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

Samantha Petti1, Carlos Martí-Gómez2, Justin B Kinney2

  • 1Department of Mathematics, Tufts University, Medford, MA, 02155.

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Summary

This study connects sequence-to-function map inference using regularized regression in weight space with Gaussian processes 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 DNA, RNA, and protein functionality.
  • Inferring these maps and decomposing them to understand subsequence contributions are key challenges.
  • Interpreting sequence-function maps requires 'gauge-fixing' to define unique representations.

Purpose of the Study:

  • To establish the relationship between regularized regression in overparameterized weight space and Gaussian process approaches in function space.
  • To disentangle how weight space regularizers influence implicit priors and restrict optimal weights to a specific gauge.
  • To enable the construction of regularizers for arbitrary Gaussian process priors and various gauges.

Main Methods:

  • Connecting L2-regularized regression in overparameterized weight space with Gaussian processes in function space.
  • Analyzing how weight space regularizers impose implicit priors and define gauges.
  • Developing methods to construct regularizers for specific Gaussian process priors and gauges.

Main Results:

  • Established a formal link between weight space regularized regression and function space Gaussian processes.
  • Characterized implicit function space priors for common weight space regularizers.
  • Derived efficient computation methods for sequence-to-function statistics, including gauge-fixed weights and epistatic coefficients, using a kernel trick for product-kernel priors.

Conclusions:

  • The study provides a unified framework for understanding sequence-to-function map inference and decomposition.
  • It offers a method to construct regularizers that align with desired Gaussian process priors and gauges.
  • Efficient computation of complex sequence-function statistics is now possible, advancing biological sequence analysis.