Geometric planted matchings beyond the Gaussian model

We consider the problem of recovering an unknown matching between a set of $n$ randomly placed points in $\mathbb{R}^d$ and random perturbations of these points. This can be seen as a model for particle tracking and more generally, entity resolution. We use matchings in random geometric graphs to derive minimax lower bounds for this problem that hold under great generality. Using these results we show that for a fixed $d$, as long as the noise distribution has finite $d$-th moment, and both initial positions and noise have bounded continuous densities, the minimax rate for the problem scales as $\Theta(n^2\sigma^d \wedge n)$. Under the stronger assumptions that the tail of the noise is sub-Gaussian, we show that the order of the number of mistakes made by an estimator that minimizes the sum of squared Euclidean distances is minimax optimal when $d$ is fixed and is optimal up to $n^{o(1)}$ factors when $d = o(\log n)$. In the high-dimensional regime we consider a setup where both initial positions and perturbations have independent sub-Gaussian coordinates. In this setup we give sufficient conditions under which the same estimator makes no mistakes with high probability. We prove an analogous result for an adapted version of this estimator that incorporates information on the covariance matrix of the perturbations.