Ensemble recovery below the partial-recovery separation threshold

Establish whether an ensemble of clusterwise-regression replicates, each operating below the partial-recovery separation threshold and therefore failing to approximate the target partition individually, can collectively recover a partition that none of the individual replicates approximates.

Background

The paper analyzes ensembles of exact solutions to clusterwise regression problems computed on small random subsamples. Its guarantees are expressed through the per-unit probabilities that an individual replicate classifies correctly. In the two-group location model, the subsample-size theorem requires each replicate to be better than random, which is described as the partial-recovery threshold.

The paper also proves that aggregation can, in principle, outperform every individual replicate: the vote may recover a partition even when no single replicate exactly equals that partition. The unresolved issue is whether this collective benefit extends below the separation threshold, where individual replicates do not even approximate the target partition. The adaptive contamination procedure is explicitly said to have no theorem, but that limitation is not itself stated as a concrete open question and is therefore excluded.