Quotient Dynamics, Effective Curvature, and Implicit Bias in Positive Quadratic Networks
Authors: Pengcheng Cheng
Organizations: School of Mathematics, Jilin University Changchun, 130012, China
Abstract
Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top. We study how this quotient structure governs training dynamics, curvature, recovery, and interpolation bias. On the full-column-rank stratum, we identify mathbb{R}^{dtimes r}_*/O(r) with the rank-r PSD manifold. For smooth objectives L(U)=ell(UU^top), the Euclidean factor gradient is horizontal. Thus, factor gradient flow projects exactly to quotient Riemannian gradient flow, while finite-step gradient descent induces an exact congruence recursion for the predictor. For quadratic regression, we derive the effective Hessian at interpolators as the empirical measurement Gram form restricted to the tangent space relative to the quotient metric. Under Gaussian rank-one measurements, we compute population curvature, prove uniform deviation bounds for the empirical normal operator, construct a spectral initializer, and establish local exponential convergence for gradient flow and linear convergence for small-step descent. Recovery guarantees are explicit but conservative due to reliance on full-space second-moment control. In underdetermined commuting regimes, factor gradient flow becomes an exact entropy mirror flow in joint spectral coordinates. Strictly positive initializations converge to Bregman projections onto the interpolation set. With isotropic initialization q(0)=varepsilon^2mathbf{1}, predictors approach the minimum-trace solution set as varepsilondownarrow0, resolving nonuniqueness via weighted entropy within the invariant joint spectral algebra. Finite-step descent selects interpolants differing from continuous-time Bregman projections by O(eta). Numerical experiments verify these quotient identities, curvature predictions, recovery behaviors, and selection laws.
We study the gradient flow dynamics of diagonal linear networks for regression tasks under infinitesimal initialization. Extending Theorem 1 from Pesme & Flammarion (2023), we generalize the analysis to both deep diagonal linear networks and a broader class of two-layer diagonal linear networks (as defined in Definition 4.1). Specifically, we demonstrate that the training trajectories of these models can be equivalently characterized by the proposed Algorithm 1. We further prove that this algorithm converges to the solution of a modified l1 norm minimization problem. As a result, we establish that the implicit bias of both network architectures corresponds to a modified l1 norm in the regime of infinitesimal initialization. Additionally, we provide insights into the underlying mechanisms governing these dynamics by identifying the Structural Invariant Manifold (SIM) (Zhao et al., 2026) as the key geometric structure that shapes the learning process.
Muon can be interpreted as optimizing a linear local objective over a spectral-norm ball. This gives a matrix-sign update that preserves the singular directions of the gradient and assigns the same magnitude to all active singular modes. We ask whether these two properties remain optimal when local curvature is taken into account. To answer this question, we keep Muon's spectral-norm constraint unchanged and replace the linear local model with a quadratic one. We call the resulting method \emph{Quadratic Spectral Descent} (QSD). We show that curvature can change both the singular values and the singular directions of the optimal update. To make QSD practical, we approximate curvature with Kronecker-factored statistics and solve the constrained quadratic with a small number of Frank--Wolfe steps, each of which has a closed-form matrix-sign subproblem. We further provide an optimality certificate, a comparison with Muon under the same quadratic surrogate, and an O(1/K) convergence rate for the inner solver. Experiments on GPT pre-training show that QSD consistently improves validation loss over Muon and recent Muon variants, and reduces wall-clock training time by up to 8.49% at matched validation loss.
The successful training of neural networks hinges on the use of first order optimization methods, yet the theoretical characterization of these methods remains incomplete. This is especially true in settings with mild overparameterization. In this work, we study the gradient flow dynamics of two-layer ReLU networks from small initialization with orthogonal training data. We prove the limiting flow converges to a saddle-to-saddle jump process as the initialization scale tends to zero, revealing an incremental learning phenomenon in which a new neuron activates at each saddle. This analysis recovers the known result of Dana et al. (2025, arXiv:2502.16977) that the network interpolates the training data with high probability as soon as m≳log(n), where m is the network width and n is the number of training samples. This incremental process characterization also allows us to derive a novel implicit bias result: the learned interpolator has a squared ℓ2-norm scaling as n, which is within a constant factor of the minimal ℓ2-norm interpolator. More broadly, our work provides the first rigorous proof of an incremental learning process for ReLU networks, whilst suggesting mildly overparameterized networks can converge to interpolating solutions whose complexity is of the same order as that of the optimal interpolator.