More sophisticated adaptive expansion constructions

Develop more sophisticated forms of adaptivity for expand-and-sparsify projection directions, including adaptivity based on clustering the data.

Background

The paper discusses data-dependent expansion mappings in which the rows of the expansion matrix are sampled from a distribution whose support matches the support of the data distribution. Prior theoretical work shows that this mild form of data dependence can improve approximation guarantees, including allowing smaller sparsity levels and removing dependence on the ambient dimension from certain constants.

The paper identifies a further unresolved direction: designing more sophisticated adaptive mechanisms for selecting projection directions, with clustering-based selection given as a concrete example. This problem would extend data dependence beyond sampling directly from the data-support distribution and could potentially yield representations better matched to the structure of the input distribution.