Principled selection of the contrastive dimension in linear contrastive dimension reduction

Develop a principled and reproducible procedure for selecting the reduced dimension d in linear contrastive dimension reduction methods that learn a low-dimensional subspace highlighting variation in a foreground dataset relative to a background dataset, so that the number of foreground-specific components can be chosen reliably without ad hoc heuristics.

Background

Determining the appropriate number of foreground-specific components is central to applying contrastive dimension reduction (CDR) methods. Existing approaches often rely on separate intrinsic dimension estimates for the foreground and background, which inherit the well-known instability and sensitivity of intrinsic dimension estimation, leading to inconsistent and irreproducible choices of d.

The paper emphasizes that this challenge persists even in the linear setting, and that moving to nonlinear (manifold) settings further complicates the notion of contrastive dimension, underscoring the need for a principled selection rule already in linear CDR.