In this work, we revisit algorithms for Tensor PCA: given an order-$r$ tensor of the form $T = G+\lambda \cdot v^{\otimes r}$ where $G$ is a random symmetric Gaussian tensor with unit variance entries and $v$ is an unknown boolean vector in $\{\pm 1\}^n$, what's the minimum $\lambda$ at which one can distinguish $T$ from a random Gaussian tensor and more generally, recover $v$? As a result of a long line of work, we know that for any $\ell \in \N$, there is a $n^{O(\ell)}$ time algorithm that succeeds when the signal strength $\lambda \gtrsim \sqrt{\log n} \cdot n^{-r/4} \cdot \ell^{1/2-r/4}$. The question of whether the logarithmic factor is necessary turns out to be crucial to understanding whether larger polynomial time allows recovering the signal at a lower signal strength. Such a smooth trade-off is necessary for tensor PCA being a candidate problem for quantum speedups[SOKB25]. It was first conjectured by [WAM19] and then, more recently, with an eye on smooth trade-offs, reiterated in a blogpost of Bandeira. In this work, we resolve these conjectures and show that spectral algorithms based on the Kikuchi hierarchy \cite{WAM19} succeed whenever $\lambda \geq \Theta_r(1) \cdot n^{-r/4} \cdot \ell^{1/2-r/4}$ where $\Theta_r(1)$ only hides an absolute constant independent of $n$ and $\ell$. A sharp bound such as this was previously known only for $\ell \leq 3r/4$ via non-asymptotic techniques in random matrix theory inspired by free probability.

翻译：在本研究中，我们重新审视了张量主成分分析（Tensor PCA）的算法问题：给定一个形式为$T = G+\lambda \cdot v^{\otimes r}$的$r$阶张量，其中$G$是一个具有单位方差项的高斯随机对称张量，$v$是$\{\pm 1\}^n$中的一个未知布尔向量。我们需要探究：在多大的最小信号强度$\lambda$下，能够将$T$与随机高斯张量区分开来，并更一般地恢复出$v$？经过一系列长期研究，我们已知对于任意$\ell \in \N$，存在一个$n^{O(\ell)}$时间算法，当信号强度$\lambda \gtrsim \sqrt{\log n} \cdot n^{-r/4} \cdot \ell^{1/2-r/4}$时该算法能够成功。对数因子是否必要的问题，对于理解更大的多项式时间是否允许在更低的信号强度下恢复信号至关重要。这种平滑的权衡是张量PCA成为量子加速潜在候选问题的必要条件[SOKB25]。这一猜想最初由[WAM19]提出，随后，着眼于平滑权衡，Bandeira在其博客文章中再次重申。在本研究中，我们解决了这些猜想，并证明基于Kikuchi层次结构\cite{WAM19}的谱算法在$\lambda \geq \Theta_r(1) \cdot n^{-r/4} \cdot \ell^{1/2-r/4}$时成功，其中$\Theta_r(1)$仅隐藏一个与$n$和$\ell$无关的绝对常数。如此尖锐的界此前仅通过受自由概率启发的随机矩阵理论中的非渐近技术，在$\ell \leq 3r/4$的情况下已知。