院分区：
文献类型：
--
作者：
Dirk A. Lorenz;Stefan Schiffler;D. Trede
研究了带罚函数的线性不适定问题的Tikhonov正则化。我们回顾结果的线性收敛速度和结果的支持精确恢复。此外，我们得到的条件，确切的支持恢复是特别适用于不适定问题的情况下，其中其他条件，例如，基于所谓的相干性或受限等距性的方法通常不适用。所得结果还表明，正则化解不仅收敛于n = 1-范数，而且收敛于向量空间n = 0（当n趋于无穷大时，作为空间的严格归纳极限）.此外，不同的条件之间的精确支持恢复和线性收敛速度的关系进行了研究。从数字全息成像的例子说明了所获得的结果的适用性，即，一个可以检查先验，如果实验设置保证准确的恢复与稀疏约束的吉洪诺夫正则化。
The Tikhonov regularization of linear ill-posed problems with an ℓ1 penalty is considered. We recall results for linear convergence rates and results on exact recovery of the support. Moreover, we derive conditions for exact support recovery which are especially applicable in the case of ill-posed problems, where other conditions, e.g., based on the so-called coherence or the restricted isometry property are usually not applicable. The obtained results also show that the regularized solutions do not only converge in the ℓ1-norm but also in the vector space ℓ0 (when considered as the strict inductive limit of the spaces as n tends to infinity). Additionally, the relations between different conditions for exact support recovery and linear convergence rates are investigated. With an imaging example from digital holography the applicability of the obtained results is illustrated, i.e. that one may check a priori if the experimental setup guarantees exact recovery with the Tikhonov regularization with sparsity constraints.