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.

翻译：近年来，矩阵仿射张量框架的发展使得在多线性数据分析中保持线性代数性质成为可能，从而获得多路数据的最优表示。矩阵仿射性源于将张量解释为可类比矩阵进行乘法运算、分解和分析的算子。张量运算的基础是一个由可逆线性变换参数化的代数框架。线性映射的选择对表示质量至关重要，在实践中通常基于数据预期相关性的启发式方法进行选择。然而在许多情况下，这些相关性是未知的，常见的启发式方法会导致次优性能。本研究在不依赖数据先验知识的情况下，同步学习最优线性映射及其对应的张量表示。我们提出的新框架通过变量投影显式捕捉变换与表示之间的耦合关系。通过黎曼优化学习正交变换，保持了线性映射的可逆性。我们建立了关于变换唯一性的原创理论，并对基于变量投影的算法进行了收敛性分析。通过在金融指数追踪、图像压缩和降阶建模等广泛应用的数值实验，我们证明了该框架的普适性。本工作相关代码已发布于 https://github.com/elizabethnewman/star-M-opt。