Tracking Through Decoupling Singularities: A Singularity-Robust Homotopy-Continuation Extension of Feedback Linearization

What the math gives to ML

The paper provides a constructive method for continuing solutions through rank-deficient input-output maps: append a homotopy direction to the singular Jacobian and solve a minimum-norm augmented system rather than directly inverting the Jacobian. The key guarantee is that the augmented matrix remains full row rank at a generic rank-one singularity when the left-null vector of the original Jacobian has nonzero projection onto the homotopy direction. This transfers naturally to implicit or equilibrium neural layers whose fixed-point Jacobians become singular, enabling bounded pseudo-arclength continuation through folds instead of Newton divergence or uncontrolled damping.