Kernel Dynamics under Path Entropy Maximization

• URL: http://arxiv.org/abs/2603.27880v1
• Date: Sun, 29 Mar 2026 21:34:08 GMT
• Title: Kernel Dynamics under Path Entropy Maximization
• Abstract summary: kernel function k : X x X -> R is treated as a dynamical variable subject to path entropy.<n>We formulate fixed-point conditions for self-consistent kernels.<n>We pose six open questions that make the program empirically and mathematically testable.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a variational framework in which the kernel function k : X x X -> R, interpreted as the foundational object encoding what distinctions an agent can represent, is treated as a dynamical variable subject to path entropy maximization (Maximum Caliber, MaxCal). Each kernel defines a representational structure over which an information geometry on probability space may be analyzed; a trajectory through kernel space therefore corresponds to a trajectory through a family of effective geometries, making the optimization landscape endogenous to its own traversal. We formulate fixed-point conditions for self-consistent kernels, propose renormalization group (RG) flow as a structured special case, and suggest neural tangent kernel (NTK) evolution during deep network training as a candidate empirical instantiation. Under explicit information-thermodynamic assumptions, the work required for kernel change is bounded below by delta W >= k_B T delta I_k, where delta I_k is the mutual information newly unlocked by the updated kernel. In this view, stable fixed points of MaxCal over kernels correspond to self-reinforcing distinction structures, with biological niches, scientific paradigms, and craft mastery offered as conjectural interpretations. We situate the framework relative to assembly theory and the MaxCal literature, separate formal results from structured correspondences and conjectural bridges, and pose six open questions that make the program empirically and mathematically testable.

Related papers

• The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics [46.539149169728624]
  We introduce the Kuramoto Neural Operator (KNO), which represents the solution through the evolution of a latent field of interacting oscillators.<n>KNO achieves strong predictive performance, with improvements over competing approaches.
  arXiv  Detail & Related papers  (2026-08-10T21:11:58Z)
• Training-Free Universal Approximation by Prompting Random Transformers [48.556633593447465]
  We show that a softmax attention network with random, untrained weights can approximate any Hlder function on a compact manifold when steered by an appropriate soft prompt.<n>We further quantify the cost of prompting, exposing a tradeoff between the norm of the constructed soft prompt tokens, prompt length, and hidden dimension.
  arXiv  Detail & Related papers  (2026-08-10T12:57:22Z)
• Beyond the Gegenbauer Paradigm: q-Orthogonal Kernels for Machine Learning [0.0]
  We introduce a class of $q$-orthogonals that generalize classical Hermite scaling through a deformation parameter $q$.<n>Our results confirm that $q$-orthogonals constitute a promising direction for kernel design with practical machine learning applications.
  arXiv  Detail & Related papers  (2026-08-04T11:18:28Z)
• Structural interpretability in SVMs with truncated orthogonal polynomial kernels [0.0]
  We study post-training interpretability for Support Vector Machines built from truncated kernels.<n>Since the associated kernel space is finite-dimensional and admits an explicit tensor-product orthonormal basis, the fitted decision function can be expanded exactly in RKHS.<n>This leads to intrinsic Orthogonal Representation Contribution Analysis (ORCA), a diagnostic framework based on normalized Orthogonal Kernel Contribution (OKC) indices.
  arXiv  Detail & Related papers  (2026-04-16T17:52:08Z)
• The Kernel Manifold: A Geometric Approach to Gaussian Process Model Selection [0.8536502059643899]
  We present a Bayesian optimization framework built on kernel-of- Kernels geometry.<n>A multidimensional scaling (MDS) embedding maps a discrete kernel library into a continuous Euclidean manifold, enabling smooth BO.<n>We demonstrate the approach on synthetic benchmarks, real-world time-series datasets, and an additive manufacturing case study.
  arXiv  Detail & Related papers  (2026-01-08T20:52:53Z)
• From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systems [7.807210884802377]
  We introduce a novel, scalable GPO that capitalizes on sparsity, locality, and structural information through judicious kernel design.<n>We demonstrate that our framework consistently achieves high accuracy across varying discretization scales.
  arXiv  Detail & Related papers  (2025-06-18T22:40:52Z)
• Generalization Bound of Gradient Flow through Training Trajectory and Data-dependent Kernel [55.82768375605861]
  We establish a generalization bound for gradient flow that aligns with the classical Rademacher complexity for kernel methods.<n>Unlike static kernels such as NTK, the LPK captures the entire training trajectory, adapting to both data and optimization dynamics.
  arXiv  Detail & Related papers  (2025-06-12T23:17:09Z)
• Understanding In-Context Learning on Structured Manifolds: Bridging Attention to Kernel Methods [45.94152084965753]
  We establish a novel connection between the attention mechanism and classical kernel methods.<n>We derive generalization error bounds in terms of the prompt length and the number of training tasks.<n>Our result characterizes how the generalization error scales with the number of training tasks.
  arXiv  Detail & Related papers  (2025-06-12T17:56:26Z)
• General Graph Random Features [42.75616308187867]
  We propose a novel random walk-based algorithm for unbiased estimation of arbitrary functions of a weighted adjacency matrix. Our algorithm enjoys subquadratic time complexity with respect to the number of nodes, overcoming the notoriously prohibitive cubic scaling of exact graph kernel evaluation.
  arXiv  Detail & Related papers  (2023-10-07T15:47:31Z)
• AERK: Aligned Entropic Reproducing Kernels through Continuous-time Quantum Walks [17.95088104970343]
  We develop an Aligned Entropic Reproducing Kernel (AERK) for graph classification. For pairwise graphs, the proposed AERK kernel is defined by computing a reproducing kernel based similarity between the quantum Shannon entropies of their each pair of aligned vertices. The experimental evaluation on standard graph datasets demonstrates that the proposed AERK kernel is able to outperform state-of-the-art graph kernels for graph classification tasks.
  arXiv  Detail & Related papers  (2023-03-04T16:48:39Z)
• The Manifold Hypothesis for Gradient-Based Explanations [55.01671263121624]
  gradient-based explanation algorithms provide perceptually-aligned explanations. We show that the more a feature attribution is aligned with the tangent space of the data, the more perceptually-aligned it tends to be. We suggest that explanation algorithms should actively strive to align their explanations with the data manifold.
  arXiv  Detail & Related papers  (2022-06-15T08:49:24Z)
• Kernel Identification Through Transformers [54.3795894579111]
  Kernel selection plays a central role in determining the performance of Gaussian Process (GP) models. This work addresses the challenge of constructing custom kernel functions for high-dimensional GP regression models. We introduce a novel approach named KITT: Kernel Identification Through Transformers.
  arXiv  Detail & Related papers  (2021-06-15T14:32:38Z)
• Flow-based Kernel Prior with Application to Blind Super-Resolution [143.21527713002354]
  Kernel estimation is generally one of the key problems for blind image super-resolution (SR) This paper proposes a normalizing flow-based kernel prior (FKP) for kernel modeling. Experiments on synthetic and real-world images demonstrate that the proposed FKP can significantly improve the kernel estimation accuracy.
  arXiv  Detail & Related papers  (2021-03-29T22:37:06Z)
• Strong Uniform Consistency with Rates for Kernel Density Estimators with General Kernels on Manifolds [11.927892660941643]
  We show how to handle kernel density estimation with intricate kernels not designed by the user. The isotropic kernels considered in this paper are different from the kernels in the Vapnik-Chervonenkis class that are frequently considered in statistics society.
  arXiv  Detail & Related papers  (2020-07-13T14:36:06Z)

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.