Tensors provide a structured representation for multidimensional data, yet discretization can obscure important information when such data originates from continuous processes. We address this limitation by introducing a functional Tucker decomposition (FTD) that embeds mode-wise continuity constraints directly into the decomposition. The FTD employs reproducing kernel Hilbert spaces (RKHS) to model continuous modes without requiring an a-priori basis, while preserving the multi-linear subspace structure of the Tucker model. Through RKHS-driven representation, the model yields adaptive and expressive factor descriptions that enable targeted modeling of subspaces. The value of this approach is demonstrated in domain-variant tensor classification. In particular, we illustrate its effectiveness with classification tasks in hyperspectral imaging and multivariate time series analysis, highlighting the benefits of combining structural decomposition with functional adaptability.

翻译：张量能够为多维数据提供结构化表示，然而当此类数据源自连续过程时，离散化可能会掩盖重要信息。我们通过引入一种函数Tucker分解（FTD）来解决这一局限，该分解直接在分解过程中嵌入了模态连续性约束。FTD利用再生核希尔伯特空间（RKHS）对连续模式进行建模，无需预设基函数，同时保留了Tucker模型的多线性子空间结构。通过RKHS驱动的表示，该模型生成了自适应且富有表达力的因子描述，从而能够实现对子空间的针对性建模。该方法的价值在域变张量分类中得到了证明。具体而言，我们通过高光谱成像和多变量时间序列分析中的分类任务展示了其有效性，凸显了结构分解与函数适应性相结合的优势。