← 研踪 PaperTrailarXiv:2611.13546 · v1
Statistical Analysis of the Structure from Noise Phenomenon in Generalized Procrustes Problems
首次提交 2026-11-10 12:07
版本更新 2026-11-10 12:07
雷达观测 2026-11-11 13:08
摘要
In cryo-electron microscopy (cryo-EM), observations are highly noisy, and scientists often use reference templates to guide matching, alignment, and averaging to reconstruct the molecular structure. However, these procedures can produce an output resembling the template even when the observations contain only noise, a phenomenon known as Einstein from noise (EfN). This phenomenon has attracted attention in statistics, where simplified observation models involving group actions have been studied to understand the resulting model bias. Characterizing the bias in the full cryo-EM pipeline remains challenging. We study template matching and averaging in the generalized orthogonal Procrustes problem, which provides an analytically tractable setting involving continuous orthogonal group action and lower-dimensional projections of ordered point clouds. We show that, for pure Gaussian noise, the mean of the aligned-and-averaged estimator admits the same left and right singular-vector bases as the template but its singular values may be distorted. This identifies the subspace structure imposed by the template even in the absence of signal. We characterize the singular values and bound their distortion, establish finite-sample operator-norm bounds, and give sufficient sample sizes for the population mean to dominate random fluctuations. In fact, this bias still exists under weak signals: when the template rank is at least the observation dimension, the change from the pure-noise mean is in signal amplitude . Our results provide an explicit characterization of model bias in a registration problem motivated by cryo-EM and show why averaging more observations can stabilize structure imposed by the template without removing the bias.
作者备注：21 pages, 4 figures, 1 table
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v1 · 2026-11-10 12:07 · 最新观测版本
Statistical Analysis of the Structure from Noise Phenomenon in Generalized Procrustes Problems
In cryo-electron microscopy (cryo-EM), observations are highly noisy, and scientists often use reference templates to guide matching, alignment, and averaging to reconstruct the molecular structure. However, these procedures can produce an output resembling the template even when the observations contain only noise, a phenomenon known as Einstein from noise (EfN). This phenomenon has attracted attention in statistics, where simplified observation models involving group actions have been studied to understand the resulting model bias. Characterizing the bias in the full cryo-EM pipeline remains challenging. We study template matching and averaging in the generalized orthogonal Procrustes problem, which provides an analytically tractable setting involving continuous orthogonal group action and lower-dimensional projections of ordered point clouds. We show that, for pure Gaussian noise, the mean of the aligned-and-averaged estimator admits the same left and right singular-vector bases as the template but its singular values may be distorted. This identifies the subspace structure imposed by the template even in the absence of signal. We characterize the singular values and bound their distortion, establish finite-sample operator-norm bounds, and give sufficient sample sizes for the population mean to dominate random fluctuations. In fact, this bias still exists under weak signals: when the template rank is at least the observation dimension, the change from the pure-noise mean is in signal amplitude . Our results provide an explicit characterization of model bias in a registration problem motivated by cryo-EM and show why averaging more observations can stabilize structure imposed by the template without removing the bias.
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