Non-uniform refinement: Adaptive regularization improves single particle cryo-EM reconstruction

Ali Punjani; Haowei Zhang; David J. Fleet

We formulate a new regularizer for cryo-EM reconstruction in terms of the minimization of a crossvalidation objective [10, 31] . Cross-validation (CV) is a general principle that is widely used in machine learning and statistics for model selection and parameter estimation with complex models. In CV, observed data are randomly partitioned into a training set and a held-out validation set. Model parameters are inferred using the training data, the quality of which is then assessed by measuring an error function applied to the validation data. In k-fold CV, the observations are partitioned into k parts. In each of k trials, one part is selected as the held-out validation set, and the remaining k − 1 parts comprise the training set. The per-trial validation errors are summed, providing the total CV error. This procedure measures agreement between the optimized model and the observations, without bias due to over-fitting. Rather, over-fitting during training is detected directly as an increase in the validation error. Importantly, formulating regularization in a cross-validation setting provides a principled way to design regularization operators that are more complex than the conventional, isotropic frequency-space filters. The CV framework is not restricted to a Fourier basis. One may consider more complex parameterizations, the use of meta-parameters, and incorporate cryo-EM domain knowledge.