Robust Bayes-like estimation of the distribution of a determinantal process

Abstract

We consider the problem of estimating the distribution of a determinantal process (DP) on a measured space from the observation of $n$ independent copies of it. Our estimator is obtained by sampling from a $T$-posterior distribution associated with a suitable prior, and we establish non-asymptotic risk bounds for the Hellinger loss which hold with high probability. These bounds remain valid when the data are only independent rather than i.i.d., and when the model is misspecified, which makes the estimator robust, for instance, to the presence of a small fraction of outliers. For determinantal projection processes of rank $k$ whose kernels are associated with $k$-dimensional subspaces of a $D$-dimensional space, Haar priors on Grassmannians lead to a rate of order $\sqrt{k(D-k)\log n/n}$, and model selection extends this result to infinite-dimensional settings. For general DPs, we introduce priors based on the spectral decomposition of the kernel that combine uniform distributions on the eigenvalues with Haar measures on Stiefel manifolds. When the underlying space is finite with $N$ elements, we obtain a rate of order $N\sqrt{\log n/n}$, without any assumption on the kernel, and derive that the $2^{N}$ principal minors of the kernel can be estimated simultaneously. Finally, for translation-invariant DPs on the torus $\mathbb{T}^d$, we obtain rates that only depend on the smoothness of the kernel at the origin and that are adaptive with respect to it. Combined with a new lower bound on the Hellinger distance between such DPs, these rates also yield estimators of the kernel in $\mathbb{L}_2$.