We present a nonlinear regression framework based on tensor algebra tailored to high dimensional contexts where data is scarce. We exploit algebraic properties of a partial tensor product, namely the m-tensor product, to leverage structured equations with separated variables. The proposed method combines kernel properties along with tensor algebra to prevent the curse of dimensionality and tackle approximations up to hundreds of parameters while avoiding the fixed point strategy. This formalism allows us to provide different regularization techniques fit for low amount of data with a high number of parameters while preserving well-known matrix-based properties. We demonstrate complexity scaling on a general benchmark and dynamical systems to show robustness for engineering problems and ease of implementation.

翻译：本文提出一种基于张量代数的非线性回归框架，专门适用于数据稀缺的高维场景。我们利用部分张量积（即m-张量积）的代数性质，构建具有分离变量结构的方程组。该方法结合核函数特性与张量代数，既能规避维度灾难，又能处理高达数百个参数的逼近问题，同时避免使用定点迭代策略。该形式体系使我们能够提供适用于"小样本-多参数"场景的不同正则化技术，同时保持经典的矩阵代数特性。我们在通用基准测试和动力系统上验证了复杂度缩放规律，证明了该方法对工程问题的鲁棒性和易于实现的特点。