One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. We demonstrate how to implement our framework on linear inverse problems, where we can further show the inverse accuracy does not depend on the ambient space dimension. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient descent method. We prove convergence of these numerical schemes under suitable assumptions on the forward problem. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.
Columbia University（哥伦比亚大学）；Getulio Vargas Foundation（热图利奥·瓦加斯基金会）
Inverse problems are paramount in Science and Engineering. In this paper, we consider the setup of Statistical Inverse Problem (SIP) and demonstrate how Stochastic Gradient Descent (SGD) algorithms can be used in the linear SIP setting. We provide consistency and finite sample bounds for the excess risk. We also propose a modification for the SGD algorithm where we leverage machine learning methods to smooth the stochastic gradients and improve empirical performance. We exemplify the algorithm in a setting of great interest nowadays: the Functional Linear Regression model. In this case we cons…
We consider Bayesian inference in inverse regression problems where the objective is to infer about unobserved covariates from observed responses and covariates. We establish posterior consistency of such unobserved covariates in Bayesian inverse regression problemsunder appropriate priors in a leave-one-out cross-validation setup. We relate this to posterior consistency of inverse reference distributions (Bhattacharya (2013)) for assessing model adequacy. We illustrate our theory and methods with various examples of Bayesian inverse regression, along with adequate simulation experiments.
In the Bayesian approach, the a priori knowledge about the input of a mathematical model is described via a probability measure. The joint distribution of the unknown input and the data is then conditioned, using Bayes' formula, giving rise to the posterior distribution on the unknown input. In this setting we prove posterior consistency for nonlinear inverse problems: a sequence of data is considered, with diminishing fluctuations around a single truth and it is then of interest to show that the resulting sequence of posterior measures arising from this sequence of data concentrates around th…
For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth function $g:\mathcal{O}\to\mathbb{R}$, we consider the statistical nonlinear inverse problem of recovering the conductivity $f>0$ in the divergence form equation $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from $N$ discrete noisy point evaluations of the solution $u=u_f$ on $\mathcal O$. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is f…
Learning sparsity-promoting regularizers for linear inverse problems
MaLGa Center, University of Genoa（基因奥纳大学马尔加中心）；Lappeenranta-Lahti University of Technology（拉佩恩拉-拉赫蒂技术大学）；University of Helsinki（赫尔辛基大学）；University of Bologna（博洛尼亚大学）
We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by \cite{Li:1991}. Under mild conditions, the asymptotic ratio $ρ= \lim p/n$ is the phase transition parameter and the SIR estimator is consistent if and only if $ρ= 0$. When dimension $p$ is greater than $n$, we propose a diagonal thresholding screening SIR (DT-SIR) algorithm. This method provides us with an estimate of the eigen-space of the covariance matrix of the conditional expectation $var(\mathbf{E}[\boldsymbol{x}|y])$. The desi…
Formulating a statistical inverse problem as one of inference in a Bayesian model has great appeal, notably for what this brings in terms of coherence, the interpretability of regularisation penalties, the integration of all uncertainties, and the principled way in which the set-up can be elaborated to encompass broader features of the context, such as measurement error, indirect observation, etc. The Bayesian formulation comes close to the way that most scientists intuitively regard the inferential task, and in principle allows the free use of subject knowledge in probabilistic model building…
Multiplicative noise arises in inverse problems when, for example, uncertainty on measurements is proportional to the size of the measurement itself. The likelihood that arises is hence more complicated than that from additive noise. We consider two multiplicative noise models: purely multiplicative noise, and a mixture of multiplicative noise and additive noise. Adopting a Bayesian approach, we provide conditions for the resulting posterior distributions on Banach space to be continuous with respect to perturbations in the data; the inclusion of additive noise in addition to multiplicative no…
Consistency of Variational Inference for Nonlinear Inverse Problems of Partial Differential Equations
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Ministry of Education Key Lab of Intelligent Networks and Network Security, Xi’an Jiaotong University（西安交通大学智能网络与网络安全教育部重点实验室）；Pazhou Laboratory (Huangpu)（琶洲实验室（黄埔））
For ill-posed inverse problems, a regularised solution can be interpreted as a mode of the posterior distribution in a Bayesian framework. This framework enriches the set the solutions, as other posterior estimates can be used as a solution to the inverse problem, such as the posterior mean that can be easier to compute in practice. In this paper we prove consistency of Bayesian solutions of an ill-posed linear inverse problem in the Ky Fan metric for a general class of likelihoods and prior distributions in a finite dimensional setting. This result can be applied to study infinite dimensional…
We consider the statistical nonlinear inverse problem of recovering the absorption term $f>0$ in the heat equation $$ \partial_tu-\frac{1}{2}Δu+fu=0 \quad \text{on $\mathcal{O}\times(0,\textbf{T})$}\quad u = g \quad \text{on $\partial\mathcal{O}\times(0,\textbf{T})$}\quad u(\cdot,0)=u_0 \quad \text{on $\mathcal{O}$}, $$ where $\mathcal{O}\in\mathbb{R}^d$ is a bounded domain, $\textbf{T}<\infty$ is a fixed time, and $g,u_0$ are given sufficiently smooth functions describing boundary and initial values respectively. The data consists of $N$ discrete noisy point evaluations of the solution $u_f$ …
We establish the consistency of a nonparametric maximum likelihood estimator for a class of stochastic inverse problems. We proceed by embedding the framework into the general settings of early results of Pfanzagl related to mixtures.
New York University（纽约大学）；University of Toronto（多伦多大学）；Courant Institute, New York University（纽约大学柯朗研究所）
Despite the ubiquity of transportation data, methods to infer the state parameters of a network either ignore sensitivity of route decisions, require route enumeration for parameterizing descriptive models of route selection, or require complex bilevel models of route assignment behavior. These limitations prevent modelers from fully exploiting ubiquitous data in monitoring transportation networks. Inverse optimization methods that capture network route choice behavior can address this gap, but they are designed to take observations of the same model to learn the parameters of that model, whic…
Functional data analysis is a growing research field as more and more practical applications involve functional data. In this paper, we focus on the problem of regression and classification with functional predictors: the model suggested combines an efficient dimension reduction procedure [functional sliced inverse regression, first introduced by Ferré & Yao (Statistics, 37, 2003, 475)], for which we give a regularized version, with the accuracy of a neural network. Some consistency results are given and the method is successfully confronted to real-life data.
