Stability and Concentration in Nonlinear Inverse Problems with Block-Structured Parameters: Lipschitz Geometry, Identifiability, and an Application to Gaussian Splatting

TL;DR

This paper introduces an operator-theoretic framework to analyze stability and concentration in nonlinear inverse problems with block-structured parameters, providing fundamental limits and explicit bounds applicable to modern imaging techniques.

Contribution

It develops a unified approach combining Lipschitz geometry, identifiability, and concentration to derive stability and error bounds, exemplified by Gaussian Splatting.

Findings

Establishes deterministic stability inequalities and Lipschitz bounds.

Derives high-probability error bounds independent of algorithms.

Verifies assumptions for Gaussian Splatting, revealing a stability-resolution tradeoff.

Abstract

We develop an operator-theoretic framework for stability and statistical concentration in nonlinear inverse problems with block-structured parameters. Under a unified set of assumptions combining blockwise Lipschitz geometry, local identifiability, and sub-Gaussian noise, we establish deterministic stability inequalities, global Lipschitz bounds for least-squares misfit functionals, and nonasymptotic concentration estimates. These results yield high-probability parameter error bounds that are intrinsic to the forward operator and independent of any specific reconstruction algorithm. As a concrete instantiation, we verify that the Gaussian Splatting rendering operator satisfies the proposed assumptions and derive explicit constants governing its Lipschitz continuity and resolution-dependent observability. This leads to a fundamental stability--resolution tradeoff, showing that estimation error is inherently constrained by the ratio between image resolution and model complexity. Overall, the analysis characterizes operator-level limits for a broad class of high-dimensional nonlinear inverse problems arising in modern imaging and differentiable rendering.