Analysis & PDE ( IF 2.2 ) Pub Date : 2021-05-18 , DOI: 10.2140/apde.2021.14.823 Leon Bungert , Martin Burger , Antonin Chambolle , Matteo Novaga
This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total variation flow have been proven previously, and by recent results on finite-dimensional polyhedral seminorms, where gradient flows can yield spectral decompositions into eigenvectors.
We provide a geometric characterization of eigenvectors via a dual unit ball and prove that they are subgradients of minimal norm. This establishes the connection to gradient flows, whose time evolution is a decomposition of the initial condition into subgradients of minimal norm. If these are eigenvectors, this implies an interesting orthogonality relation and the equivalence of the gradient flow to a variational regularization method and an inverse scale space flow. Indeed we verify that all scenarios where these equivalences were known before by other arguments — such as one-dimensional total variation, multidimensional generalizations to vector fields, or certain polyhedral seminorms — yield spectral decompositions, and we provide further examples. We also investigate extinction times and extinction profiles, which we characterize as eigenvectors in a very general setting, generalizing several results from literature.
中文翻译:
一阶泛函的梯度流引起的非线性频谱分解
本文通过考虑与无限维希尔伯特空间中的绝对一齐泛函有关的特征值问题,建立了非线性光谱分解的理论。这种方法的动机是针对总变化,其中特征值问题及其与总变化流之间的关系的有趣结果先前已得到证明,而最近有限元多面体半范数的结果已经得到证明,其中梯度流可以将频谱分解为特征向量
我们通过对偶单位球提供特征向量的几何特征,并证明它们是最小范数的子梯度。这就建立了与梯度流的联系,梯度流的时间演变是将初始条件分解为最小范数的子梯度。如果这些是特征向量,则这意味着有趣的正交关系以及梯度流与变分正则化方法和反比例空间流的等价关系。的确,我们验证了所有其他等价物之前都知道这些等价物的所有情况(例如,一维总变数,矢量场的多维概括或某些多面体半范数)会产生频谱分解,并提供了更多示例。我们还研究了灭绝时间和灭绝概况,