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This study links sequence-to-function map inference using regularized regression in weight space to Gaussian process models in function space. It clarifies how regularizers define sequence function representations and enables efficient computation of sequence-function statistics.

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Area of Science:

  • Computational Biology
  • Machine Learning
  • Genomics

Background:

  • Biological sequence-to-function maps are crucial for understanding gene and protein activity.
  • Interpreting these maps requires unique representations, often achieved through gauge-fixing in weight space.
  • Existing methods link gauge-fixed representations to L2-regularized regression in overparameterized models.

Purpose of the Study:

  • To establish the theoretical connection between weight space regularized regression and function space Gaussian processes.
  • To disentangle the roles of regularizers in imposing priors and selecting gauges for sequence-function maps.
  • To develop methods for constructing arbitrary Gaussian process priors and gauges, and to derive posterior distributions for sequence-function statistics.

Main Methods:

  • Connecting L2-regularized regression in overparameterized weight space to Gaussian process models in function space.
  • Analyzing how weight space regularizers influence implicit priors and gauge selection.
  • Constructing novel regularizers for specific Gaussian process priors and gauges.
  • Deriving posterior distributions for sequence-to-function statistics using a kernel trick for product-kernel priors.

Main Results:

  • Demonstrated that weight space regularizers correspond to specific Gaussian process priors and gauges in function space.
  • Characterized the implicit function space priors of common weight space regularizers.
  • Derived efficient methods for computing posterior distributions of gauge-fixed sequence-function statistics, including epistatic coefficients.
  • Showcased the utility of a kernel trick for product-kernel priors.

Conclusions:

  • The study provides a unified framework linking weight space regression and function space Gaussian processes for sequence-to-function mapping.
  • This framework allows for flexible specification of priors and gauges, enhancing the interpretability of biological sequence data.
  • Efficient computational methods are presented for analyzing sequence-function relationships and their uncertainties.