Overview
A novel algorithm, LOT Wassmap, has been introduced to identify low-dimensional structures within the Wasserstein space. This algorithm addresses scenarios where datasets are conceptualized as probability measures rather than discrete points in $\mathbb{R}^n$, necessitating manifold learning approaches in the Wasserstein domain. The development of LOT Wassmap aims to overcome computational challenges associated with existing algorithms that typically rely on computing pairwise Wasserstein distance matrices, particularly for large, high-dimensional datasets.
Research Context
Many datasets are inherently understood as probability measures, not as individual points in a Euclidean space. To derive low-dimensional descriptions from such datasets, the application of manifold learning algorithms tailored for the Wasserstein space becomes necessary. A common computational bottleneck in this context involves the calculation of pairwise Wasserstein distance matrices, which can be demanding for extensive datasets characterized by high dimensionality.
Approach
LOT Wassmap's design incorporates approximation schemes, specifically Sinkhorn distances and linearized optimal transport, to enhance computational efficiency. A key aspect of its methodology is the avoidance of computing a complete pairwise distance matrix, which distinguishes it from many other available algorithms. The framework also extends to situations where explicit descriptions of probability measures are unavailable, requiring handling of finite samples.
Findings
- LOT Wassmap was observed to achieve correct embeddings in experimental evaluations.
- The quality of these embeddings demonstrated improvement with an increase in sample size.
- The algorithm significantly reduced computational costs when compared against methods that depend on pairwise distance computations.
- Guarantees on the quality of the embedding are provided, even under conditions of approximation and when working with finite samples instead of explicit probability measure descriptions.