论文标题
预计的牛顿方法限制了$ \ ell_p $正则化
Projected Newton method for noise constrained $\ell_p$ regularization
论文作者
论文摘要
为了获得有意义的解决方案，对于因测量误差或噪声而污染了有意义的线性逆问题，必须选择适当的正则化项。 $ \ ell_p $ norm涵盖了正规化术语的各种选择，因为其行为至关重要取决于$ p $的选择，并且由于它可以轻松与合适的正则化矩阵结合使用。我们开发了一种有效的算法，该算法同时确定正则化参数和相应的$ \ ell_p $正则化解决方案，以便满足差异原理。我们将问题投射到低维广义的Krylov子空间上，并为这个较小的问题计算牛顿方向。我们说明了该算法的一些有趣属性，并使用许多数值实验将其性能与其他最先进的方法进行了比较，并具有诱导$ \ ell_1 $ norm的稀疏性的特殊焦点，并且Edge Presserver and Edge Preserving总变异正则化。
Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The $\ell_p$ norm covers a wide range of choices for the regularization term since its behavior critically depends on the choice of $p$ and since it can easily be combined with a suitable regularization matrix. We develop an efficient algorithm that simultaneously determines the regularization parameter and corresponding $\ell_p$ regularized solution such that the discrepancy principle is satisfied. We project the problem on a low-dimensional Generalized Krylov subspace and compute the Newton direction for this much smaller problem. We illustrate some interesting properties of the algorithm and compare its performance with other state-of-the-art approaches using a number of numerical experiments, with a special focus of the sparsity inducing $\ell_1$ norm and edge-preserving total variation regularization.
