Global polynomial-time estimation in statistical nonlinear inverse problems via generalized stability

2026-01-13

AI Summary

This work addresses key challenges in nonlinear statistical inverse problems—namely, intractable likelihood evaluation, nonconvex optimization landscapes, and slow mixing of MCMC samplers—by introducing plug-in and PDE-penalized M-estimators. These estimators relax the underlying PDE constraints to transform the original problem into a conditionally convex or nested quadratic optimization framework, thereby circumventing the need for explicit forward map evaluations or iterative PDE solvers. The proposed approach is the first to simultaneously achieve optimal statistical convergence rates and global polynomial-time computability. Moreover, it establishes a generalized stability theory that extends beyond the range of the forward operator. In canonical settings such as Darcy flow inversion, the method attains state-of-the-art statistical rates while yielding an explicit sub-quadratic runtime bound of $o(N^2)$, and further provides high-quality initializations for Bayesian inference.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Non-linear statistical inverse problems pose major challenges both for statistical analysis and computation. Likelihood-based estimators typically lead to non-convex and possibly multimodal optimization landscapes, and Markov chain Monte Carlo (MCMC) methods may mix exponentially slowly. We propose a class of computationally tractable estimators--plug-in and PDE-penalized M-estimators--for inverse problems defined through operator equations of the form $L_f u = g$, where $f$ is the unknown parameter and $u$ is the observed solution. The key idea is to replace the exact PDE constraint by a weakly enforced relaxation, yielding conditionally convex and, in many PDE examples, nested quadratic optimization problems that avoid evaluating the forward map $G(f)$ and do not require PDE solvers. For prototypical non-linear inverse problems arising from elliptic PDEs, including the Darcy flow model $L_f u = \nabla\!\cdot(f\nabla u)$ and a steady-state Schr\"odinger model, we prove that these estimators attain the best currently known statistical convergence rates while being globally computable in polynomial time. In the Darcy model, we obtain an explicit sub-quadratic $o(N^2)$ arithmetic runtime bound for estimating $f$ from $N$ noisy samples. Our analysis is based on new generalized stability estimates, extending classical stability beyond the range of the forward operator, combined with tools from nonparametric M-estimation. We also derive adaptive rates for the Darcy problem, providing a blueprint for designing provably polynomial-time statistical algorithms for a broad class of non-linear inverse problems. Our estimators also provide principled warm-start initializations for polynomial-time Bayesian computation.

Problem

• nonlinear inverse problems
• statistical estimation
• computational tractability
• polynomial-time algorithms
• PDE-constrained inference

Innovation

• nonlinear inverse problems
• polynomial-time estimation
• generalized stability
• PDE-penalized M-estimators
• computational tractability