Beyond Negative-Ridge Endpoints: Mixed-Sign Spectral Regularization via Negative-Shifted Gradient Descent

What the math gives to ML

The paper gives a constructive spectral regularization path that deliberately shifts the curvature operator in the negative direction, then relies on finite-time stopping before the resulting noncontractive dynamics diverge. Unlike a stable negative-ridge endpoint, whose pole must lie below the smallest eigenvalue, the finite-time filter has a removable singularity and can anti-shrink high-eigenvalue directions while retaining shrinkage or limited exposure on weak directions. The most practical neural-network transfer is an early-stopped negative-shifted optimizer applied to a linear head, LoRA factors, or a late-stage fine-tuning subspace, with the shift and stopping time selected by a small validation grid.