This paper introduces and analyzes a setup for linear independent component analysis (ICA) in Wasserstein space. This is motivated by applications in which an instance of data is naturally interpreted as a probability measure or point-cloud, such as gene expression data, and the need to meaningfully analyze this type of data. ICA is a well-developed method for identifying independent components in multivariate data, but mainly focuses on data in Euclidean space. We propose an extension to Wasserstein space by viewing the linear ICA problem in this space as a deviation of the “classical” Euclidean setting. We then show how spectral methods based on the Wasserstein distance can be used to identify independent components in point-cloud data.