Phase Transitions in Approximate Ranking

TL;DR

This paper investigates the fundamental limits of approximate ranking from pairwise interaction data, revealing phase transition boundaries in the optimal error rates depending on the signal-to-noise ratio.

Contribution

It characterizes the exact optimal statistical error rates for approximate ranking and uncovers a novel phase transition phenomenon in these rates.

Findings

Optimal error rates depend on SNR, showing trivial, polynomial, exponential, or zero behavior.

Discovered a new phase transition boundary between polynomial and exponential error regimes.

Provides a comprehensive framework for understanding the limits of approximate ranking.

Abstract

We study the problem of approximate ranking from observations of pairwise interactions. The goal is to estimate the underlying ranks of $n$ objects from data through interactions of comparison or collaboration. Under a general framework of approximate ranking models, we characterize the exact optimal statistical error rates of estimating the underlying ranks. We discover important phase transition boundaries of the optimal error rates. Depending on the value of the signal-to-noise ratio (SNR) parameter, the optimal rate, as a function of SNR, is either trivial, polynomial, exponential or zero. The four corresponding regimes thus have completely different error behaviors. To the best of our knowledge, this phenomenon, especially the phase transition between the polynomial and the exponential rates, has not been discovered before.