On minimizers and convolutional filters: theoretical connections and applications to genome analysis

TL;DR

This paper reveals a mathematical connection between minimizers and CNN filters in biological sequence analysis, showing that random CNN initialization mimics minimizer selection and demonstrating practical implications for genome analysis.

Contribution

It provides a theoretical link between minimizers and CNN filters, and empirically explores their effects on sequence analysis and genome data.

Findings

Random CNN filters with max-pooling emulate minimizer selection.

Decreased density in repetitive regions observed in experiments.

CNN embedding of SARS-CoV-2 reads captures sequence distances.

Abstract

Minimizers and convolutional neural networks (CNNs) are two quite distinct popular techniques that have both been employed to analyze categorical biological sequences. At face value, the methods seem entirely dissimilar. Minimizers use min-wise hashing on a rolling window to extract a single important k-mer feature per window. CNNs start with a wide array of randomly initialized convolutional filters, paired with a pooling operation, and then multiple additional neural layers to learn both the filters themselves and how they can be used to classify the sequence. Here, our main result is a careful mathematical analysis of hash function properties showing that for sequences over a categorical alphabet, random Gaussian initialization of convolutional filters with max-pooling is equivalent to choosing a minimizer ordering such that selected k-mers are (in Hamming distance) far from the k-mers within the sequence but close to other minimizers. In empirical experiments, we find that this property manifests as decreased density in repetitive regions, both in simulation and on real human telomeres. We additionally train from scratch a CNN embedding of synthetic short-reads from the SARS-CoV-2 genome into 3D Euclidean space that locally recapitulates the linear sequence distance of the read origins, a modest step towards building a deep learning assembler, though it is at present too slow to be practical. In total, this manuscript provides a partial explanation for the effectiveness of CNNs in categorical sequence analysis.