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：本文研究从随机含噪张量（即所谓的尖峰张量模型）中恢复具有可能相关分量的低秩信号张量问题。当潜在分量正交时，可通过连续秩一近似执行的张量缩减方法有效恢复，而非正交分量可能会改变张量缩减机制，从而阻碍有效恢复。本文借助近期发展的随机张量工具，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，精确处理非正交情形。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，该算法经构造性证明对所研究的张量模型具有最优性。相同的思路可扩展至更一般的低秩张量模型（例如更高阶和更高秩），从而开发出更高效的张量方法，对机器学习及其他领域产生更广泛的影响。