Inter-Layer Hessian Analysis of Neural Networks with DAG Architectures

• URL: http://arxiv.org/abs/2604.11639v1
• Date: Mon, 13 Apr 2026 15:48:41 GMT
• Title: Inter-Layer Hessian Analysis of Neural Networks with DAG Architectures
• Abstract summary: Modern automatic differentiation frameworks (JAX, PyTorch) return the Hessian of the loss function as a monolithic tensor, without exposing the internal structure of inter-layer interactions.<n>The theoretical analysis explains exponential decay of resonance in empirical networks and its connections under skip skip.
• Score: 45.88028371034407
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Modern automatic differentiation frameworks (JAX, PyTorch) return the Hessian of the loss function as a monolithic tensor, without exposing the internal structure of inter-layer interactions. This paper presents an analytical formalism that explicitly decomposes the full Hessian into blocks indexed by the DAG of an arbitrary architecture. The canonical decomposition $H = H^{GN} + H^T$ separates the Gauss--Newton component (convex part) from the tensor component (residual curvature responsible for saddle points). For piecewise-linear activations (ReLU), the tensor component of the input Hessian vanishes ($H^{T}_{v,w}\!\equiv\!0$ a.e., $H^f_{v,w}\!=\!H^{GN}_{v,w}\!\succeq\!0$); the full parametric Hessian contains residual terms that do not reduce to the GGN. Building on this decomposition, we introduce diagnostic metrics (inter-layer resonance~$\mathcal{R}$, geometric coupling~$\mathcal{C}$, stable rank~$\mathcal{D}$, GN-Gap) that are estimated stochastically in $O(P)$ time and reveal structural curvature interactions between layers. The theoretical analysis explains exponential decay of resonance in vanilla networks and its preservation under skip connections; empirical validation spans fully connected MLPs (Exp.\,1--5) and convolutional architectures (ResNet-18, ${\sim}11$M~parameters, Exp.\,6). When the architecture reduces to a single node, all definitions collapse to the standard Hessian $\nabla^2_θ\mathcal{L}(θ)\in\mathbb{R}^{p\times p}$.

Related papers

• Spectral Asymptotics of Neural Network Loss Landscapes: An Exact Decomposition of the Curvature Exponent [0.0]
  We prove the Spectral Alignment Decomposition: $= dlog_k / dlog_k$, where $_k$ measures alignment between Kronecker factor eigenbases and gradient singular directions.<n>This reduces "why does $$ vary?" to a geometric question we answer for LayerNorm, residual connections, and softmax heads.
  arXiv  Detail & Related papers  (2026-05-22T23:31:38Z)
• Nodal algebraic curves and entropy diagnostics in degenerate two-dimensional harmonic-oscillator shells [0.0]
  We show that degenerate quantum eigenspaces can support substantial changes in nodal geometry at fixed energy.<n>We combine these criteria with entropy diagnostics to identify strata where topology-changing events can occur.<n>The framework suggests experimentally reconstructible signatures for real-phase Hermite--Gaussian structured light and approximately isotropic trapped motional systems.
  arXiv  Detail & Related papers  (2026-04-30T17:15:26Z)
• On the Superlinear Relationship between SGD Noise Covariance and Loss Landscape Curvature [1.6773271875801752]
  Gradient Descent (SGD) introduces anisotropic noise that is correlated with the local curvature of the loss landscape, thereby biasing optimization toward flat minima.<n>We show that this assumption holds only under restrictive conditions that are typically violated in deep neural networks.<n>Experiments across datasets, architectures, and loss functions validate these bounds, providing a unified characterization of the noise-curvature relationship in deep learning.
  arXiv  Detail & Related papers  (2026-02-05T12:35:13Z)
• Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation [2.815765641180636]
  We show that a constant-depth $mathrmReLUk-1$ network with bounded weights can approximate any function in the Sobolev space.<n>We also prove that our construction is nearly optimal by showing the required number of parameters matches up to a logarithmic factor.
  arXiv  Detail & Related papers  (2025-09-11T11:28:20Z)
• Tensor Decomposition Networks for Fast Machine Learning Interatomic Potential Computations [48.46721044282335]
  tensor decomposition networks (TDNs) achieve competitive performance with dramatic speedup in computations.<n>We evaluate TDNs on PubChemQCR, a newly curated molecular relaxation dataset containing 105 million DFT-calculated snapshots.<n>Results show that TDNs achieve competitive performance with dramatic speedup in computations.
  arXiv  Detail & Related papers  (2025-07-01T18:46:27Z)
• Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit [75.4661041626338]
  We study the problem of gradient descent learning of a single-index target function $f_*(boldsymbolx) = textstylesigma_*left(langleboldsymbolx,boldsymbolthetarangleright)$<n>We prove that a two-layer neural network optimized by an SGD-based algorithm learns $f_*$ with a complexity that is not governed by information exponents.
  arXiv  Detail & Related papers  (2024-06-03T17:56:58Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Generalization and Stability of Interpolating Neural Networks with Minimal Width [37.908159361149835]
  We investigate the generalization and optimization of shallow neural-networks trained by gradient in the interpolating regime. We prove the training loss number minimizations $m=Omega(log4 (n))$ neurons and neurons $Tapprox n$. With $m=Omega(log4 (n))$ neurons and $Tapprox n$, we bound the test loss training by $tildeO (1/)$.
  arXiv  Detail & Related papers  (2023-02-18T05:06:15Z)
• Detection-Recovery Gap for Planted Dense Cycles [72.4451045270967]
  We consider a model where a dense cycle with expected bandwidth $n tau$ and edge density $p$ is planted in an ErdHos-R'enyi graph $G(n,q)$. We characterize the computational thresholds for the associated detection and recovery problems for the class of low-degree algorithms.
  arXiv  Detail & Related papers  (2023-02-13T22:51:07Z)
• Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data [12.225207401994737]
  We show that complex systems with massive amounts of parameters exhibit the same structural properties when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of vectors.
  arXiv  Detail & Related papers  (2023-01-01T16:29:56Z)
• Agnostic Learning of a Single Neuron with Gradient Descent [92.7662890047311]
  We consider the problem of learning the best-fitting single neuron as measured by the expected square loss. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$.
  arXiv  Detail & Related papers  (2020-05-29T07:20:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.