This paper tackles the problem of recovering a low-rank signal tensor with possibly correlated components from a random noisy tensor, or so-called spiked tensor model. When the underlying components are orthogonal, they can be recovered efficiently using tensor deflation which consists of successive rank-one approximations, while non-orthogonal components may alter the tensor deflation mechanism, thereby preventing efficient recovery. Relying on recently developed random tensor tools, this paper deals precisely with the non-orthogonal case by deriving an asymptotic analysis of a parameterized deflation procedure performed on an order-three and rank-two spiked tensor. Based on this analysis, an efficient tensor deflation algorithm is proposed by optimizing the parameter introduced in the deflation mechanism, which in turn is proven to be optimal by construction for the studied tensor model. The same ideas could be extended to more general low-rank tensor models, e.g., higher ranks and orders, leading to more efficient tensor methods with a broader impact on machine learning and beyond.

翻译：本文研究了从含随机噪声的张量中恢复具有潜在相关分量的低秩信号张量的问题，即所谓的尖峰张量模型。当潜在分量正交时，可通过由连续秩一逼近构成的张量缩减方法高效恢复；而非正交分量可能改变张量缩减机制，从而阻碍高效恢复。基于近期发展的随机张量工具，本文通过推导在三阶秩二尖峰张量上执行的参数化缩减过程的渐近分析，精确处理了非正交情况。基于该分析，通过优化缩减机制中引入的参数，提出了一种高效的张量缩减算法，该算法被证明对所研究的张量模型具有构造性最优性。相同思想可推广至更一般的低秩张量模型（例如更高阶次与秩数），从而开发对机器学习等领域具有更广泛影响的更高效张量方法。