state-space-ntk-collapse-bifurcations

name	state-space-ntk-collapse-bifurcations
description	Analysis of Neural Tangent Kernel (NTK) collapse near dynamical bifurcations in state-space models. Studies how the NTK spectrum degrades as recurrent networks approach critical transitions. Activation: NTK collapse, bifurcation analysis, state-space NTK, critical transitions neural networks, dynamical systems deep learning.

State-Space NTK Collapse Near Bifurcations

Analysis of Neural Tangent Kernel (NTK) behavior near dynamical bifurcations in state-space neural networks, revealing how training dynamics change as models approach critical phase transitions.

Metadata

• Source: arXiv:2605.12763
• Authors: James Hazelden, Eric Shea-Brown
• Published: 2026-05-14
• Categories: Machine Learning (cs.LG); Dynamical Systems (math.DS); Optimization and Control (math.OC); Neurons and Cognition (q-bio.NC)

Core Methodology

Key Innovation

Analyzes the behavior of the Neural Tangent Kernel (NTK) as state-space recurrent neural networks approach dynamical bifurcations. The NTK provides a linearized view of neural network training dynamics, and this work reveals how the NTK spectrum degrades near critical transitions, affecting trainability and generalization.

Technical Framework

1. State-Space RNN Formulation: Analyzes recurrent networks through their continuous-time state-space dynamics
2. NTK Computation: Computes the Neural Tangent Kernel for state-space models
3. Bifurcation Analysis: Studies how NTK eigenvalues change as network parameters approach bifurcation points
4. Critical Transition Theory: Connects dynamical systems bifurcation theory with deep learning training dynamics

Key Findings

• NTK spectrum collapses as the network approaches bifurcation points
• Eigenvalue structure reveals which directions in parameter space become ill-conditioned
• Different bifurcation types (saddle-node, Hopf, etc.) produce distinct NTK signatures
• Training dynamics slow down near critical transitions due to NTK degradation
• Has implications for understanding critical brain dynamics and phase transitions in neural systems

Implementation Guide

Prerequisites

• PyTorch/JAX for neural network implementation
• Linear algebra libraries for eigenvalue computation
• Bifurcation analysis tools (e.g., PyDSTool, PyAuto)

Step-by-Step

1. Define State-Space Model: Implement recurrent network as continuous-time dynamical system
2. Compute NTK: Calculate Neural Tangent Kernel for the state-space model
3. Parameter Sweep: Vary parameters to approach bifurcation points
4. Eigenvalue Analysis: Track NTK eigenvalue spectrum during parameter changes
5. Bifurcation Detection: Identify critical transition points from NTK behavior

Code Concept

# Conceptual framework
def compute_state_space_ntk(model, inputs):
    """Compute NTK for state-space recurrent model."""
    # Linearize model around current parameters
    # Compute Jacobian of outputs w.r.t. parameters
    # NTK = J @ J.T
    jacobian = torch.autograd.functional.jacobian(model, inputs)
    ntk = jacobian @ jacobian.T
    return ntk

def analyze_ntk_collapse(ntk, bifurcation_param):
    """Analyze NTK eigenvalue spectrum near bifurcation."""
    eigenvalues = torch.linalg.eigvalsh(ntk)
    condition_number = eigenvalues.max() / eigenvalues.min()
    return eigenvalues, condition_number

Applications

• Understanding training dynamics of recurrent neural networks near critical points
• Predicting when RNN training will stall due to NTK collapse
• Designing initialization schemes that avoid bifurcation-adjacent regions
• Connecting deep learning theory with critical brain dynamics
• Analyzing stability of learned dynamical systems

Pitfalls

• NTK analysis assumes linearized training dynamics, which may not hold far from initialization
• Computing full NTK is O(N²) in data size — requires approximations for large datasets
• Bifurcation detection requires careful parameter sweep design

Related Skills

• geodynamics-geometric-state-space
• neural-dynamics-universal-translator
• neural-critical-dynamics-theory
• nonlinear-rnn-fixed-connectivity-solution
• renormalization-scaling-brain-activity