We quantify the robust value lost by restricting Wasserstein transport to a common low-dimensional subspace. Affine transport
sets and outer moment sets yield exact bounds for quadratic losses with nonsingular nominal covariance. A spectral rule augments
a prescribed base space and attains a sharp, base-dependent approximation factor for the aggregate first-order restriction loss.
Eigenvalue tails yield dimension and finite-radius decision certificates. Gradient principal component analysis admits no bounded
approximation factor. Exact examples and synthetic reduced decision problems illustrate the guarantees.