The Geometry of Abstraction: Continual Learning via Recursive Quotienting

• URL: http://arxiv.org/abs/2512.18471v1
• Date: Sat, 20 Dec 2025 19:10:38 GMT
• Title: The Geometry of Abstraction: Continual Learning via Recursive Quotienting
• Authors: Xin Li,
• Abstract summary: Continual learning systems face a fundamental geometric barrier: the flat manifold problem.<n>We propose a geometric resolution to this paradox based on Recursive Metric Contraction.<n>We show that tokens in neural architectures are physically realizable as singularities or wormholes.
• Score: 6.0044467881527614
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: Continual learning systems operating in fixed-dimensional spaces face a fundamental geometric barrier: the flat manifold problem. When experience is represented as a linear trajectory in Euclidean space, the geodesic distance between temporal events grows linearly with time, forcing the required covering number to diverge. In fixed-dimensional hardware, this volume expansion inevitably forces trajectory overlap, manifesting as catastrophic interference. In this work, we propose a geometric resolution to this paradox based on Recursive Metric Contraction. We formalize abstraction not as symbolic grouping, but as a topological deformation: a quotient map that collapses the metric tensor within validated temporal neighborhoods, effectively driving the diameter of local sub-manifolds to zero. We substantiate our framework with four rigorous results. First, the Bounded Capacity Theorem establishes that recursive quotient maps allow the embedding of arbitrarily long trajectories into bounded representational volumes, trading linear metric growth for logarithmic topological depth. Second, the Topological Collapse Separability Theorem, derived via Urysohn's Lemma, proves that recursive quotienting renders non-linearly separable temporal sequences linearly separable in the limit, bypassing the need for infinite-dimensional kernel projections. Third, the Parity-Partitioned Stability Theorem solves the catastrophic forgetting problem by proving that if the state space is partitioned into orthogonal flow and scaffold manifolds, the metric deformations of active learning do not disturb the stability of stored memories. Our analysis reveals that tokens in neural architectures are physically realizable as singularities or wormholes, regions of extreme positive curvature that bridge distant points in the temporal manifold.

Related papers

• Kinematic budget of quantum correlations [0.0]
  We introduce a local-unitary-invariant budget split of symmetrised second moments into local and nonlocal sectors.<n>By coarse-graining over gauge-like first moments, the budget geometry acts as a thermodynamic phase diagram.
  arXiv  Detail & Related papers  (2026-03-04T09:40:02Z)
• Riemannian Flow Matching for Disentangled Graph Domain Adaptation [51.98961391065951]
  Graph Domain Adaptation (GDA) typically uses adversarial learning to align graph embeddings in Euclidean space.<n>DisRFM is a geometry-aware GDA framework that unifies embedding and flow-based transport.
  arXiv  Detail & Related papers  (2026-01-31T11:05:35Z)
• Manifold Percolation: from generative model to Reinforce learning [0.26905021039717986]
  Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task becomes disentangling the geometric support from the probability distribution.<n>We propose that continuum percolation is uniquely suited to this support analysis, as the sampling process effectively projects high-dimensional density estimation onto a geometric counting problem on the support.
  arXiv  Detail & Related papers  (2025-11-25T17:12:42Z)
• TokenBlowUp: Resolving Representational Singularities in LLM Token Spaces via Monoidal Transformations [0.9179857807576733]
  Recent work has provided compelling evidence challenging the foundational manifold hypothesis for the token embedding spaces of Large Language Models.<n>We formalize this problem in the language of scheme theory and propose a rigorous resolution by applying the scheme-theoretic blow-up at each singular point.<n>We prove a formal theorem guaranteeing the geometric regularization of this new space, showing that the original pathologies are resolved.
  arXiv  Detail & Related papers  (2025-07-26T02:39:54Z)
• Learning Beyond Euclid: Curvature-Adaptive Generalization for Neural Networks on Manifolds [0.0]
  Existing generalization theories often rely on complexity measures derived from Euclidean geometry, which fail to account for intrinsic structure of non-Euclidean spaces.<n>We derive number bounds that explicitly incorporate manifold-specific properties such as sectional curvature, volume growth, and injectivity radius.<n>This framework provides a principled understanding of how intrinsic geometry affects learning capacity, offering both theoretical insight and practical implications for deep learning on structured data domains.
  arXiv  Detail & Related papers  (2025-07-01T23:16:49Z)
• Non-orientable Exceptional Points in Twisted Boundary Systems [1.188383832081829]
  We study the adiabatic transport of exceptional points along non-orientable closed loops.<n>We reveal distinct topological responses arising from the lack of global orientation.<n>We numerically implement these ideas in a microdisk cavity with embedded scatterers.
  arXiv  Detail & Related papers  (2025-04-16T11:26:05Z)
• Nonlinearity-driven Topology via Spontaneous Symmetry Breaking [79.16635054977068]
  We consider a chain of parametrically-driven quantum resonators coupled only via weak nearest-neighbour cross-Kerr interaction.<n>Topology is dictated by the structure of the Kerr nonlinearity, yielding a non-trivial bulk-boundary correspondence.
  arXiv  Detail & Related papers  (2025-03-15T00:20:45Z)
• Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation [52.73824786627612]
  This paper establishes new convergence results for textitgeodesic strongly monotone games.<n>Our key result shows that RGD attains last-iterate linear convergence in a textitgeometry-agnostic fashion.<n>Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Dynamical chaos in nonlinear Schr\"odinger models with subquadratic power nonlinearity [137.6408511310322]
  We deal with a class of nonlinear Schr"odinger lattices with random potential and subquadratic power nonlinearity. We show that the spreading process is subdiffusive and has complex microscopic organization. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border.
  arXiv  Detail & Related papers  (2023-01-20T16:45:36Z)
• A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations [77.86290991564829]
  Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. We study a particular sequence of maps between manifold, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between implementing neural networks of practical interest.
  arXiv  Detail & Related papers  (2021-12-17T11:43:30Z)
• Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields [53.31927549039624]
  We show that a piecewise discretization preserves better contrast from existing discretization problems. We apply this theory to the problem of matching two images.
  arXiv  Detail & Related papers  (2021-07-13T12:31:06Z)
• Manifold Learning via Manifold Deflation [105.7418091051558]
  dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. Many popular methods can fail dramatically, even on simple two-dimensional Manifolds. This paper presents an embedding method for a novel, incremental tangent space estimator that incorporates global structure as coordinates. Empirically, we show our algorithm recovers novel and interesting embeddings on real-world and synthetic datasets.
  arXiv  Detail & Related papers  (2020-07-07T10:04:28Z)

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.