用凸松弛方法从噪声中恢复低维流形数据，理论保证强。
Denoising data using convex relaxations
- 先降维再投影到潜在流形的凸包上
- 在低质量条件下仍能给出有限样本误差界
- 适合生物成像等高噪声低维数据场景
研究观测值 $Y_i = X_i + Z_i$ 的去噪问题，其中潜变量 $X_i$ 来自 $ ^n$ 中的低维流形，噪声 $Z_i$ 为各向同性高斯。提出一种凸松弛估计器：先通过主成分分析降维，再将观测值投影到投影后潜变量流形的凸包上。构建一个统计代理，通过经验高斯尾概率估计其支撑超平面。在潜分布满足低质量条件时，证明了该代理的有限样本保证，并推导出去噪器的误差界。分析结合了凸约束下最小二乘投影的风险界与凸包的熵界。同时验证了冷冻电镜观测模型的框架假设，建立了相关群作用和成像算子的合适覆盖数与Lipschitz估计。
原文摘要 · Abstract (English)
We study the problem of denoising observations \(Y_i=X_i+Z_i\), where the latent variables \(X_i\) are sampled from a low-dimensional manifold in \(\mathbb{R}^n\) and the noise variables \(Z_i\) are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.
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