  Identifying low-dimensional structure in high-dimensional probability
measures is an essential pre-processing step for efficient sampling. We
introduce a method for identifying and approximating a target measure $\pi$ as
a perturbation of a given reference measure $\mu$ along a few significant
directions of $\mathbb{R}^{d}$. The reference measure can be a Gaussian or a
nonlinear transformation of a Gaussian, as commonly arising in generative
modeling. Our method extends prior work on minimizing majorizations of the
Kullback--Leibler divergence to identify optimal approximations within this
class of measures. Our main contribution unveils a connection between the
\emph{dimensional} logarithmic Sobolev inequality (LSI) and approximations with
this ansatz. Specifically, when the target and reference are both Gaussian, we
show that minimizing the dimensional LSI is equivalent to minimizing the KL
divergence restricted to this ansatz. For general non-Gaussian measures, the
dimensional LSI produces majorants that uniformly improve on previous majorants
for gradient-based dimension reduction. We further demonstrate the
applicability of this analysis to the squared Hellinger distance, where
analogous reasoning shows that the dimensional Poincar\'e inequality offers
improved bounds.
