During the inversion of discrete linear systems, noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion, which is a process called regularization. The influence of provided prior information is controlled by non-negative regularization parameter(s). There are a number of methods used to select appropriate regularization parameters, as well as a number of methods used for inversion. In this paper, we consider the unbiased risk estimator, generalized cross validation, and the discrepancy principle as the means of selecting regularization parameters. When multiple data sets describing the same physical phenomena are available, the use of multiple regularization parameters can enhance results. Here we demonstrate that it is possible to learn multiple parameter regularization parameters using regularization parameter estimators that are modified to handle multiple parameters and multiple data. The results demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to methods based on true data for learning the relevant parameters.
翻译:在离散线性系统倒置期间,数据中的噪音可以放大,从而产生毫无意义的解决办法。为了消除这一影响,在倒置过程中,数学上执行被认为是可取的解决办法的特点,这是一个称为正规化的过程。先前提供的信息的影响由非消极的正规化参数控制。有多种方法用来选择适当的正规化参数,以及用于倒置的一些方法。在本文件中,我们认为,选择正规化参数的手段是无偏见的风险估计值、通用交叉验证和差异原则。当有多个数据集描述相同的物理现象时,使用多重正规化参数可以加强结果。在这里,我们表明,使用经修改的正规化参数估计值来处理多个参数和多个数据是有可能学习多个参数的。结果显示,这些修改方法不需要使用真正的数据来学习正规化参数,是有效和高效的,并且与根据真实数据学习相关参数的方法相匹配。