Dimensionality reduction of longitudinal ’omics data using modern tensor factorizations

Longitudinal ’omics analytical methods are extensively used in the evolving field of precision medicine, by enabling ‘big data’ recording and high-resolution interpretation of complex datasets, driven by individual variations in response to perturbations such as disease pathogenesis, medical treatment or changes in lifestyle. However, inherent technical limitations in biomedical studies often result in the generation of feature-rich and sample-limited datasets. Analyzing such data using conventional modalities often proves to be challenging since the repeated, high-dimensional measurements overload the outlook with inconsequential variations that must be filtered from the data in order to find the true, biologically relevant signal. Tensor methods for the analysis and meaningful representation of multiway data may prove useful to the biological research community by their advertised ability to tackle this challenge. In this study, we present tcam—a new unsupervised tensor factorization method for the analysis of multiway data. Building on top of cutting-edge developments in the field of tensor-tensor algebra, we characterize the unique mathematical properties of our method, namely, 1) preservation of geometric and statistical traits of the data, which enable uncovering information beyond the inter-individual variation that often takes over the focus, especially in human studies. 2) Natural and straightforward out-of-sample extension, making tcam amenable for integration in machine learning workflows. A series of re-analyses of real-world, human experimental datasets showcase these theoretical properties, while providing empirical confirmation of tcam’s utility in the analysis of longitudinal ’omics data.Author summary: Tensor methods have proven useful for exploration of high-dimensional, multiway data that is produced in longitudinal ’omics studies. However, even the most recent applications of these methods to ’omics data are based on the canonical polyadic tensor-rank factorization whose results heavily depend on the choice of target rank, lack any guarantee for optimal approximation, and do not allow for out-of-sample extension in a straightforward manner. In this paper, we present a method for tensor component analysis for the analysis of longitudinal ’omics data, built on top of cutting-edge developments in the field of tensor-tensor algebra. We show that our method, in contrast to existing tensor-methods, enjoys provable optimal properties on the distortion and variance in the embedding space, enabling direct and meaningful interpretation, supporting traditional multivariate statistical analysis to be performed in the embedding space. Due to the method’s construction using tensor-tensor products, the procedure of mapping a point to the embedding space of a pre-trained factorization is simple and scalable, giving rise to the application of our method as a feature engineering step in standard machine learning workflows.