Impossibility of latent inner product recovery via rate distortion

TL;DR

This paper proves that recovering inner products of latent positions in certain random graph models is impossible when the latent dimension exceeds a specific threshold, using rate-distortion theory and Wishart distribution bounds.

Contribution

It establishes a fundamental impossibility result for inner product recovery in latent space models based on rate-distortion theory.

Findings

Inner product recovery is impossible if latent dimension d exceeds n times the entropy function h(p).

The proof uses lower bounds on the rate-distortion function of the Wishart distribution.

Results match known conditions for positive recovery in the literature.

Abstract

In this largely expository note, we present an impossibility result for inner product recovery in a random geometric graph or latent space model using the rate-distortion theory. More precisely, suppose that we observe a graph $A$ on $n$ vertices with average edge density $p$ generated from Gaussian or spherical latent locations $z_1, \dots, z_n \in \mathbb{R}^d$ associated with the $n$ vertices. It is of interest to estimate the inner products $\langle z_i, z_j \rangle$ which represent the geometry of the latent points. We prove that it is impossible to recover the inner products if $d \gtrsim n h(p)$ where $h(p)$ is the binary entropy function. This matches the condition required for positive results on inner product recovery in the literature. The proof follows the well-established rate-distortion theory with the main technical ingredient being a lower bound on the rate-distortion function of the Wishart distribution which is interesting in its own right.