Daren Wang

Shivam Kumar, Haotian Xu, Carlos Misael Madrid Padilla et al.

We study denoising of a third-order tensor when the ground-truth tensor is not necessarily Tucker low-rank. Specifically, we observe $$ Y=X^\ast+Z\in \mathbb{R}^{p_{1} \times p_{2} \times p_{3}}, $$ where $X^\ast$ is the ground-truth tensor, and $Z$ is the noise tensor. We propose a simple variant of the higher-order tensor SVD estimator $\widetilde{X}$. We show that uniformly over all user-specified Tucker ranks $(r_{1},r_{2},r_{3})$, $$ \| \widetilde{X} - X^* \|_{ \mathrm{F}}^2 = O \Big( κ^2 \Big\{ r_{1}r_{2}r_{3}+\sum_{k=1}^{3} p_{k} r_{k} \Big\} \; + \; ξ_{(r_{1},r_{2},r_{3})}^2\Big) \quad \text{ with high probability.} $$ Here, the bias term $ξ_{(r_1,r_2,r_3)}$ corresponds to the best achievable approximation error of $X^\ast$ over the class of tensors with Tucker ranks $(r_1,r_2,r_3)$; $κ^2$ quantifies the noise level; and the variance term $κ^2 \{r_{1}r_{2}r_{3}+\sum_{k=1}^{3} p_{k} r_{k}\}$ scales with the effective number of free parameters in the estimator $\widetilde{X}$. Our analysis achieves a clean rank-adaptive bias--variance tradeoff: as we increase the ranks of estimator $\widetilde{X}$, the bias $ξ(r_{1},r_{2},r_{3})$ decreases and the variance increases. As a byproduct we also obtain a convenient bias-variance decomposition for the vanilla low-rank SVD matrix estimators.

Yifan Peng, Yuehaw Khoo, Daren Wang

Multivariate density estimation is of great interest in various scientific and engineering disciplines. In this work, we introduce a new framework called Variance-Reduced Sketching (VRS), specifically designed to estimate multivariate density functions with a reduced curse of dimensionality. Our VRS framework conceptualizes multivariate functions as infinite-size matrices/tensors, and facilitates a new sketching technique motivated by the numerical linear algebra literature to reduce the variance in density estimation problems. We demonstrate the robust numerical performance of VRS through a series of simulated experiments and real-world data applications. Notably, VRS shows remarkable improvement over existing neural network density estimators and classical kernel methods in numerous distribution models. Additionally, we offer theoretical justifications for VRS to support its ability to deliver density estimation with a reduced curse of dimensionality.