Tensor-on-tensor regression: Riemannian optimization, over-parameterization, statistical-computational gap and their interplay
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张量对张量回归：黎曼优化、过参数化与统计计算鸿沟的相互作用
- 研究问题
- 当连接张量响应与张量协变量的低 Tucker 秩参数之真实秩未知时，如何兼顾可行计算、统计精度与秩过参数化？
- 方法与思路
- 作者提出黎曼梯度下降与黎曼高斯—牛顿方法，分析正确设秩和过参数化，并用低次多项式论证统计计算鸿沟。
- 材料陈述的贡献
- 摘要称两种方法分别线性和二次收敛至统计最优估计；在广泛的三阶及以上张量情形中，适度过参数化不增加所需样本量。
- 阅读关注点
- 核对收敛所需的初始化、设计及噪声条件，统计最优性的度量，以及过参数化范围和计算下界所依赖的论证框架。
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- 仅据 OpenAlex 索引摘要，稿件版本未注明；不能核验“首次”及最优性主张，或将样本量结论推广至全部张量模型与后续刊本。
原文摘要
We study the tensor-on-tensor regression, where the goal is to connect tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without prior knowledge of its intrinsic rank. We propose the Riemannian gradient descent (RGD) and Riemannian Gauss–Newton (RGN) methods and cope with the challenge of unknown rank by studying the effect of rank over-parameterization. We provide the first convergence guarantee for the general tensor-on-tensor regression by showing that RGD and RGN respectively converge linearly and quadratically to a statistically optimal estimate in both rank correctly-parameterized and over-parameterized settings. Our theory reveals an intriguing phenomenon: Riemannian optimization methods naturally adapt to over-parameterization without modifications to their implementation. We also prove the statistical-computational gap in scalar-on-tensor regression by a direct low-degree polynomial argument. Our theory demonstrates a “blessing of statistical-computational gap” phenomenon: in a wide range of scenarios in tensor-on-tensor regression for tensors of order three or higher, the computationally required sample size matches what is needed by moderate rank over-parameterization when considering computationally feasible estimators, while there are no such benefits in the matrix settings. This shows moderate rank over-parameterization is essentially “cost-free” in terms of sample size in tensor-on-tensor regression of order three or higher. Finally, we conduct simulation studies to show the advantages of our proposed methods and to corroborate our theoretical findings.