Quantification of model bias underlying the phenomenon of Einstein from Noise

Arising from cryogenic electron microscopy image analysis, “Einstein from noise” is a phenomenon of significant statistical interest because spurious patterns could easily emerge by averaging a large number of white-noise images aligned to a reference image through rotation and translation. While this phenomenon is often attributed to model bias, quantitative studies on such a bias are lacking. Here, we introduce a simple framework under which an image of p pixels is treated as a vector of dimension p and a white-noise image is a random vector uniformly sampled from the (p − 1)-dimensional unit sphere. Moreover, we adopt the cross correlation of two images which is a similarity measure based on the dot product of image pixels. This framework geometrically explains how the bias results from averaging a properly chosen set of white-noise images that are most highly cross-correlated with the reference image. We quantify the bias in terms of three parameters: the number of white-noise images (n), the image dimension (p), and the size of the selection set (m). Under the conditions that n, p and m are all large and (lnn)/p and m/n are both small, we show that the bias is approximately √ 2γ 1+2γ where γ = m p ln ( n m ) . 1 Statistica Sinica: Newly accepted Paper (accepted author-version subject to English editing)