关于Hotelling型张量收缩的精度：一个随机张量分析
On the Accuracy of Hotelling-Type Tensor Deflation: A Random Tensor Analysis
摘要：
利用随机张量理论的最新进展，本文考虑了一个秩- $r$非对称尖峰张量模型，其形式为$\sum_{i=1}^r β_i A_i + W$，其中$β_i\geq 0$和$A_i$是秩一张量，使得$\langle A_i, A_j \rangle\in [0, 1]$对$i\neq j$，基于此我们提供了在高维区域中Hotelling型张量收缩的渐近研究。具体来说，我们的分析特点的奇异值和对齐在每一步的放气过程中，渐近大张量尺寸。这可以用于构建涉及潜在问题的不同数量的一致估计，例如信噪比$β_i$或不同信号分量$\langle A_i, A_j \rangle$之间的对齐。
英文摘要：
Leveraging on recent advances in random tensor theory, we consider in this paper a rank-$r$ asymmetric spiked tensor model of the form $\sum_{i=1}^r β_i A_i + W$ where $β_i\geq 0$ and the $A_i$'s are rank-one tensors such that $\langle A_i, A_j \rangle\in [0, 1]$ for $i\neq j$, based on which we provide an asymptotic study of Hotelling-type tensor deflation in the large dimensional regime. Specifically, our analysis characterizes the singular values and alignments at each step of the deflation procedure, for asymptotically large tensor dimensions. This can be used to construct consistent estimators of different quantities involved in the underlying problem, such as the signal-to-noise ratios $β_i$ or the alignments between the different signal components $\langle A_i, A_j \rangle$.
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