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

Joe-Mei Feng, Hsin-Hsiung Kao

TL;DR

This work develops an operator-theoretic framework for nonlinear inverse problems with block-structured parameters and applies it to Gaussian Splatting. By combining blockwise Lipschitz geometry, local identifiability, and sub-Gaussian noise, it derives deterministic stability inequalities, nonasymptotic concentration bounds, and high-probability error estimates that are intrinsic to the forward operator. For Gaussian Splatting, explicit constants are obtained, and a fundamental stability–resolution tradeoff is established: increasing image resolution improves local observability while increasing model complexity worsens global sensitivity, yielding an intrinsic error floor that scales as $\sqrt{M/N}$. The results provide operator-level, algorithm-agnostic limits for high-dimensional nonlinear inverse problems in differentiable rendering and modern imaging, with clear guidance on how resolution and parameterization interact to bound achievable accuracy.

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.

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

TL;DR

This work develops an operator-theoretic framework for nonlinear inverse problems with block-structured parameters and applies it to Gaussian Splatting. By combining blockwise Lipschitz geometry, local identifiability, and sub-Gaussian noise, it derives deterministic stability inequalities, nonasymptotic concentration bounds, and high-probability error estimates that are intrinsic to the forward operator. For Gaussian Splatting, explicit constants are obtained, and a fundamental stability–resolution tradeoff is established: increasing image resolution improves local observability while increasing model complexity worsens global sensitivity, yielding an intrinsic error floor that scales as . The results provide operator-level, algorithm-agnostic limits for high-dimensional nonlinear inverse problems in differentiable rendering and modern imaging, with clear guidance on how resolution and parameterization interact to bound achievable accuracy.

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.
Paper Structure (87 sections, 11 theorems, 114 equations)

This paper contains 87 sections, 11 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Problem Formulation and Global Assumptions
  4. Parameterization and Forward Model
  5. Noise Model and Observation Space
  6. Global Assumptions
  7. Notation Summary
  8. Noise and concentration.
  9. Identifiability and observability.
  10. Geometric Sensitivity and Blockwise Lipschitz Structure
  11. Coordinatewise Operator Sensitivity
  12. Coordinatewise Misfit Lipschitz
  13. Global Misfit Lipschitz Constant
  14. Local Identifiability and Deterministic Stability
  15. Local Identifiability
  16. ...and 72 more sections

Key Result

Lemma 1

Under Assumptions assump:global(A2)--(A5), for each block index $i\in\{1,\dots,N\}$ there exists a constant $G_i>0$ such that for all $Z,Z'\in\mathcal{Z}$ that differ only in the $i$-th block,

Theorems & Definitions (25)

  • Remark 1: Identifiability vs. Observability
  • Remark 2: Resolution scaling
  • Lemma 1: Coordinatewise Operator Lipschitz
  • Lemma 2: Coordinatewise Misfit Lipschitz
  • Theorem 1: Global Misfit Lipschitz
  • Definition 1: Local Identifiability
  • Theorem 2: Deterministic Stability
  • Corollary 1: Deterministic Error Bound
  • proof
  • Lemma 3: Misfit Moment Bound
  • ...and 15 more