The Capacity of Associated Subsequence Retrieval

TL;DR

This paper introduces an information-theoretic framework for associated subsequence retrieval in genomic data, establishing the capacity and thresholds for accurately identifying relevant subsequences linked to observable traits.

Contribution

It formulates the associated subsequence retrieval problem, derives its capacity, and provides achievable schemes and converses for zero-error and epsilon-error scenarios.

Findings

Threshold effect in error probability versus rate curve.

Capacity characterized for zero-error and epsilon-error cases.

Achievable schemes and converses match, establishing optimality.

Abstract

The objective of a genome-wide association study (GWAS) is to associate subsequences of individuals' genomes to the observable characteristics called phenotypes (e.g., high blood pressure). Motivated by the GWAS problem, in this paper we introduce the information-theoretic problem of \emph{associated subsequence retrieval}, where a dataset of $N$ (possibly high-dimensional) sequences of length $G$, and their corresponding observable (binary) characteristics is given. The sequences are chosen independently and uniformly at random from $\mathcal{X}^G$, where $\mathcal{X}$ is a finite alphabet. The observable (binary) characteristic is only related to a specific unknown subsequence of length $L$ of the sequences, called \textit{associated subsequence}. For each sequence, if the associated subsequence of it belongs to a universal finite set, then it is more likely to display the observable characteristic (i.e., it is more likely that the observable characteristic is one). The goal is to retrieve the associated subsequence using a dataset of $N$ sequences and their observable characteristics. We demonstrate that as the parameters $N$, $G$, and $L$ grow, a threshold effect appears in the curve of probability of error versus the rate which is defined as ${Gh(L/G)}/{N}$, where $h(\cdot)$ is the binary entropy function. This effect allows us to define the capacity of associated subsequence retrieval. We develop an achievable scheme and a matching converse for this problem, and thus characterize its capacity in two scenarios: the zero-error-rate and the $\epsilon$-error-rate.