On learning functions over biological sequence space: relating Gaussian process priors, regularization, and gauge fixing

Mappings from biological sequences (DNA, RNA, protein) to quantitative measures of sequence functionality play an important role in contemporary biology. We are interested in the related tasks of (i) inferring predictive sequence-to-function maps and (ii) decomposing sequence-function maps to elucidate the contributions of individual subsequences. Because each sequence-function map can be written as a weighted sum over subsequences in multiple ways, meaningfully interpreting these weights requires ``gauge-fixing,'' i.e., defining a unique representation for each map. Recent work has established that most existing gauge-fixed representations arise as the unique solutions to $L_2$-regularized regression in an overparameterized ``weight space'' where the choice of regularizer defines the gauge. Here, we establish the relationship between regularized regression in overparameterized weight space and Gaussian process approaches that operate in ``function space,'' i.e.~the space of all real-valued functions on a finite set of sequences. We disentangle how weight space regularizers both impose an implicit prior on the learned function and restrict the optimal weights to a particular gauge. We show how to construct regularizers that correspond to arbitrary explicit Gaussian process priors combined with a wide variety of gauges and characterize the implicit function space priors associated with the most common weight space regularizers. Finally, we derive the posterior distribution of a broad class of sequence-to-function statistics, including gauge-fixed weights and multiple systems for expressing higher-order epistatic coefficients. We show that such distributions can be efficiently computed for product-kernel priors using a kernel trick.