High-Dimensional Spectral Limits for Gaussian KL-Unbalanced Optimal Transport

What the math gives to ML

The paper develops a nonstandard Gaussian KL-unbalanced optimal-transport covariance functional whose equal-penalty form is expressed through nonlinear ridge products and log-determinants, with a positive-semidefinite extension that remains finite for singular covariances. Its most transferable asset is dimension-aware spectral regularization: in high-dimensional sample-covariance regimes, the KL penalty must scale critically as \(\tau_p\asymp p\) to control covariance noise. This suggests replacing unstable minibatch covariance matching or Bures-style feature losses with a ridge-logdet discrepancy calibrated to feature dimension. The first test should measure whether this improves representation stability and downstream accuracy when feature dimension is comparable to batch size.