稀疏约束下Tikhonov正则化的收敛速度和源条件
Convergence rates and source conditions for Tikhonov regularization with sparsity constraints
摘要：
本文通过加权$\ell^p$惩罚$0\leq p\leq 2$的稀疏约束的正则化。对于$1\leq p\leq 2$，特别注意在范数和源条件下的收敛速度。作为主要结果，证明了当未知解是稀疏的时，对于$1\leq p\leq 2$，在$\sqrtδ$范数下有收敛速度。案例$p=1$需要一种特殊的技术，其中不仅必须使用Bregman距离，还必须使用所谓的Bregman-Taylor距离。对于$p<1$，仅显示初步结果。这些结果表明，与$p\geq 1$不同，正则化性质依赖于算子的相互作用和稀疏性基础。$p=0$的一个反例表明正则化不需要发生。
英文摘要：
This paper addresses the regularization by sparsity constraints by means of weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main result it is proven that one gets a convergence rate in norm of $\sqrtδ$ for $1\leq p\leq 2$ as soon as the unknown solution is sparse. The case $p=1$ needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed. For $p<1$ only preliminary results are shown. These results indicate that, different from $p\geq 1$, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for $p=0$ shows that regularization need not to happen.
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