A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition. We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data. The proposed framework allows for flexible incorporation of separate regularisers for spatial and temporal features. For the special case of a stationary operator, we can effectively use the decomposition to reduce the computational complexity and obtain a substantial speed-up. The proposed methods are evaluated for two simulated phantoms and we compare the obtained results to a separate low-rank reconstruction and subsequent decomposition approach based on the widely used principal component analysis.
Trading off 1-norm and sparsity against rank for linear models using mathematical optimization: 1-norm minimizing partially reflexive ah-symmetric generalized inverses
Inverse scattering beyond Born approximation via rotation-equivariance-aware neural network and low-rank structure
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Department of Mathematical Sciences, The University of Texas at Dallas（德克萨斯大学达拉斯分校数学系）
A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs
Institute for Computational Engineering & Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
A theoretical and computational framework for three dimensional inverse medium scattering using the linearized low-rank structure
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；PRISME, EDF R&D（EDF研发部PRISME）；UMA, Inria, ENSTA Paris, Institut Polytechnique de Paris（巴黎理工学院ENSTA巴黎大学应用数学实验室，法国国家数字科学研究所）；Department of Mathematical Sciences, The University of Texas at Dallas（德克萨斯大学达拉斯分校数学系）
Jointly Low-Rank and Bisparse Recovery: Questions and Partial Answers
我们调查的问题，恢复联合$r$秩和$s$双解析矩阵从尽可能少的线性测量，考虑任意测量以及秩一测量。在这两种情况下，我们表明，$m \asymp r s \ln(en/s)$测量使恢复在理论上是可能的，这意味着通过一个不切实际的算法。在任意测量的情况下，我们研究了当$m \asymp r s^γ\ln(en/s)$为某个指数$γ> 0$时，通过迭代硬阈值算法实现实际恢复的可能性。我们表明，这是可行的$γ= 2$，提出的分析不能涵盖的情况下$γ\leq 1$。最优指数$γ\in [1,2]$的精确值是本文提出但未解决的关于联合低秩双解析结构的头部投影的问题的目标。一些相关的问题在这里得到了部分的回答。对于秩一测量，我们建议在《双城之战》神秘的理由迭代硬阈值算法修改，以利用非标准的限制等距属性服从这类测量。
