We introduce a novel framework for uncertainty quantification in clustering. By combining the martingale posterior paradigm with density-based clustering, uncertainty in the estimated density is naturally propagated to the clustering structure. The approach scales effectively to high-dimensional and irregularly shaped data by leveraging modern neural density estimators and GPU-friendly parallel computation. We establish frequen-tist consistency guarantees and validate the methodology on synthetic and real data.

Participants are not allowed to modify the training procedure.

This paper presents Jigsaw, a novel framework for assembling physically broken 3D objects from multiple pieces.

Sparse recovery is among the most well-studied problems in learning theory and high-dimensional statistics. In this work, we investigate the statistical and computational landscapes of sparse recovery with $\ell_\infty$ error guarantees. This variant of the problem is motivated by \emph{variable selection} tasks, where the goal is to estimate the support of a $k$-sparse signal in $\mathbb{R}^d$. Our main contribution is a provable separation between the \emph{oblivious} (``for each'') and \emph{adaptive} (``for all'') models of $\ell_\infty$ sparse recovery. We show that under an oblivious model, the optimal $\ell_\infty$ error is attainable in near-linear time with $\approx k\log d$ samples, whereas in an adaptive model, $\gtrsim k^2$ samples are necessary for any algorithm to achieve this bound. This establishes a surprising contrast with the standard $\ell_2$ setting, where $\approx k \log d$ samples suffice even for adaptive sparse recovery. We conclude with a preliminary examination of a \emph{partially-adaptive} model, where we show nontrivial variable selection guarantees are possible with $\approx k\log d$ measurements.