Emphasis in the tensor literature on random embeddings (tools for low-distortion dimension reduction) for the canonical polyadic (CP) tensor decomposition has left analogous results for the more expressive Tucker decomposition comparatively lacking. This work establishes general Johnson-Lindenstrauss (JL) type guarantees for the estimation of Tucker decompositions when an oblivious random embedding is applied along each mode. When these embeddings are drawn from a JL-optimal family, the decomposition can be estimated within $\varepsilon$ relative error under restrictions on the embedding dimension that are in line with recent CP results. We implement a higher-order orthogonal iteration (HOOI) decomposition algorithm with random embeddings to demonstrate the practical benefits of this approach and its potential to improve the accessibility of otherwise prohibitive tensor analyses. On moderately large face image and fMRI neuroimaging datasets, empirical results show that substantial dimension reduction is possible with minimal increase in reconstruction error relative to traditional HOOI ($\leq$5% larger error, 50%-60% lower computation time for large models with 50% dimension reduction along each mode). Especially for large tensors, our method outperforms traditional higher-order singular value decomposition (HOSVD) and recently proposed TensorSketch methods.

翻译：张量文献中对用于规范多分量（CP）张量分解的随机嵌入（低失真降维工具）的强调，使得针对表达能力更强的Tucker分解的类似结果相对缺乏。本研究建立了当沿每个模态应用遗忘随机嵌入时，Tucker分解估计的广义Johnson-Lindenstrauss（JL）型保证。当这些嵌入从JL最优族中抽取时，可以在嵌入维度的限制下以$\varepsilon$相对误差估计分解，这些限制与近期的CP结果一致。我们实现了一种结合随机嵌入的高阶正交迭代（HOOI）分解算法，以证明该方法的实际优势及其提升原本难以进行的张量分析的可及性的潜力。在中等规模的人脸图像和fMRI神经影像数据集上的实证结果表明，相对于传统HOOI（误差增加≤5%，对于沿每个模态降维50%的大型模型，计算时间降低50%-60%），可以在重建误差最小增长的情况下实现显著的维度降低。特别是对于大型张量，我们的方法优于传统的高阶奇异值分解（HOSVD）和最近提出的TensorSketch方法。