揭示高维非线性反问题中参数稳定性的根本限制，适用于高分辨率渲染。
Stability and Concentration in Nonlinear Inverse Problems with Block-Structured Parameters: Lipschitz Geometry, Identifiability, and an Application to Gaussian Splatting
- 基于块结构参数建立统一分析框架，结合利普希茨几何与可识别性假设。
- 导出不依赖算法的高概率参数误差上界，明确分辨率与模型复杂度的权衡。
- 首次严格分析高斯点渲染的稳定性，给出可计算的连续性常数与可观测性界限。
我们为具有块结构参数的非线性反问题构建了算子理论框架，用于分析稳定性和统计集中性。在结合块内利普希茨几何、局部可识别性与亚高斯噪声的统一假设下，建立了确定性稳定不等式、最小二乘损失函数的全局利普希茨界以及非渐近集中估计。这些结果给出了高概率参数误差上界，其本质由前向算子决定，不依赖具体重建算法。作为具体应用，我们验证了高斯点渲染算子满足上述假设，并推导出其利普希茨连续性与分辨率相关可观测性的显式常数。由此揭示了根本的稳定性-分辨率权衡：估计误差本质上受限于图像分辨率与模型复杂度之比。整体分析刻画了现代成像与可微渲染中一大类高维非线性反问题的算子级极限。
原文摘要 · Abstract (English)
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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