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The paper introduces a finite‑width geometric framework that explains how learned feature geometries are organized, transported, and selectively aligned in deep neural networks. It quantifies incompatibilities among weight‑generated covariance, gates, and backward sensitivities using three families of commutators, and provides exact layerwise identities that decompose these commutators into sources such as downstream transport, adjacent‑layer imbalance, and nonlinear gate‑covariance interactions. The study demonstrates that spectral alignment is a layer‑ and scale‑dependent compatibility phenomenon governed by transport, interaction, cancellation, and possible damping, rather than a universal consequence of training.
arXiv:2610.32849v1 Announce Type: cross Abstract: During joint training, backpropagation allows later layers to reshape earlier representations, a feedback mechanism absent from purely layer-wise lea...
The paper investigates how exact automatic differentiation behaves under hard‑ReLU gradient descent. It shows that while gradient‑descent states converge to the piecewise‑smooth gradient flow, the derivative of the training map does not, due to missing event‑time sensitivities captured by saltation matrices. The study demonstrates that for convex objectives, activation events can create large sensitivity gaps, and provides empirical evidence that event‑aware corrections are necessary for accurate flow derivatives.
In implicit neural representations (INRs), 3D signals are encoded directly in network weights, making stable weight-space operations essential. However, permutation symmetry causes independently train...
The paper introduces a progressive permutation alignment technique for implicit neural representations, using layer-wise correlation estimation, Hungarian matching, and temperature-controlled spectral annealing to gradually sharpen the alignment toward a permutation matrix. This deterministic, lightweight method avoids costly gradient optimization and is followed by a final refinement stage that optimizes layer-by-layer weight matching. Experiments on MLPs, CNNs, and NeRFs demonstrate reliable neuron correspondence recovery and reduced loss barriers during weight interpolation.
The paper analyzes gradient flow in a two‑layer linear neural network as the input, hidden, and output dimensions grow proportionally. Using a closed‑form solution combined with random matrix theory, it derives exact asymptotic spectral distributions for the weight matrices at all times and tracks how the learned subspace evolves. In the limit of small initialization, the study reveals a silent alignment phase where singular directions align quickly while singular values and loss stay unchanged, followed by a visible learning phase on a logarithmic timescale where singular values increase and loss decreases through a saddle‑to‑saddle dynamic that becomes continuous in the high‑dimensional limit.