Topological Obstructions and How to Avoid Them

• URL: http://arxiv.org/abs/2312.07529v1
• Date: Tue, 12 Dec 2023 18:56:14 GMT
• Title: Topological Obstructions and How to Avoid Them
• Abstract summary: We show that local optima can arise due to singularities or an incorrect degree or winding number. We propose a new flow-based model that maps data points to multimodal distributions over geometric spaces.
• Score: 22.45861345237023
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Incorporating geometric inductive biases into models can aid interpretability and generalization, but encoding to a specific geometric structure can be challenging due to the imposed topological constraints. In this paper, we theoretically and empirically characterize obstructions to training encoders with geometric latent spaces. We show that local optima can arise due to singularities (e.g. self-intersection) or due to an incorrect degree or winding number. We then discuss how normalizing flows can potentially circumvent these obstructions by defining multimodal variational distributions. Inspired by this observation, we propose a new flow-based model that maps data points to multimodal distributions over geometric spaces and empirically evaluate our model on 2 domains. We observe improved stability during training and a higher chance of converging to a homeomorphic encoder.

Related papers

• A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds [0.05729426778193397]
  Speciation in generative diffusion models denotes the emergence of distinct stable branches during denoising.<n>We go beyond existing theoretical descriptions, which usually identify speciation with a symmetric pitchfork bifurcation.<n>A chart-based intrinsic score-learning scheme based on neural networks contrasts the theoretically predicted transitions on prototypal.
  arXiv  Detail & Related papers  (2026-08-24T20:03:31Z)
• Representational Capacity: Geometric Limits on Feature Representation in Transformer Language Models [51.56484100374058]
  We develop a framework for estimating how many near-orthogonal directions a model can support.<n>Applying this metric across dozens of open-source models reveals two classes: models with high $varepsilon$ whose embeddings lack near-orthogonal structure, and models with low $varepsilon$ that maintain it.
  arXiv  Detail & Related papers  (2026-06-01T18:28:56Z)
• The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models [11.106770083374288]
  Large Language Models exhibit paradoxical hallucination in fundamental arithmetic, implying a disconnect between internal computation and discrete output.<n>We propose the Noisy Quantization Model to explain this geometry, framing arithmetic errors as Geometric Slippages caused by internal neural noise pushing a continuous, latent Carry Potential across quantization thresholds.<n>We validate these insights through a geometric consistency check method that effectively detects and corrects these quantization failures during inference.
  arXiv  Detail & Related papers  (2026-05-29T10:55:50Z)
• A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning [52.452902154360565]
  We introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning.<n>MCBP explicitly models the intrinsic curvature of the data manifold and computes pointwise mean curvature without requiring principled manifold parametrization.<n>Experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance.
  arXiv  Detail & Related papers  (2026-05-05T20:19:09Z)
• The Geometric Price of Discrete Logic: Context-driven Manifold Dynamics of Number Representations [6.414826816896125]
  Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries.<n>We argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion"<n>We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing.
  arXiv  Detail & Related papers  (2026-03-24T13:41:57Z)
• Latent Semantic Manifolds in Large Language Models [0.0]
  We develop a mathematical framework that interprets Large Language Models hidden states as points on a latent semantic manifold.<n>We prove two theorems: a rate-distortion lower bound on distortion for any finite vocabulary, and a linear volume scaling law for the expressibility gap.<n>We discuss implications for architecture design, model compression, decoding strategies, and scaling laws.
  arXiv  Detail & Related papers  (2026-03-17T13:05:56Z)
• Riemannian Consistency Model [57.933800575074535]
  We propose the Riemannian Consistency Model (RCM), which, for the first time, enables few-step consistency modeling.<n>We derive the closed-form solutions for both discrete- and continuous-time training objectives for RCM.<n>We provide a unique kinematics perspective for interpreting the RCM objective, offering new theoretical angles.
  arXiv  Detail & Related papers  (2025-10-01T14:57:25Z)
• Learning Latent Graph Geometry via Fixed-Point Schrödinger-Type Activation: A Theoretical Study [1.1745324895296467]
  We develop a unified theoretical framework for neural architectures with internal representations evolving as stationary states of dissipative Schr"odinger-type dynamics on learned latent graphs.<n>We prove existence, uniqueness, and smooth dependence of equilibria, and show that the dynamics are equivalent under the Bloch map to norm-preserving Landau--Lifshitz flows.<n>The resulting model class provides a compact, geometrically interpretable, and analytically tractable foundation for learning latent graph geometry via fixed-point Schr"odinger-type activations.
  arXiv  Detail & Related papers  (2025-07-27T00:35:15Z)
• Can Diffusion Models Disentangle? A Theoretical Perspective [52.360881354319986]
  This paper presents a novel theoretical framework for understanding how diffusion models can learn disentangled representations. We establish identifiability conditions for general disentangled latent variable models, analyze training dynamics, and derive sample complexity bounds for disentangled latent subspace models.
  arXiv  Detail & Related papers  (2025-03-31T20:46:18Z)
• Geometric Trajectory Diffusion Models [58.853975433383326]
  Generative models have shown great promise in generating 3D geometric systems. Existing approaches only operate on static structures, neglecting the fact that physical systems are always dynamic in nature. We propose geometric trajectory diffusion models (GeoTDM), the first diffusion model for modeling the temporal distribution of 3D geometric trajectories.
  arXiv  Detail & Related papers  (2024-10-16T20:36:41Z)
• Decoder ensembling for learned latent geometries [15.484595752241122]
  We show how to easily compute geodesics on the associated expected manifold. We find this simple and reliable, thereby coming one step closer to easy-to-use latent geometries.
  arXiv  Detail & Related papers  (2024-08-14T12:35:41Z)
• Geometric Neural Diffusion Processes [55.891428654434634]
  We extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We show that with these conditions, the generative functional model admits the same symmetry.
  arXiv  Detail & Related papers  (2023-07-11T16:51:38Z)
• Latent Traversals in Generative Models as Potential Flows [113.4232528843775]
  We propose to model latent structures with a learned dynamic potential landscape. Inspired by physics, optimal transport, and neuroscience, these potential landscapes are learned as physically realistic partial differential equations. Our method achieves both more qualitatively and quantitatively disentangled trajectories than state-of-the-art baselines.
  arXiv  Detail & Related papers  (2023-04-25T15:53:45Z)
• Geometric Scattering on Measure Spaces [12.0756034112778]
  We introduce a general, unified model for geometric scattering on measure spaces. We consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold.
  arXiv  Detail & Related papers  (2022-08-17T22:40:09Z)
• Moser Flow: Divergence-based Generative Modeling on Manifolds [49.04974733536027]
  Moser Flow (MF) is a new class of generative models within the family of continuous normalizing flows (CNF) MF does not require invoking or backpropagating through an ODE solver during training. We demonstrate for the first time the use of flow models for sampling from general curved surfaces.
  arXiv  Detail & Related papers  (2021-08-18T09:00:24Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)
• Continuous normalizing flows on manifolds [0.342658286826597]
  We describe how the recently introduced Neural ODEs and continuous normalizing flows can be extended to arbitrary smooth manifold. We propose a general methodology for parameterizing vector fields on these spaces and demonstrate how gradient-based learning can be performed.
  arXiv  Detail & Related papers  (2021-03-14T15:35:19Z)
• GELATO: Geometrically Enriched Latent Model for Offline Reinforcement Learning [54.291331971813364]
  offline reinforcement learning approaches can be divided into proximal and uncertainty-aware methods. In this work, we demonstrate the benefit of combining the two in a latent variational model. Our proposed metrics measure both the quality of out of distribution samples as well as the discrepancy of examples in the data.
  arXiv  Detail & Related papers  (2021-02-22T19:42:40Z)
• Generative Model without Prior Distribution Matching [26.91643368299913]
  Variational Autoencoder (VAE) and its variations are classic generative models by learning a low-dimensional latent representation to satisfy some prior distribution. We propose to let the prior match the embedding distribution rather than imposing the latent variables to fit the prior.
  arXiv  Detail & Related papers  (2020-09-23T09:33:24Z)
• Neural Ordinary Differential Equations on Manifolds [0.342658286826597]
  Recently normalizing flows in Euclidean space based on Neural ODEs show great promise, yet suffer the same limitations. We show how vector fields provide a general framework for parameterizing a flexible class of invertible mapping on these spaces.
  arXiv  Detail & Related papers  (2020-06-11T17:56:34Z)

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.