具有光滑Wasserstein损失的非负矩阵和张量分解的统一框架
A unified framework for non-negative matrix and tensor factorisations with a smoothed Wasserstein loss
摘要：
非负矩阵和张量分解是寻找高维数据集的低维表示的经典工具。在诸如成像的应用中，数据集可以被认为是在具有度量结构的空间上支持的分布。在这种情况下，基于最优运输理论的瓦瑟斯坦距离的损失函数是一个自然的选择，因为它包含了数据的基本几何形状。我们介绍了一个通用的数学框架计算非负因式分解的矩阵和张量相对于最佳的运输损失。我们推导出一个有效的计算方法，其解决方案使用凸对偶制定，并证明了这种方法的适用性与矩阵和张量值数据的几个数值例子。
英文摘要：
Non-negative matrix and tensor factorisations are a classical tool for finding low-dimensional representations of high-dimensional datasets. In applications such as imaging, datasets can be regarded as distributions supported on a space with metric structure. In such a setting, a loss function based on the Wasserstein distance of optimal transportation theory is a natural choice since it incorporates the underlying geometry of the data. We introduce a general mathematical framework for computing non-negative factorisations of both matrices and tensors with respect to an optimal transport loss. We derive an efficient computational method for its solution using a convex dual formulation, and demonstrate the applicability of this approach with several numerical illustrations with both matrix and tensor-valued data.
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