Approximate Homomorphisms and Convergent Representations in Transducers

• URL: http://arxiv.org/abs/2608.20428v1
• Date: Wed, 19 Aug 2026 19:56:22 GMT
• Title: Approximate Homomorphisms and Convergent Representations in Transducers
• Abstract summary: We study the stability of minimal representations of controlled processes (in particular, transducers) under perturbations.<n>We consider standard, linear and predictive transducers.
• Score: 0.913755431537592
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the stability of minimal representations of controlled stochastic processes (in particular, transducers) under perturbations. This question is motivated by recent experiments finding predictive-state structure in the latent representations of neural networks. We consider standard, linear and predictive transducers. We introduce notions of approximate homomorphism capturing local structural similarity between them, together with metrics comparing their induced dynamics (which we refer to as interfaces), and prove properties such as composability of the approximate homomorphisms. For standard transducers, we show that there exist simple interfaces for which there is no approximate homomorphism between the different implementations of the dynamics. In contrast, for every finite-rank interface $\mathcal I$, we prove that all minimal linear transducers implementing interfaces sufficiently close to $\mathcal I$ have an approximate homomorphism to the minimal implementation of $\mathcal I$, with error linear in the perturbation size. We prove an analogous stability result for predictive transducers under a residual metric using some mild hypothesis regarding the indistinguishability of the belief states. These results identify conditions under which canonical transducer representations are robust to perturbations, while showing that such convergence fails without additional structural restrictions. Under the assumption that these type of abstractions are embedded into the hidden layers of modern AI models, this gives some theoretical support to the hypothesis that their latent representations exhibit structural convergence.

Related papers

• Contraction Analysis of Holomorphic Dynamical Systems via the Intrinsic Kobayashi Metric [0.0]
  This paper studies incremental stability of holomorphic dynamical systems through the infinitesimal Kobayashi metric.<n>Contraction is formalized as an upper Dini-derivative inequality on the Kobayashi metric along trajectories.
  arXiv  Detail & Related papers  (2026-08-01T17:06:21Z)
• SIGMA: Structure-Invariant Generative Molecular Alignment for Chemical Language Models via Autoregressive Contrastive Learning [8.06177122578894]
  We propose Structure-Invariant Generative Molecular Alignment ( SIGMA)<n>Rather than altering the linear representation, SIGMA enables the model to strictly recognize geometric symmetries.<n> Empirical evaluations on standard benchmarks demonstrate that SIGMA bridges the gap between sequence scalability and graph fidelity.
  arXiv  Detail & Related papers  (2026-03-26T05:55:17Z)
• Scale-Consistent State-Space Dynamics via Fractal of Stationary Transformations [9.983526161001997]
  Recent deep learning models increasingly rely on depth without structural guarantees on the validity of intermediate representations.<n>We address this limitation by formulating a structural requirement for state-space model's scale-consistent latent dynamics.<n>We empirically verify the predicted scale-consistent behavior, showing that adaptive efficiency emerges from the aligned latent geometry.
  arXiv  Detail & Related papers  (2026-01-27T12:44:20Z)
• Loss-Complexity Landscape and Model Structure Functions [53.92822954974537]
  We develop a framework for dualizing the Kolmogorov structure function $h_x(alpha)$.<n>We establish a mathematical analogy between information-theoretic constructs and statistical mechanics.<n>We explicitly prove the Legendre-Fenchel duality between the structure function and free energy.
  arXiv  Detail & Related papers  (2025-07-17T21:31:45Z)
• Emergence of Quantised Representations Isolated to Anisotropic Functions [0.0]
  This paper presents a novel methodology for determining representational structure, which builds upon the existing Spotlight Resonance method.<n>It shows how discrete representations can emerge and organise in autoencoder models, through a controlled ablation study in which only the activation function is altered.<n>Using this technique, the validity of whether function-driven symmetries can act as implicit inductive biases on representations is determined.
  arXiv  Detail & Related papers  (2025-07-16T09:27:54Z)
• Relative Representations: Topological and Geometric Perspectives [50.85040046976025]
  Relative representations are an established approach to zero-shot model stitching.<n>We introduce a normalization procedure in the relative transformation, resulting in invariance to non-isotropic rescalings and permutations.<n>Second, we propose to deploy topological densification when fine-tuning relative representations, a topological regularization loss encouraging clustering within classes.
  arXiv  Detail & Related papers  (2024-09-17T08:09:22Z)
• DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion [66.21290235237808]
  We introduce an energy constrained diffusion model which encodes a batch of instances from a dataset into evolutionary states. We provide rigorous theory that implies closed-form optimal estimates for the pairwise diffusion strength among arbitrary instance pairs. Experiments highlight the wide applicability of our model as a general-purpose encoder backbone with superior performance in various tasks.
  arXiv  Detail & Related papers  (2023-01-23T15:18:54Z)
• Unifying Causal Inference and Reinforcement Learning using Higher-Order Category Theory [4.119151469153588]
  We present a unified formalism for structure discovery of causal models and predictive state representation models in reinforcement learning. Specifically, we model structure discovery in both settings using simplicial objects.
  arXiv  Detail & Related papers  (2022-09-13T19:04:18Z)
• Identifying Weight-Variant Latent Causal Models [82.14087963690561]
  We find that transitivity acts as a key role in impeding the identifiability of latent causal representations. Under some mild assumptions, we can show that the latent causal representations can be identified up to trivial permutation and scaling. We propose a novel method, termed Structural caUsAl Variational autoEncoder, which directly learns latent causal representations and causal relationships among them.
  arXiv  Detail & Related papers  (2022-08-30T11:12:59Z)
• Convolutional Filtering and Neural Networks with Non Commutative Algebras [153.20329791008095]
  We study the generalization of non commutative convolutional neural networks. We show that non commutative convolutional architectures can be stable to deformations on the space of operators.
  arXiv  Detail & Related papers  (2021-08-23T04:22:58Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)

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.