We develop and analyze a new ‘relax-and-split’ (RS) approach for inverse problems modeled using nonsmooth nonconvex optimization formulations. RS uses a relaxation technique together with partial minimization, and brings classic techniques including direct factorization, matrix decompositions, and fast iterative methods to bear on nonsmooth nonconvex problems. We also extend the approach to robustify any such inverse problem through trimming, a mechanism that robustifies inverse problems to measurement outliers. We then show practical performance of RS and trimmed RS (TRS) on a diverse set of problems, including: (1) phase retrieval, (2) semi-supervised classification, (3) stochastic shortest path problems, and (4) nonconvex clustering. RS/TRS are easy to implement, competitive with existing methods, and show promising results on difficult inverse problems with nonsmooth and nonconvex features.

中文翻译：

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非凸反问题的松弛分裂方法

我们针对使用非光滑非凸优化公式建模的逆问题，开发并分析了一种新的“松弛与分裂”（RS）方法。RS将松弛技术与部分最小化结合使用，并带来了包括直接分解，矩阵分解和快速迭代方法在内的经典技术来解决非光滑非凸问题。我们还扩展了通过修整来增强任何此类反问题的方法，该机制可将反问题增强到测量异常值。然后，我们展示了RS和调整后的RS（TRS）在各种问题上的实际性能，这些问题包括：（1）相位检索，（2）半监督分类，（3）随机最短路径问题和（4）非凸聚类。RS / TRS易于实施，可与现有方法竞争，