Hessian Spectrum is Constant Across Minimizers in Regularized Deep Scalar Factorization

We characterize the full Hessian spectrum of $\ell^2$-regularized deep scalar factorization problems across all minimizers. We prove that the spectrum is constant across all minimizers and, in particular, that the maximum eigenvalue depends on the depth, the (shared) magnitude of optimal layers, and the regularization parameter. Setting the regularization parameter to zero recovers the unregularized case for the flattest minima. To the best of our knowledge, our results offer the first complete characterization of Hessian spectra across minimizers in deep-factorization-type problems.