Kernel techniques are among the most influential approaches in data science
and statistics. Under mild conditions, the reproducing kernel Hilbert space
associated to a kernel is capable of encoding the independence of $M\ge 2$
random variables. Probably the most widespread independence measure relying on
kernels is the so-called Hilbert-Schmidt independence criterion (HSIC; also
referred to as distance covariance in the statistics literature). Despite
various existing HSIC estimators designed since its introduction close to two
decades ago, the fundamental question of the rate at which HSIC can be
estimated is still open. In this work, we prove that the minimax optimal rate
of HSIC estimation on $\mathbb R^d$ for Borel measures containing the Gaussians
with continuous bounded translation-invariant characteristic kernels is
$\mathcal O\!\left(n^{-1/2}\right)$. Specifically, our result implies the
optimality in the minimax sense of many of the most-frequently used estimators
(including the U-statistic, the V-statistic, and the Nystr\"om-based one) on
$\mathbb R^d$.

在本研究中，我们证明了具有连续有界平移不变特征核的 Borel 测度中，Hilbert-Schmidt 独立性准则（HSIC）在 R^d 上的最优最小化估计率为 O (n^(-1/2))，从而证实了许多经常使用的估计器（包括 U 统计量、V 统计量和 Nystrom-based 统计量）在 R^d 上的最小化最优性。

平移不变核 HSIC 估计的极小最大速率

The Minimax Rate of HSIC Estimation for Translation-Invariant Kernels

In this paper, we conduct a comprehensive analysis of generalization
properties of Kernel Ridge Regression (KRR) in the noiseless regime, a scenario
crucial to scientific computing, where data are often generated via computer
simulations. We prove that KRR can attain the minimax optimal rate, which
depends on both the eigenvalue decay of the associated kernel and the relative
smoothness of target functions. Particularly, when the eigenvalue decays
exponentially fast, KRR achieves the spectral accuracy, i.e., a convergence
rate faster than any polynomial. Moreover, the numerical experiments well
corroborate our theoretical findings. Our proof leverages a novel extension of
the duality framework introduced by Chen et al. (2023), which could be useful
in analyzing kernel-based methods beyond the scope of this work.

我们对核岭回归的泛化性质在无噪声情况下进行了综合分析，证明了核岭回归能够达到最小最优速率，该速率取决于相关核函数的特征值衰减和目标函数的相对平滑度。

噪声无关情况下核岭回归的对偶分析

A Duality Analysis of Kernel Ridge Regression in the Noiseless Regime

Covariance matrix estimation is one of the most important problems in
statistics. To accommodate the complexity of modern datasets, it is desired to
have estimation procedures that not only can incorporate the structural
assumptions of covariance matrices, but are also robust to outliers from
arbitrary sources. In this paper, we define a new concept called matrix depth
and then propose a robust covariance matrix estimator by maximizing the
empirical depth function. The proposed estimator is shown to achieve minimax
optimal rate under Huber's $\epsilon$-contamination model for estimating
covariance/scatter matrices with various structures including bandedness and
sparsity.

本文引入了矩阵深度的概念，提出了一种鲁棒的协方差矩阵估计方法，该方法具有稳健性和最小化最坏情况下的风险率的特点，能够适应包括有带状结构和稀疏结构在内的多种结构的协方差 / 散布矩阵估计问题。