Statistical Optimality of Deep Wide Neural Networks

In this paper, we consider the generalization ability of deep wide feedforward ReLU neural networks defined on a bounded domain $\mathcal X \subset \mathbb R^{d}$. We first demonstrate that the generalization ability of the neural network can be fully characterized by that of the corresponding deep neural tangent kernel (NTK) regression. We then investigate on the spectral properties of the deep NTK and show that the deep NTK is positive definite on $\mathcal{X}$ and its eigenvalue decay rate is $(d+1)/d$. Thanks to the well established theories in kernel regression, we then conclude that multilayer wide neural networks trained by gradient descent with proper early stopping achieve the minimax rate, provided that the regression function lies in the reproducing kernel Hilbert space (RKHS) associated with the corresponding NTK. Finally, we illustrate that the overfitted multilayer wide neural networks can not generalize well on $\mathbb S^{d}$. We believe our technical contributions in determining the eigenvalue decay rate of NTK on $\mathbb R^{d}$ might be of independent interests

Robust estimation of bacterial cell count from optical density

Optical density (OD) is widely used to estimate the density of cells in liquid culture, but cannot be compared between instruments without a standardized calibration protocol and is challenging to relate to actual cell count. We address this with an interlaboratory study comparing three simple, low-cost, and highly accessible OD calibration protocols across 244 laboratories, applied to eight strains of constitutive GFP-expressing E. coli. Based on our results, we recommend calibrating OD to estimated cell count using serial dilution of silica microspheres, which produces highly precise calibration (95.5% of residuals <1.2-fold), is easily assessed for quality control, also assesses instrument effective linear range, and can be combined with fluorescence calibration to obtain units of Molecules of Equivalent Fluorescein (MEFL) per cell, allowing direct comparison and data fusion with flow cytometry measurements: in our study, fluorescence per cell measurements showed only a 1.07-fold mean difference between plate reader and flow cytometry data

Significant Sparse Polygenic Risk Scores across 813 traits in UK Biobank

We present a systematic assessment of polygenic risk score (PRS) prediction across more than 1,500 traits using genetic and phenotype data in the UK Biobank. We report 813 sparse PRS models with significant (p < 2.5 x 10−5) incremental predictive performance when compared against the covariate-only model that considers age, sex, types of genotyping arrays, and the principal component loadings of genotypes. We report a significant correlation between the number of genetic variants selected in the sparse PRS model and the incremental predictive performance (Spearman’s ⍴ = 0.61, p = 2.2 x 10−59 for quantitative traits, ⍴ = 0.21, p = 9.6 x 10−4 for binary traits). The sparse PRS model trained on European individuals showed limited transferability when evaluated on non-European individuals in the UK Biobank. We provide the PRS model weights on the Global Biobank Engine (https://biobankengine.stanford.edu/prs)