Fundamental Limits on Data Acquisition: Trade-offs between Sample Complexity and Query Difficulty

TL;DR

This paper investigates the fundamental trade-offs in query-based data acquisition, demonstrating how the number of measurements and query difficulty influence the ability to reliably recover information bits using Fountain code principles.

Contribution

It establishes the necessary and sufficient conditions for sample complexity in relation to query difficulty for reliable data recovery.

Findings

Sample complexity scales as n=c*max{k,(k log k)/d̄}

Trade-offs between query difficulty and measurement count are characterized

High probability recovery is achievable under the derived bounds

Abstract

We consider query-based data acquisition and the corresponding information recovery problem, where the goal is to recover $k$ binary variables (information bits) from parity measurements of those variables. The queries and the corresponding parity measurements are designed using the encoding rule of Fountain codes. By using Fountain codes, we can design potentially limitless number of queries, and corresponding parity measurements, and guarantee that the original $k$ information bits can be recovered with high probability from any sufficiently large set of measurements of size $n$. In the query design, the average number of information bits that is associated with one parity measurement is called query difficulty ($\bar{d}$) and the minimum number of measurements required to recover the $k$ information bits for a fixed $\bar{d}$ is called sample complexity ($n$). We analyze the fundamental trade-offs between the query difficulty and the sample complexity, and show that the sample complexity of $n=c\max\{k,(k\log k)/\bar{d}\}$ for some constant $c>0$ is necessary and sufficient to recover $k$ information bits with high probability as $k\to\infty$.