发表机构
Korea Advanced Institute of Science and Technology（韩国科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在未知高斯噪声下研究流形假设，提出条件证书方法，改进维数估计的样本条件，并证明一致维数估计，实验验证了方法的有效性。
AI 中文摘要
我们研究了在明确的识别和正则性条件下，含噪数据能对流形假设确立什么结论。一个总体残差证书结合了独立视图定位、高斯集中、成员不确定性以及总体转移。现有的可求长性准则随后产生一个覆盖尺度结论。对于具有正Hölder密度的局部光滑流形，实际球协方差极限将谱交叉与几何维数识别。我们证明了在重复观测下几乎必然的最终恢复。重用精确的定位平均值将充分的点样本条件从$Nr^{d+4}\gg\log N$改进为$Nr^d\gg\log N$，且复制次数满足$kr^2\gg\log N$。一个双质量证书在声明的类别界限下控制错误的几何维数发射。对于具有未知高斯噪声的单次观测，仿射支撑或已知坐标界限制提供噪声区间和一致的高斯相关维数估计器。Ahlfors正则性将该指数与Hausdorff维数以及齐次光滑类的几何维数等同。精确的Cantor计算描绘了整数谱计数和相邻半径斜率的极限。在指定我们的构造之前，我们引用了已建立的局部PCA、可求长性、集中、二项推断和反卷积结果。可复现的实验区分了点估计、有限尺度覆盖和证书发射。
英文摘要
We study what noisy data can establish about the Manifold Hypothesis under explicit identification and regularity conditions. A population residual certificate combines independent-view localization, Gaussian concentration, membership uncertainty, and population transfer. Existing rectifiability criteria then yield a covered-scale consequence. For a local smooth manifold with positive Hölder density, the actual-ball covariance limit identifies the spectral crossing with geometric dimension. We prove almost-sure eventual recovery under repeated observations. Reusing accurate localization averages improves the sufficient point-sample condition from $Nr^{d+4}\gg\log N$ to $Nr^d\gg\log N$, with replication $kr^2\gg\log N$. A two-mass certificate controls incorrect geometric-dimension emissions under declared class bounds. For single observations with unknown Gaussian noise, affine-support or known coordinate-bound restrictions provide noise intervals and consistent Gaussian correlation-dimension estimators. Ahlfors regularity identifies this exponent with Hausdorff dimension and with the geometric dimension of a homogeneous smooth class. Exact Cantor calculations delineate the limits of integer spectral counts and adjacent-radius slopes. We credit established local PCA, rectifiability, concentration, binomial inference, and deconvolution results before specifying our constructions. Reproducible experiments distinguish point estimation, finite-scale coverage, and certificate emission.
Comments116 page, 21 figures
