Optimal quantization of the mean measure and applications to statistical learning

统计估计中重尾数据的量化：（近）极小极大速率、协变量量化与一致恢复

Quantizing Heavy-tailed Data in Statistical Estimation: (Near) Minimax Rates, Covariate Quantization, and Uniform Recovery

本文提出截断与抖动预处理结合均匀量化的方案，证明在协方差估计、压缩感知和矩阵补全中可实现近极小极大误差速率，并针对协变量量化下的非凸恢复及重尾噪声建立近最优一致恢复保证。

High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization

In this paper, we propose a uniformly dithered 1-bit quantization scheme for high-dimensional statistical estimation. The scheme contains truncation, dithering, and quantization as typical steps. As canonical examples, the quantization scheme is applied to the estimation problems of sparse covariance matrix estimation, sparse linear regression (i.e., compressed sensing), and matrix completion. We study both sub-Gaussian and heavy-tailed regimes, where the underlying distribution of heavy-tailed data is assumed to have bounded moments of some order. We propose new estimators based on 1-bit quan…

Conditional quantile estimation through optimal quantization

In this paper, we use quantization to construct a nonparametric estimator of conditional quantiles of a scalar response $Y$ given a d-dimensional vector of covariates $X$. First we focus on the population level and show how optimal quantization of $X$, which consists in discretizing $X$ by projecting it on an appropriate grid of $N$ points, allows to approximate conditional quantiles of $Y$ given $X$. We show that this is approximation is arbitrarily good as $N$ goes to infinity and provide a rate of convergence for the approximation error. Then we turn to the sample case and define an estimat…

Convex Quantization Preserves Logconcavity

A logconcave likelihood is as important to proper statistical inference as a convex cost function is important to variational optimization. Quantization is often disregarded when writing likelihood models, ignoring the limitations of the physical detectors used to collect the data. These two facts call for the question: would including quantization in likelihood models preclude logconcavity? are the true data likelihoods logconcave? We provide a general proof that the same simple assumption that leads to logconcave continuous-data likelihoods also leads to logconcave quantized-data likelihoods…

Optimal quantization applied to Sliced Inverse Regression

In this paper we consider a semiparametric regression model involving a $d$-dimensional quantitative explanatory variable $X$ and including a dimension reduction of $X$ via an index $β'X$. In this model, the main goal is to estimate the euclidean parameter $β$ and to predict the real response variable $Y$ conditionally to $X$. Our approach is based on sliced inverse regression (SIR) method and optimal quantization in $\mathbf{L}^p$-norm. We obtain the convergence of the proposed estimators of $β$ and of the conditional distribution. Simulation studies show the good numerical behavior of the pr…

Convergence rate of optimal quantization grids and application to empirical measure

We study the convergence rate of the optimal quantization for a probability measure sequence $(μ_{n})_{n\in\mathbb{N}^{*}}$ on $\mathbb{R}^{d}$ converging in the Wasserstein distance in two aspects: the first one is the convergence rate of optimal quantizer $x^{(n)}\in(\mathbb{R}^{d})^{K}$ of $μ_{n}$ at level $K$; the other one is the convergence rate of the distortion function valued at $x^{(n)}$, called the "performance" of $x^{(n)}$. Moreover, we also study the mean performance of the optimal quantization for the empirical measure of a distribution $μ$ with finite second moment but possibly…