Non-asymptotic deviation inequalities for smoothed additive functionals in non-linear state-space models

Distribution-robust mean estimation via smoothed random perturbations

We consider the problem of mean estimation assuming only finite variance. We study a new class of mean estimators constructed by integrating over random noise applied to a soft-truncated empirical mean estimator. For appropriate choices of noise, we show that this can be computed in closed form, and utilizing relative entropy inequalities, these estimators enjoy deviations with exponential tails controlled by the second moment of the underlying distribution. We consider both additive and multiplicative noise, and several noise distribution families in our analysis. Furthermore, we empirically …

Analyzing the discrepancy principle for kernelized spectral filter learning algorithms

We investigate the construction of early stopping rules in the nonparametric regression problem where iterative learning algorithms are used and the optimal iteration number is unknown. More precisely, we study the discrepancy principle, as well as modifications based on smoothed residuals, for kernelized spectral filter learning algorithms including gradient descent. Our main theoretical bounds are oracle inequalities established for the empirical estimation error (fixed design), and for the prediction error (random design). From these finite-sample bounds it follows that the classical discre…

Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional Gaussian time series

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} \rightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covari…

多元椭圆扩散过程的集中度分析

Concentration analysis of multivariate elliptic diffusion processes

本文证明了多元不可逆扩散过程无界函数加性泛函的集中度不等式与PAC界，并应用于验证Lasso估计的约束特征值条件及分析未调整Langevin MCMC算法的采样估计。

马尔可夫链线性统计量的Rosenthal型不等式

Rosenthal-type inequalities for linear statistics of Markov chains

本文为几何遍历马尔可夫链的可加泛函建立了类似Rosenthal不等式的集中不等式，显式界依赖于混合时间，并采用基于泊松分解的新证明方法。

Penalized Estimation in Additive Regression with High-Dimensional Data

Additive regression provides an extension of linear regression by modeling the signal of a response as a sum of functions of covariates of relatively low complexity. We study penalized estimation in high-dimensional nonparametric additive regression where functional semi-norms are used to induce smoothness of component functions and the empirical $L_2$ norm is used to induce sparsity. The functional semi-norms can be of Sobolev or bounded variation types and are allowed to be different amongst individual component functions. We establish new oracle inequalities for the predictive performance o…