Parity Queries for Binary Classification

TL;DR

This paper investigates the optimal number of parity-based measurements needed to recover binary variables efficiently, revealing fundamental trade-offs between measurement complexity, query difficulty, and recovery accuracy.

Contribution

It introduces a method for designing queries that achieve near-optimal recovery of binary variables with minimal measurements, establishing key theoretical bounds.

Findings

Sample complexity scales as max{k, (k log k)/d̄} for full recovery.

Sample complexity scales as max{k, (k log(1/δ))/d̄} for partial recovery.

Fundamental trade-offs between recovery accuracy, query difficulty, and number of measurements.

Abstract

Consider a query-based data acquisition problem that aims to recover the values of $k$ binary variables from parity (XOR) measurements of chosen subsets of the variables. Assume the response model where only a randomly selected subset of the measurements is received. We propose a method for designing a sequence of queries so that the variables can be identified with high probability using as few ($n$) measurements as possible. We define the query difficulty $\bar{d}$ as the average size of the query subsets and the sample complexity $n$ as the minimum number of measurements required to attain a given recovery accuracy. We obtain fundamental trade-offs between recovery accuracy, query difficulty, and sample complexity. In particular, the necessary and sufficient sample complexity required for recovering all $k$ variables with high probability is $n = c_0 \max\{k, (k \log k)/\bar{d}\}$ and the sample complexity for recovering a fixed proportion $(1-\delta)k$ of the variables for $\delta=o(1)$ is $n = c_1\max\{k, (k \log(1/\delta))/\bar{d}\}$, where $c_0, c_1>0$.