We study matrix and tensor denoising when the underlying signal is \textbf{not} necessarily low-rank. In the tensor setting, we observe \[ Y = X^\ast + Z \in \mathbb{R}^{p_1 \times p_2 \times p_3}, \] where $X^\ast$ is an unknown signal tensor and $Z$ is a noise tensor. We propose a one-step variant of the higher-order SVD (HOSVD) estimator, denoted $\widetilde X$, and show that, uniformly over any user-specified Tucker ranks $(r_1,r_2,r_3)$, with high probability, \[ \|\widetilde X - X^\ast\|_{\mathrm F}^2 = O\Big( κ^2\Big\{r_1r_2r_3 + \sum_{k=1}^3 p_k r_k\Big\} + ξ_{(r_1,r_2,r_3)}^2 \Big). \] Here, $ξ_{(r_1,r_2,r_3)}$ is the best achievable Tucker rank-$(r_1,r_2,r_3)$ approximation error of $X^\ast$ (bias), $κ^2$ quantifies the noise level, and $κ^2\{r_1r_2r_3+\sum_{k=1}^3 p_k r_k\}$ is the variance term scaling with the effective degrees of freedom of $\widetilde X$. This yields a rank-adaptive bias-variance tradeoff: increasing $(r_1,r_2,r_3)$ decreases the bias $ξ_{(r_1,r_2,r_3)}$ while increasing variance. In the matrix setting, we show that truncated SVD achieves an analogous bias-variance tradeoff for arbitrary signal matrices. Notably, our matrix result requires \textbf{no} assumptions on the signal matrix, such as finite rank or spectral gaps. Finally, we complement our upper bounds with matching information-theoretic lower bounds, showing that the resulting bias-variance tradeoff is minimax optimal up to universal constants in both the matrix and tensor settings.

翻译：本文研究当底层信号**未必**低秩时的矩阵与张量去噪问题。在张量设定中，我们观测到 \[ Y = X^\ast + Z \in \mathbb{R}^{p_1 \times p_2 \times p_3}, \] 其中 $X^\ast$ 是未知信号张量，$Z$ 是噪声张量。我们提出高阶奇异值分解（HOSVD）估计量的一步变体，记为 $\widetilde X$，并证明在任意用户指定的Tucker秩 $(r_1,r_2,r_3)$ 上一致地，以高概率成立：\[ \|\widetilde X - X^\ast\|_{\mathrm F}^2 = O\Big( κ^2\Big\{r_1r_2r_3 + \sum_{k=1}^3 p_k r_k\Big\} + ξ_{(r_1,r_2,r_3)}^2 \Big). \] 此处 $ξ_{(r_1,r_2,r_3)}$ 是 $X^\ast$ 的最佳可达Tucker秩-$(r_1,r_2,r_3)$ 逼近误差（偏差），$κ^2$ 量化噪声水平，而 $κ^2\{r_1r_2r_3+\sum_{k=1}^3 p_k r_k\}$ 是随 $\widetilde X$ 有效自由度缩放的方差项。这产生了秩自适应的偏差-方差权衡：增大 $(r_1,r_2,r_3)$ 会降低偏差 $ξ_{(r_1,r_2,r_3)}$ 但增加方差。在矩阵设定中，我们证明截断奇异值分解（SVD）对任意信号矩阵可实现类似的偏差-方差权衡。值得注意的是，我们的矩阵结果**无需**对信号矩阵作任何假设（例如有限秩或谱间隙）。最后，我们通过匹配的信息论下界补充了上界结果，表明所得偏差-方差权衡在矩阵和张量设定中均达到极小极大最优性（相差通用常数）。