稀疏加低秩矩阵嵌入及其在癌症放疗优化中的应用
Sparse plus low-rank matrix embedding with applications in cancer radiotherapy optimization
- School of Informatics, the University of Edinburgh（爱丁堡大学信息学院）
- Department of Medical Physics, Memorial Sloan Kettering Cancer Center（纪念斯隆-凯特琳癌症中心医学物理系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出稀疏加低秩矩阵嵌入（SLME）及R3-Trust算法，用于高效近似大型稠密矩阵，在癌症放疗优化中平衡精度与计算成本，并在恢复任务中优于现有方法。
AI中文摘要：
将矩阵分解为稀疏和低秩分量是鲁棒主成分分析的核心，并在机器学习、信号处理和计算机视觉中有广泛应用。经典公式旨在恢复潜在的稀疏和低秩结构。我们转而引入稀疏加低秩矩阵嵌入（SLME），其目标是构造一个大型稠密矩阵的计算高效替代品，而不要求其分量可解释。给定$A\in\mathbb{R}^{m\times n}$，SLME近似$A \approx S+HW$，其中$S$为稀疏矩阵，$H\in\mathbb{R}^{m\times r}$，$W\in\mathbb{R}^{r\times n}$，且$r\ll \min\{m,n\}$。所得的矩阵-向量乘积可计算为$Sx+H(Wx)$，其运算量为$\mathrm{nnz}(S)+r(m+n)$，而非$Ax$所需的$mn$次运算。我们的主要动机来自癌症放疗治疗计划中的优化问题，其中大型稠密的剂量影响矩阵是主要的计算瓶颈。我们将SLME表述为一个双目标非凸优化问题，在近似误差与下游任务的计算成本之间取得平衡。然后，我们开发了R3-Trust，一种高效的信任域算法，可在单次无参数运行中近似帕累托前沿。所得前沿上的每个点提供具有不同精度与下游计算成本平衡的稀疏加低秩表示。在临床放疗矩阵上的实验表明，从现有恢复导向公式获得的分解对于嵌入目标可能不是最优的。相反，在合成实例上的实验表明，SLME和R3-Trust也可应用于稀疏加低秩恢复，在重建精度和计算时间方面均与最先进的恢复方法相比具有竞争力。
英文摘要：
Decomposing a matrix into sparse and low-rank components is central to robust principal component analysis and has broad applications in machine learning, signal processing, and computer vision. Classical formulations seek to recover the underlying sparse and low-rank structure. We instead introduce \emph{sparse-plus-low-rank matrix embedding} (SLME), whose goal is to construct a computationally efficient surrogate for a large dense matrix, without requiring its components to be interpretable. Given $A\in\mathbb{R}^{m\times n}$, SLME approximates $A \approx S+HW$, where $S$ is sparse, $H\in\mathbb{R}^{m\times r}$, $W\in\mathbb{R}^{r\times n}$, and $r\ll \min\{m,n\}$. The resulting matrix-vector product can be evaluated as $Sx+H(Wx)$ in $\mathrm{nnz}(S)+r(m+n)$ operations, rather than the $mn$ operations required by $Ax$. Our primary motivation arises from optimization problems in cancer radiotherapy treatment planning, where a large dense \emph{dose-influence matrix} is a major computational bottleneck. We formulate SLME as a bi-objective nonconvex optimization problem that balances approximation error against the computational cost of the downstream tasks. We then develop \emph{R3-Trust}, an efficient trust-region algorithm that approximates the Pareto frontier in a single parameter-free run. Each point on the resulting frontier provides a sparse-plus-low-rank representation with a different balance between accuracy and downstream computational cost. Experiments on clinical radiotherapy matrices show that decompositions obtained from existing recovery-oriented formulations can be suboptimal for the embedding objective. Conversely, experiments on synthetic instances demonstrate that SLME and R3-Trust can also be applied to sparse-plus-low-rank recovery, where they compare favorably with state-of-the-art recovery methods in both reconstruction accuracy and computational time.
