We study rank selection for low-rank tensor regression under random covariates design. Under a Gaussian random-design model and some mild conditions, we derive population expressions for the expected training-testing discrepancy (optimism) for both CP and Tucker decomposition. We further demonstrate that the optimism is minimized at the true tensor rank for both CP and Tucker regression. This yields a prediction-oriented rank-selection rule that aligns with cross-validation and extends naturally to tensor-model averaging. We also discuss conditions under which under- or over-ranked models may appear preferable, thereby clarifying the scope of the method. Finally, we showcase its practical utility on a real-world image regression task and extend its application to tensor-based compression of neural network, highlighting its potential for model selection in deep learning.

翻译：我们研究了随机协变量设计下低秩张量回归的秩选择问题。在高斯随机设计模型及若干温和条件下，我们推导了CP分解和Tucker分解下期望训练-测试差异（乐观度）的总体表达式。进一步证明，在CP回归和Tucker回归中，乐观度在真实张量秩处达到最小值。这产生了一种预测导向的秩选择准则，该准则与交叉验证一致，并可自然推广至张量模型平均。我们还讨论了欠秩或过秩模型可能表现更优的条件，从而明确了该方法的适用范围。最后，我们通过真实图像回归任务展示了其实际效用，并将应用拓展至基于张量的神经网络压缩，凸显了该方法在深度学习模型选择中的潜力。