Interaction Order Estimation in Tensor Curie-Weiss Models

TL;DR

This paper investigates the fundamental limits of estimating the interaction order in tensor Curie-Weiss models, revealing conditions under which such estimation is impossible or feasible depending on temperature and known parameters.

Contribution

It establishes impossibility results for joint parameter estimation and identifies thresholds for consistent estimation of the interaction order when temperature is known.

Findings

Estimation of $p$ is impossible if $eta$ is unknown.

Consistent estimation of $p$ is possible when $eta$ exceeds a certain threshold.

Joint estimation of $eta$ and $p$ is fundamentally impossible from a single observation.

Abstract

In this paper, we consider the problem of estimating the interaction parameter $p$ of a $p$-spin Curie-Weiss model at inverse temperature $\beta$, given a single observation from this model. We show, by a contiguity argument, that joint estimation of the parameters $\beta$ and $p$ is impossible, which implies that estimation of $p$ is impossible if $\beta$ is unknown. These impossibility results are also extended to the more general $p$-spin Erd\H{o}s-R\'enyi Ising model. The situation is more delicate when $\beta$ is known. In this case, we show that there exists an increasing threshold function $\beta^*(p)$, such that for all $\beta$, consistent estimation of $p$ is impossible when $\beta^*(p) > \beta$, and for almost all $\beta$, consistent estimation of $p$ is possible for $\beta^*(p)<\beta$.