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Minimizers and convolutional neural networks (CNNs) are two quite distinctpopular techniques that have both been employed to analyze categoricalbiological sequences. At face value, the methods seem entirely dissimilar.Minimizers use min-wise hashing on a rolling window to extract a singleimportant k-mer feature per window. CNNs start with a wide array of randomlyinitialized convolutional filters, paired with a pooling operation, and thenmultiple additional neural layers to learn both the filters themselves and howthey can be used to classify the sequence.  Here, our main result is a careful mathematical analysis of hash functionproperties showing that for sequences over a categorical alphabet, randomGaussian initialization of convolutional filters with max-pooling is equivalentto choosing a minimizer ordering such that selected k-mers are (in Hammingdistance) far from the k-mers within the sequence but close to otherminimizers. In empirical experiments, we find that this property manifests asdecreased density in repetitive regions, both in simulation and on real humantelomeres. We additionally train from scratch a CNN embedding of syntheticshort-reads from the SARS-CoV-2 genome into 3D Euclidean space that locallyrecapitulates the linear sequence distance of the read origins, a modest steptowards building a deep learning assembler, though it is at present too slow tobe practical. In total, this manuscript provides a partial explanation for theeffectiveness of CNNs in categorical sequence analysis.

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