We present the first systematic analysis of low‑degree polynomial threshold functions (PTFs) for the natural hypothesis‑testing problem of distinguishing a noisy random lift of a base $d$‑regular graph $G$ from a uniformly random $d$‑regular graph. The detection task is formalized as a binary test between two distributions:
- $\mathcal{D}_0$: a graph drawn uniformly from all $d$‑regular graphs;
- $\mathcal{D}_1$: a graph obtained by first taking a random lift of $G$ and then adding edge‑wise noise.
Within this framework we show that any fixed‑degree PTF fails to separate $\mathcal{D}_0$ and $\mathcal{D}_1$ with probability better than a constant, i.e., the problem is information‑theoretically indistinguishable for low‑degree polynomial tests. The core technical contribution is a precise characterization of the short‑cycle count distribution in noisy random lifts: up to logarithmic cycle lengths, the distribution matches that of a purely random $d$‑regular graph. This generalizes the classic results of McKay, Wormald, and Wysocka as well as Johnson on short cycles in random regular graphs, and unifies them with the work of Fortin and Rudinsky on random lifts.
Our proof relies on combinatorial probability, generating‑function techniques, and spectral properties of regular graphs to control how noise perturbs cycle structures. The result not only delineates the limitations of low‑degree PTFs in graph‑based learning but also lays groundwork for investigating whether higher‑order or non‑linear detectors can overcome the identified barrier.
Blogger's Review: The paper offers a fresh perspective at the intersection of spectral graph theory and computational learning, especially with its fine‑grained analysis of short‑cycle statistics. Extending these techniques to broader graph families could further illuminate the computational‑statistical trade‑offs in graph inference problems.