正则化的几何分位数与通用线性分布函数

Robust Estimation of High-Dimensional Mean Regression

Data subject to heavy-tailed errors are commonly encountered in various scientific fields, especially in the modern era with explosion of massive data. To address this problem, procedures based on quantile regression and Least Absolute Deviation (LAD) regression have been devel- oped in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultra-high dimensional setting wi…

比较多元分布：一种基于最优传输图的新方法

Comparing Multivariate Distributions: A Novel Approach Using Optimal Transport-based Plots

本文提出利用最优传输和熵正则化最优传输构建多元 Q-Q 图及势函数图，以捕捉高维数据依赖关系和分布差异，并给出相应检验统计量。

具有近确定性感知矩阵的正则化线性回归的谱普适性

Spectral Universality of Regularized Linear Regression with Nearly Deterministic Sensing Matrices

该研究针对加性高斯噪声下线性回归的凸正则化最小二乘估计器，通过确定性条件定义感知矩阵普适性类，证明同类矩阵的近端梯度下降动力学及估计器性能渐近一致，可借助旋转不变系综表征有限随机结构化矩阵的回归性能。

A universal robustification procedure

We develop a procedure that transforms any asymptotically normal estimator into an asymptotically normal estimator whose distribution is robust to arbitrary data contamination. More generally, our procedure transforms any estimator whose asymptotic distribution has positive and continuous density at the origin into an asymptotically normal estimator whose distribution is robust to arbitrary contamination. In developing such a procedure we prove new general properties of componentwise and geometric quantiles in both finite and infinite dimensions.

分块依赖线性模型中的普适性及其在非参数回归中的应用

Universality in block dependent linear models with applications to nonparametric regression

本文研究高维回归中正则化估计量风险的普适性，提出分块依赖设计类，证明其下Lasso和岭回归风险与独立同分布高斯设计相同，并首次实现非参数回归的精确风险刻画。

Regularized Learning Schemes in Feature Banach Spaces

This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with totally convex functions. This broad class of regularizers provides a flexible model for various priors on the features, including in particular hard constraints and powers of Banach norms. In such context, the main results establish a new general form of the representer theorem and the consistency of the corresponding learning schemes under general conditions on …

Optimistic lower bounds for convex regularized least-squares

Minimax lower bounds are pessimistic in nature: for any given estimator, minimax lower bounds yield the existence of a worst-case target vector $β^*_{worst}$ for which the prediction error of the given estimator is bounded from below. However, minimax lower bounds shed no light on the prediction error of the given estimator for target vectors different than $β^*_{worst}$. A characterization of the prediction error of any convex regularized least-squares is given. This characterization provide both a lower bound and an upper bound on the prediction error. This produces lower bounds that are app…

通用私有估计器

Universal Private Estimators

本文提出在纯差分隐私下适用于任意未知连续分布的均值、方差和尺度通用估计器，移除了先验有界假设，在特定分布族上匹配或改进现有专门估计器，并开发了无界整数域上的实例最优经验估计器作为核心技术工具。

一般损失函数下不可微正则项诱导的低维模式的渐近分布

Asymptotic Distribution of Low-Dimensional Patterns Induced by Non-Differentiable Regularizers under General Loss Functions

针对不可微正则项诱导的低维模式的渐近分布问题，本文提出基于随机Lipschitz可微性的分析框架，拓展了M估计的经验过程理论结论，可适用于多类损失函数场景。

半尾单位：检验统计量与效率的通用尺度

Semi-tail Units: A Universal Scale for Test Statistics and Efficiency

提出基于分位数的 $ζ$ 值和 $s$ 值，通过半尾单位为检验统计量提供通用对数尺度，统一了统计量解释并简化了临界值计算、证据合并与渐近效率测度。