arXiv subjects

Connall Garrod

Publications and source records attributed to Connall Garrod.

4 recordsLinked to original sources

arXiv2026

The Implicit Bias of Depth: From Neural Collapse to Softmax Codes

Neural collapse (NC) describes the structured geometry that emerges in the features and weights of trained classifiers. Recent theory suggests NC can be suboptimal in deep architectures, attributing this to an explicit low-rank bias from L2 regularization. We study the deep unconstrained feature model (UFM)-equivalent to a deep linear network with orthogonal inputs-trained without regularization, to isolate how gradient descent and depth alone shape NC. We show that depth induces an implicit low-rank bias: low-rank matrices propagate norm more efficiently through successive multiplications, promoting low-rank alternatives to NC. These alternatives, we argue, correspond to softmax codes: max-margin solutions previously found in width-bottlenecked networks. Analyzing training dynamics under spectral initialization, we identify an early-time repulsion among singular values that drives low-rank emergence, and characterize how depth shrinks NC's basin of attraction. Finally, we show that some effects act in the opposite direction: for randomly initialized networks, increasing width biases training toward higher-rank solutions. Our results provide the first asymptotic and dynamic characterization of implicit bias in deep UFMs trained with unregularized multiclass cross-entropy.

cs.LG↗

arXiv2025

Diagonalizing the Softmax: Hadamard Initialization for Tractable Cross-Entropy Dynamics

Cross-entropy (CE) training loss dominates deep learning practice, yet existing theory often relies on simplifications, either replacing it with squared loss or restricting to convex models, that miss essential behavior. CE and squared loss generate fundamentally different dynamics, and convex linear models cannot capture the complexities of non-convex optimization. We provide an in-depth characterization of multi-class CE optimization dynamics beyond the convex regime by analyzing a canonical two-layer linear neural network with standard-basis vectors as inputs: the simplest non-convex extension for which the implicit bias remained unknown. This model coincides with the unconstrained features model used to study neural collapse, making our work the first to prove that gradient flow on CE converges to the neural collapse geometry. We construct an explicit Lyapunov function that establishes global convergence, despite the presence of spurious critical points in the non-convex landscape. A key insight underlying our analysis is an inconspicuous finding: Hadamard Initialization diagonalizes the softmax operator, freezing the singular vectors of the weight matrices and reducing the dynamics entirely to their singular values. This technique opens a pathway for analyzing CE training dynamics well beyond our specific setting considered here.

cs.LG↗

arXiv2024

The Persistence of Neural Collapse Despite Low-Rank Bias

Neural collapse (NC) and its multi-layer variant, deep neural collapse (DNC), describe a structured geometry that occurs in the features and weights of trained deep networks. Recent theoretical work by Sukenik et al. using a deep unconstrained feature model (UFM) suggests that DNC is suboptimal under mean squared error (MSE) loss. They heuristically argue that this is due to low-rank bias induced by L2 regularization. In this work, we extend this result to deep UFMs trained with cross-entropy loss, showing that high-rank structures, including DNC, are not generally optimal. We characterize the associated low-rank bias, proving a fixed bound on the number of non-negligible singular values at global minima as network depth increases. We further analyze the loss surface, demonstrating that DNC is more prevalent in the landscape than other critical configurations, which we argue explains its frequent empirical appearance. Our results are validated through experiments in deep UFMs and deep neural networks.

cs.LG↗

arXiv2024

Unifying Low Dimensional Spectra in Deep Learning

Low dimensional structures appear ubiquitously in the eigenspectra of deep learning matrices in classification networks trained in the overparameterized regime. While theoretical advances have aimed to explain this phenomenology, they typically succeed only in capturing subsets of the full behavior or rely on assumptions that cannot hold in practice. In this work, we provide an analytic explanation for the bulk plus outlier structure of several canonical deep learning matrices, including the Hessian, gradients, and weights. We achieve this using unconstrained feature models (UFMs), a now-common tool for studying the emergence of deep neural collapse (DNC). We show that DNC is the source of these low dimensional eigenspectra, in each case, the eigenvalues and eigenvectors can be constructed from feature means, the characterizing objects of DNC. This provides a unifying analytic explanation for a wide range of spectral phenomena in deep learning and goes beyond empirical characterizations, which typically focus on eigenvalues, by providing a detailed analysis of eigenvectors. We prove that our results hold for both linear and ReLU networks and provide numerical validation in both the modeling context and standard deep-network architectures on canonical datasets.

cs.LG↗