We study parameter estimation and asymptotic inference for sparse nonlinear
regression. More specifically, we assume the data are given by $y = f( x^\top
\beta^* ) + \epsilon$, where $f$ is nonlinear. To recover $\beta^*$, we propose
an $\ell_1$-regularized least-squares estimator. Unlike classical linear
regression, the corresponding optimization problem is nonconvex because of the
nonlinearity of $f$. In spite of the nonconvexity, we prove that under mild
conditions, every stationary point of the objective enjoys an optimal
statistical rate of convergence. In addition, we provide an efficient algorithm
that provably converges to a stationary point. We also access the uncertainty
of the obtained estimator. Specifically, based on any stationary point of the
objective, we construct valid hypothesis tests and confidence intervals for the
low dimensional components of the high-dimensional parameter $\beta^*$.
Detailed numerical results are provided to back up our theory.

研究了稀疏非线性回归的参数估计和渐近推断，提出了一种 $\ell_1$ 正则化最小二乘估计器，并证明了在温和条件下，目标函数的每个驻点都享有最优的统计收敛速率，并提供了有效的算法来获得估计器的不确定性，同时用数字结果支持了我们的理论。

稀疏非线性回归：参数估计与渐近推断

Sparse Nonlinear Regression: Parameter Estimation and Asymptotic  Inference

We provide a general theory of the expectation-maximization (EM) algorithm
for inferring high dimensional latent variable models. In particular, we make
two contributions: (i) For parameter estimation, we propose a novel high
dimensional EM algorithm which naturally incorporates sparsity structure into
parameter estimation. With an appropriate initialization, this algorithm
converges at a geometric rate and attains an estimator with the (near-)optimal
statistical rate of convergence. (ii) Based on the obtained estimator, we
propose new inferential procedures for testing hypotheses and constructing
confidence intervals for low dimensional components of high dimensional
parameters. For a broad family of statistical models, our framework establishes
the first computationally feasible approach for optimal estimation and
asymptotic inference in high dimensions. Our theory is supported by thorough
numerical results.

本文提供了一个关于期望最大化 (EM) 算法应用于高维潜变量模型推断的一般理论。作者提出了一个新的高维 EM 算法，自然地融入了稀疏结构到参数估计中。基于估计值，作者提出了新的推论程序来测试假设并构建置信区间。这个算法针对广泛的统计模型，提供了高维最先进的估计和渐近推断的第一个可计算的方法。

高维期望最大化算法：统计优化和渐近正态性

High Dimensional Expectation-Maximization Algorithm: Statistical  Optimization and Asymptotic Normality

We propose a generalized functional linear regression model for a regression
situation where the response variable is a scalar and the predictor is a random
function. A linear predictor is obtained by forming the scalar product of the
predictor function with a smooth parameter function, and the expected value of
the response is related to this linear predictor via a link function. If, in
addition, a variance function is specified, this leads to a functional
estimating equation which corresponds to maximizing a functional
quasi-likelihood. This general approach includes the special cases of the
functional linear model, as well as functional Poisson regression and
functional binomial regression. The latter leads to procedures for
classification and discrimination of stochastic processes and functional data.
We also consider the situation where the link and variance functions are
unknown and are estimated nonparametrically from the data, using a
semiparametric quasi-likelihood procedure. An essential step in our proposal is
dimension reduction by approximating the predictor processes with a truncated
Karhunen-Loeve expansion. We develop asymptotic inference for the proposed
class of generalized regression models. In the proposed asymptotic approach,
the truncation parameter increases with sample size, and a martingale central
limit theorem is applied to establish the resulting increasing dimension
asymptotics. We establish asymptotic normality for a properly scaled distance
between estimated and true functions that corresponds to a suitable L^2 metric
and is defined through a generalized covariance operator.

本文提出了一种广义的函数线性回归模型，包括函数线性模型、泊松回归和二项回归，并使用截断 Karhunen-Loeve 展开逼近预测过程进行降维，开发了一种适用于广义回归模型的渐近推断方法，其中截断参数随样本大小而增加，并应用控制变量的中心极限定理来建立维数的渐近增加，建立合适的 L^2 度量下的估计和真实函数之间的适当缩放距离的渐近正态性。