Recent advances in {matrix-mimetic} tensor frameworks have made it possible to preserve linear algebraic properties for multilinear data analysis and, as a result, to obtain optimal representations of multiway data. Matrix mimeticity arises from interpreting tensors as operators that can be multiplied, factorized, and analyzed analogous to matrices. Underlying the tensor operation is an algebraic framework parameterized by an invertible linear transformation. The choice of linear mapping is crucial to representation quality and, in practice, is made heuristically based on expected correlations in the data. However, in many cases, these correlations are unknown and common heuristics lead to suboptimal performance. In this work, we simultaneously learn optimal linear mappings and corresponding tensor representations without relying on prior knowledge of the data. Our new framework explicitly captures the coupling between the transformation and representation using variable projection. We preserve the invertibility of the linear mapping by learning orthogonal transformations with Riemannian optimization. We provide original theory of uniqueness of the transformation and convergence analysis of our variable-projection-based algorithm. We demonstrate the generality of our framework through numerical experiments on a wide range of applications, including financial index tracking, image compression, and reduced order modeling. We have published all the code related to this work at https://github.com/elizabethnewman/star-M-opt.

使用矩阵拟态张量框架实现多线性数据分析优化的同时学习最佳线性映射和相应张量表示，通过变量投影捕捉变换和表示之间的耦合关系，并通过黎曼优化学习正交变换以保持线性映射的可逆性。

通过变量投影实现最优矩阵模仿张量代数