非线性逆问题中块结构参数的稳定性与集中性：Lipschitz几何、可辨识性及高斯点渲染应用
Stability and Concentration in Nonlinear Inverse Problems with Block-Structured Parameters: Lipschitz Geometry, Identifiability, and an Application to Gaussian Splatting
AI总结：
本文提出一个算子理论框架，研究非线性逆问题中块结构参数的稳定性与统计集中性，通过统一的假设建立确定性稳定性不等式、全局Lipschitz界和非渐近集中估计，应用于高斯点渲染问题，揭示了估计误差与图像分辨率和模型复杂度之间的基本稳定性-分辨率权衡。
摘要：
我们开发了一个算子理论框架，用于研究具有块结构参数的非线性逆问题中的稳定性与统计集中性。在结合块状Lipschitz几何、局部可辨识性和亚高斯噪声的统一假设下，我们建立了确定性稳定性不等式、全局Lipschitz界以及非渐近集中估计。这些结果导致了具有内在前向算子特性且与特定重建算法无关的高概率参数误差界。作为具体实例，我们验证了高斯点渲染算子满足所提出的假设，并推导了其Lipschitz连续性和分辨率依赖的可观测性所支配的显式常数。这导致了基本的稳定性-分辨率权衡，表明估计误差本质上受限于图像分辨率与模型复杂度之间的比率。总体而言，分析为现代成像和可微渲染中广泛出现的高维非线性逆问题的算子层面限制进行了表征。
英文摘要：
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.
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