The Projection Transfers: Spectral Geometry

Learned representations encode global geometric structure-shared spectral directions, intrinsic dimensionality, variance hierarchies invisible to any single downstream task. Standard transfer learning evaluates compatibility through fine-tuning performance, ignoring a complementary signal: spectral overlap between source and target representations. We propose Spectral Projection, a transfer mechanism restricting source access via rank-limited projection onto its covariance spectrum. The method constructs a xed, gradient-free projector, ex-tracts dominant spectral directions through SVD, and applies them as hard geometric constraints. This occupies a previously unoccupied axis in the transfer design space: target structure preservation under limited spectral bandwidth, independent of task adaptation. We dene Transfer Fidelity as the target variance fraction captured by a rank-k source-derived projector. Transfer Fidelity exhibits a spectral knee at shared intrinsic dimensionality. Below the knee, transfer is bandwidth-limited; above it, a spanning regime emergesenabling broad-to-narrow transfer while constraining the reversewhere sucient rank yields near-perfect delity without ne-grained alignment. On CIFAR-10 domain splits, 0.27 Transfer Fidelity gaps predict 2.3×downstream error increases without additional training. The diagnostic requires one forward pass plus SVD, oering pre-training transfer assessment grounded in geometry, not empirical luck. The module is plug-and-play, runs on consumer hardware, and preserves model weights.