arXiv:2507.10383v5 Announce Type: replace-cross Abstract: Recurrent neural networks are canonical models of biological memory. In these models, memories are represented by distributed patterns of neu...
arXiv:2606. 05326v1 Announce Type: cross Abstract: We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the loss and the sharpness.
arXiv:2506. 08764v3 Announce Type: replace Abstract: Deep neural networks are known to suffer from exploding or vanishing gradients as depth increases, a phenomenon closely tied to the spectral behavior of the input-output Jacobian.
The paper investigates associative memory in a bipartite Hopfield–Krotov architecture, termed class H, where hidden neurons serve as the retrieval order parameter. Using the replica method, it derives replica‑symmetric phase diagrams and closed‑form capacities for polynomial load, showing that crosstalk statistics are similar for Ising and spherical visible neurons. With a softmax hidden layer, the load becomes exponential, mapping the thermodynamics onto a random‑energy‑model that exhibits paramagnetic, condensed, and frozen phases, and revealing that heating destabilizes retrieval through quantized attention reassignments while Gaussian patterns remain metastable at all loads.
arXiv:2605. 28983v2 Announce Type: replace-cross Abstract: In this paper, training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems: each gradient step selects the initial data of a viscous Hamilton--Jacobi equation whose Hopf--Cole propagator best fits the observations; at inference, the input is the spatial point at which that solution is evaluated and the initial condition is already encoded in the weights.
arXiv:2606. 04031v1 Announce Type: new Abstract: Coupled gradient descent--where the update of one parameter block depends on another--underlies bilevel optimization, two-time-scale stochastic approximation, and adversarial training.
arXiv:2607. 10439v2 Announce Type: replace-cross Abstract: We model human motor cortex, recorded during rest and motor-imagery BCI conditions, as a port-Hamiltonian system: a conservative interconnection (skew-symmetric coupling between band-limited neural phasors) together with a dissipative port whose state-dependent decay is set by a graph-neural-network surrogate.
The paper investigates how gradient descent behaves near codimension‑one bifurcations in recurrent neural networks by analyzing the global empirical Neural Tangent Kernel (GeNTK). Under local center‑manifold conditions, the parameter‑to‑state Jacobian is approximated by a low‑rank normal‑form operator, causing the GeNTK and Fisher information matrix to become strongly amplified and anisotropic, concentrating on a rank‑one or rank‑two channel depending on the bifurcation type. Experiments on high‑dimensional RNNs confirm that this low‑rank concentration coincides with abrupt loss changes, subtask interference, and aligns with changes in memory dynamics in a 15‑task LeakyRNN.
arXiv:2609. 09589v1 Announce Type: new Abstract: Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces.
arXiv:2606. 30384v1 Announce Type: new Abstract: Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape.
Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape. However, the large number of trainable parameters makes the direct analysis of these dynamics challenging.
The paper investigates the training dynamics of attention mechanisms in high-dimensional settings, focusing on attention-indexed models that encompass multi-layer and multi-head architectures. It shows that while the loss landscape can be described by a finite set of trace order parameters, the online stochastic gradient descent dynamics involve an infinite hierarchy of matrix moments that can be accurately approximated by a finite truncated system. The study further reveals that the choice of attention parameterization acts as an implicit bias: untied attention can get trapped in uninformative states, whereas tied attention induces symmetry breaking and enables weak recovery with θ(d² log d) samples, and untied attention exhibits a fast-slow dynamic leading to weak recovery when symmetry is broken.