Geometric and Spectral Alignment for Deep Neural Network I

• URL: http://arxiv.org/abs/2605.02108v1
• Date: Mon, 04 May 2026 00:07:24 GMT
• Title: Geometric and Spectral Alignment for Deep Neural Network I
• Abstract summary: We prove deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors.<n>Full-rank factors are mapped from $mathrmGL(d)$ to the positive cone by $Amapsto Atop A$, then to ordered eigenvalue data.<n>We prove approximate-power-law and metric-chart versions, converse lower bounds, Fisher--KL/Bures action estimates, and near-identity expansions for normalized residual chains.
• Score: 12.227949990332363
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Deep residual architectures are modeled as products of near-identity Jacobians. This paper proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors, emphasizing a normalized top-radial Cartan coordinate and fitted power-law chart. Full-rank factors are mapped from $\mathrm{GL}(d)$ to the positive cone by $A\mapsto A^\top A$, then to ordered eigenvalue data. Under Frobenius normalization, exact power-law spectra form a trace-normalized Cartan orbit. This orbit is a Gibbs family on ranks, a Fisher information line, and a Bures--Wasserstein curve with line element $d/4$ times Fisher information. The main rigidity theorem is a slack-aware margin inequality: interface radial amplitude, non-backtracking slack, and signed residual variation control displacement of the fitted Cartan coordinate. In the exact-chart zero-slack case, a depth-$L$ budget gives exponent drift of order $(\log M)/L$; generally, slack and residual increments augment the bound. We separate scalar top-radial from full-Cartan spectral control, which also needs Bures/Hellinger residual variation. We prove approximate-power-law and metric-chart versions, converse lower bounds, Fisher--KL/Bures action estimates, and near-identity expansions for normalized residual chains. Near-identity results verify transport budgets; chart quality remains measurable. Effective rank is a spectral-energy quantile, giving finite-width power-law tail bounds and robust rank-window transition estimates. Empirical static-weight exponent profiles serve as diagnostics; full verification also requires interface budgets, slacks, and residuals for the same operator chain.

Related papers

• Loss-Parameterized Fisher Width Along Learning Trajectories [0.0]
  We first derive an exact trace--shape factorization and a deterministic stability bound for fixed compact probes.<n>Results support a branchwise, rather than universal, loss parametrization of Fisher width.
  arXiv  Detail & Related papers  (2026-08-21T18:52:04Z)
• Intrinsic Structure: Spectral Identifiability for Mechanistic Interpretability [51.56484100374058]
  We prove the first identifiability theorem for a mechanistic-interpretability primitive.<n>The spectrum is an identifiable, model-intrinsic fingerprint with a stated error bar, not a legible decomposition.
  arXiv  Detail & Related papers  (2026-08-10T19:42:01Z)
• Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise [49.730496294398726]
  We develop a novel quantum mean estimator for heavy-tailed random variables.<n>We derive stronger lower bounds for tail index>4/3$, showing that a nontrivial dependence on a dimension is unavoidable.<n>For a convex objective function, we propose a quantum projected gradient descent method.
  arXiv  Detail & Related papers  (2026-07-28T09:29:37Z)
• Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity [5.67642958049511]
  Singular learning theory characterises the complexity of a deep network through the geometry of its loss singularities.<n>We introduce DeadDirection Signatures (DDS), a family of cheap closed-form spectral readings of singular structure.<n>Each reads a network's activation matrix or per-sample-gradient FisherGram at a chosen layer, replacing the posterior chain spectral linear algebra.
  arXiv  Detail & Related papers  (2026-06-19T06:49:09Z)
• Scale-Equivariant Generative Forecasting: Weight-Tied Dilated Convolutions, Wavelet Scattering Inputs, and Spectral-Consistency Training for Self-Similar Time Series [0.0]
  We give a precise definition of discrete scale equivariance for 1D causal networks.<n>We prove that dyadic dilation commutes with any dilated-convolution stack whose kernel weights are shared across levels.<n>On 30 years of S&P 500 daily log-returns, SE-WaveNet samples reproduce the empirical scaling-collapse diagnostic on the Allan-Variance top-25 universe.
  arXiv  Detail & Related papers  (2026-05-17T18:21:30Z)
• How Does Attention Help? Insights from Random Matrices on Signal Recovery from Sequence Models [6.758792133077052]
  We derive exact characterizations of the limiting eigenvalue distribution, outlier eigenvalues, and eigenvector alignment with the hidden signal.<n>We show that parameter-free causal self-attention with $/d$ score scaling yields deterministic harmonic weights that improve signal recovery over mean pooling.
  arXiv  Detail & Related papers  (2026-05-07T18:28:01Z)
• Minimax Optimality and Spectral Routing for Majority-Vote Ensembles under Markov Dependence [6.908972852063454]
  Majority-vote ensembles achieve variance reduction by averaging over diverse, approximately independent base learners.<n>We provide a minimax characterization of this phenomenon for discrete classification in a fixed-dimensional Markov setting.<n>Experiments on synthetic Markov chains, 2D spatial grids, the 128-dataset UCR archive, and Atari DQN ensembles validate the theoretical predictions.
  arXiv  Detail & Related papers  (2026-04-15T02:32:30Z)
• The Spectral Edge Thesis: A Mathematical Framework for Intra-Signal Phase Transitions in Neural Network Training [0.0]
  We show that phase transitions in neural network training are controlled by the spectral gap of the rolling-window Gram matrix of parameter updates.<n>Adiabatic parameter $mathcalA = |G|_F / (, g2)$ controls circuit stability: $mathcalA ll 1$ (plateau), $mathcalA sim 1$ (phase transition), $mathcalA gg 1$ (forgetting)
  arXiv  Detail & Related papers  (2026-03-30T20:10:22Z)
• Regularized Online RLHF with Generalized Bilinear Preferences [68.44113000390544]
  We consider the problem of contextual online RLHF with general preferences.<n>We adopt the Generalized Bilinear Preference Model to capture preferences via low-rank, skew-symmetric matrices.<n>We prove that the dual gap of the greedy policy is bounded by the square of the estimation error.
  arXiv  Detail & Related papers  (2026-02-26T15:27:53Z)
• Stability and Generalization of Push-Sum Based Decentralized Optimization over Directed Graphs [55.77845440440496]
  Push-based decentralized communication enables optimization over communication networks, where information exchange may be asymmetric.<n>We develop a unified uniform-stability framework for the Gradient Push (SGP) algorithm.<n>A key technical ingredient is an imbalance-aware generalization bound through two quantities.
  arXiv  Detail & Related papers  (2026-02-24T05:32:03Z)
• Stabilizing Fixed-Point Iteration for Markov Chain Poisson Equations [49.702772230127465]
  We study finite-state Markov chains with $n$ states and transition matrix $P$.<n>We show that all non-decaying modes are captured by a real peripheral invariant subspace $mathcalK(P)$, and that the induced operator on the quotient space $mathbbRn/mathcalK(P) is strictly contractive, yielding a unique quotient solution.
  arXiv  Detail & Related papers  (2026-01-31T02:57:01Z)
• Singular Bayesian Neural Networks [1.2891210250935148]
  Bayesian neural networks promise calibrated uncertainty but require $O(mn)$ parameters for standard mean-field Gaussian posteriors.<n>We induce a posterior that is singular with respect to the Lebesgue measure, concentrating on the rank-$r$ manifold.<n>We derive PAC-Bayes generalization bounds whose complexity term scales as $sqrtr(m+n)$ instead of $sqrtm n$, and prove loss bounds that decompose the error into optimization and rank-induced bias.
  arXiv  Detail & Related papers  (2026-01-30T23:06:34Z)
• Robust Layerwise Scaling Rules by Proper Weight Decay Tuning [50.11170157029911]
  In modern scale-invariant architectures, training quickly enters an degrading-governed steady state.<n>We introduce a weight-decay scaling rule for AdamW that preserves sublayer gain across widths.<n>Our results extend $mu$P beyond the near-init regime by explicitly controlling the steady-state scales set by parameters.
  arXiv  Detail & Related papers  (2025-10-17T02:58:35Z)
• Convergence Rate Analysis of LION [54.28350823319057]
  LION converges iterations of $cal(sqrtdK-)$ measured by gradient Karush-Kuhn-T (sqrtdK-)$. We show that LION can achieve lower loss and higher performance compared to standard SGD.
  arXiv  Detail & Related papers  (2024-11-12T11:30:53Z)
• Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise [10.418647759223965]
  Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis. We prove the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian with rates. When the manifold data are corrupted by outlier noise, we theoretically prove the graph Laplacian point-wise consistency.
  arXiv  Detail & Related papers  (2022-06-22T21:08:24Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)

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.