Elastic-net regularization versus ℓ1-regularization for linear inverse problems with quasi-sparse solutions
Elastic-net regularization versus ℓ1-regularization for linear inverse problems with quasi-sparse solutions
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DOI：
10.1088/1361-6420/33/1/015004
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发表时间：
2016-04
期刊：
影响因子：
2.1
通讯作者：
De-Han Chen;B. Hofmann;J. Zou
中科院分区：
文献类型：
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作者：
De-Han Chen;B. Hofmann;J. Zou
我们考虑不适定算子方程 Ax = y，其具有 ℓ2 和希尔伯特空间 Y 之间的单射有界线性算子 A 映射，具有唯一解 x†={x†k}k=1∞。对于稀疏性 x†εℓ0 是预期的但在实践中经常轻微违反的情况，我们与 ℓ1 正则化进行比较，研究弹性网络正则化，其中惩罚是 ℓ1-范数和 ℓ2-范数平方的加权叠加，假设 x†εℓ1。该方法中出现了两个正参数，权重参数 η 和正则化参数作为吉洪诺夫泛函中整个惩罚的乘数，而 ℓ1-正则化中只出现了一个正则化参数。基于描述前向算子 A 的解平滑度的变分不等式方法，并利用近似源条件的方法，我们提出了一些估计弹性网络正则化收敛速度的结果。发生率函数包含 k→∞ 时 x†k→0 的衰减率以及 x† 作为 ℓ2 中元素的经典平滑特性。
We consider the ill-posed operator equation Ax = y with an injective and bounded linear operator A mapping between ℓ2 and a Hilbert space Y, possessing the unique solution x†={x†k}k=1∞. For the cases that sparsity x†∈ℓ0 is expected but often slightly violated in practice, we investigate in comparison with the ℓ1-regularization the elastic-net regularization, where the penalty is a weighted superposition of the ℓ1-norm and the ℓ2-norm square, under the assumption that x†∈ℓ1. There occur two positive parameters in this approach, the weight parameter η and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in ℓ1-regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator A and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay x†k→0 for k→∞ and the classical smoothness properties of x† as an element in ℓ2.