We propose a novel non-negative spherical relaxation for optimization problems over binary matrices with injectivity constraints, which in particular has applications in multi-matching and clustering. We relax respective binary matrix constraints to the (high-dimensional) non-negative sphere. To optimize our relaxed problem, we use a conditional power iteration method to iteratively improve the objective function, while at same time sweeping over a continuous scalar parameter that is (indirectly) related to the universe size (or number of clusters). Opposed to existing procedures that require to fix the integer universe size before optimization, our method automatically adjusts the analogous continuous parameter. Furthermore, while our approach shares similarities with spectral multi-matching and spectral clustering, our formulation has the strong advantage that we do not rely on additional post-processing procedures to obtain binary results. Our method shows compelling results in various multi-matching and clustering settings, even when compared to methods that use the ground truth universe size (or number of clusters).

翻译：我们提出了一种新颖的非负球面松弛方法，用于解决具有单射约束的二元矩阵优化问题，该方法在多匹配与聚类中具有重要应用。我们将相应的二元矩阵约束松弛至（高维）非负球面。为优化松弛后的问题，我们采用条件幂迭代方法迭代改进目标函数，同时连续扫描一个与宇宙规模（或聚类数）间接相关的连续标量参数。与现有方法需在优化前固定整数宇宙规模不同，我们的方法能自动调整这一类比连续参数。此外，尽管我们的方法与谱多匹配和谱聚类具有相似性，但本方法具备显著优势：无需依赖额外的后处理程序来获得二元结果。在多种多匹配与聚类场景中，即便与使用真实宇宙规模（或聚类数）的方法相比，我们的方法仍展现出极具竞争力的效果。