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

arXiv: https://arxiv.org/abs/2111.08452 · HTML: https://mlsanity.com/paper/2111.08452/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 54 references that are in the map.
- First posted on arXiv 9 Nov 2021.

## Builds on (most cited first, 9 of 9)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Clustering for Unsupervised Learning of Visual Features](https://mlsanity.com/paper/1807.05520/) (2018)
- [In-Datacenter Performance Analysis of a Tensor Processing Unit](https://mlsanity.com/paper/1704.04760/) (2017)
- [Methods for Interpreting and Understanding Deep Neural Networks](https://mlsanity.com/paper/1706.07979/) (2017)
- [High Speed Hashing for Integers and Strings](https://mlsanity.com/paper/1504.06804/) (2015)
- [Approximately Minwise Independence with Twisted Tabulation](https://mlsanity.com/paper/1404.6724/) (2014)
- [Entropy-scaling search of massive biological data](https://mlsanity.com/paper/1503.05638/) (2015)
- [Optimal-Time Text Indexing in BWT-runs Bounded Space](https://mlsanity.com/paper/1705.10382/) (2017)
- [Efficient Construction of a Complete Index for Pan-Genomics Read Alignment](https://mlsanity.com/paper/1811.06933/) (2018)

## Abstract (arXiv)

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.