Uria Mor

Haim Avron, Uria Mor

Developed in a series of seminal papers in the early 2010s, the tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features such as a tensor Singular Value Decomposition and Eckart-Young-like optimality results. It has proven to be a powerful tool for analyzing inherently multilinear data arising in hyperspectral imaging, medical imaging, neural dynamics, scientific simulations, and more. At the heart of tubal tensor algebra lies a special tensor-tensor product: originally the t-product, later generalized into a full family of products via the $\star_M$-product. Though initially defined through the multiplication of a block-circulant unfolding of one tensor by a matricization of another, it was soon observed that the t-product can be interpreted as standard matrix multiplication where the scalars are tubes-i.e., real vectors twisted ``inward.'' Yet, a fundamental question remains: why is this the ``right'' way to define a tensor-tensor product in the tubal setting? In this paper, we show that the t-product and its $\star_M$ generalization arise naturally when viewing third-order tensors as matrices of tubes, together with a small set of desired algebraic properties. Furthermore, we prove that the $\star_M$-product is, in fact, the only way to define a tubal product satisfying these properties. Thus, while partly expository in nature - aimed at presenting the foundations of tubal tensor algebra in a cohesive and accessible way - this paper also addresses theoretical gaps in the tubal tensor framework, proves new results, and provides justification for the tubal tensor framework central constructions, thereby shedding new light on it.

Uria Mor

A valuable feature of the tubal tensor framework is that many familiar constructions from matrix algebra carry over to tensors, including SVD and notions of rank. Importantly, it has been shown that for a specific family of tubal products, an Eckart-Young type theorem holds, i.e., the best low-rank approximation of a tensor under the Frobenius norm is obtained by truncating its tubal SVD. In this paper, we provide a complete characterization of the family of tubal products that yield an Eckart-Young type result. We demonstrate the practical implications of our theoretical findings by conducting experiments with video data and data-driven dynamical systems.

Uria Mor, Yotam Cohen, Rafael Valdes-Mas et al.

Precision medicine is a clinical approach for disease prevention, detection and treatment, which considers each individual's genetic background, environment and lifestyle. The development of this tailored avenue has been driven by the increased availability of omics methods, large cohorts of temporal samples, and their integration with clinical data. Despite the immense progression, existing computational methods for data analysis fail to provide appropriate solutions for this complex, high-dimensional and longitudinal data. In this work we have developed a new method termed TCAM, a dimensionality reduction technique for multi-way data, that overcomes major limitations when doing trajectory analysis of longitudinal omics data. Using real-world data, we show that TCAM outperforms traditional methods, as well as state-of-the-art tensor-based approaches for longitudinal microbiome data analysis. Moreover, we demonstrate the versatility of TCAM by applying it to several different omics datasets, and the applicability of it as a drop-in replacement within straightforward ML tasks.