We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in Rn, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

中文翻译：

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具有近似保证的线性化 Wasserstein 降维


我们介绍了 LOT Wassmap，这是一种计算上可行的算法，用于揭示 Wasserstein 空间中的低维结构。该算法的动机是观察到许多数据集自然地被解释为概率度量而不是 Rn 中的点，并且找到此类数据集的低维描述需要在 Wasserstein 空间中使用多种学习算法。大多数可用的算法都基于成对 Wasserstein 距离矩阵的计算，这对于高维的大型数据集来说在计算上可能具有挑战性。我们的算法利用 Sinkhorn 距离和线性最优传输等近似方案来加快计算速度，特别是避免计算成对距离矩阵。我们为这种近似下的嵌入质量提供了保证，包括当概率度量的明确描述不可用并且必须处理有限样本时。实验表明，LOT Wassmap 获得了正确的嵌入，并且质量随着样本量的增加而提高。我们还展示了与依赖于成对距离计算的算法相比，LOT Wassmap 如何显著降低计算成本。