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Approximate Homomorphisms and Convergent Representations in Transducers
Santiago Cifuentes
TL;DR
The paper asks when minimal transducer representations remain structurally similar under perturbations, motivated by convergent structure observed in neural-network hidden layers. It introduces approximate homomorphisms and interface metrics, finding convergence for finite-rank linear interfaces and under a suitable predictive-transducer metric, but not generally for standard transducers.
Problem
The paper studies whether minimal representations of similar controlled stochastic-process dynamics share structure, motivated by observed similarities between neural-network hidden layers across models and architectures.
Method
The paper defines approximate homomorphisms and interface metrics, then analyzes their algebraic properties and uses them to prove convergence or non-convergence results for standard, linear, and predictive transducers.
Results
Minimal linear implementations of sufficiently close finite-rank interfaces admit approximate homomorphisms with error scaling as Γ_Iε, while standard transducers can lack such convergence and predictive transducers have a positive result under a suitable metric.
Takeaways & Limitations
Canonical transducer representations can be robust to perturbations under structural restrictions, providing theoretical support for convergent latent structure in some neural-network abstractions.
Takeaways & Limitations
The predictive-transducer stability result is restricted to scenarios where the residual distance is meaningful, and standard-transducer convergence can fail even for simple interfaces.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper studies whether different neural networks converge to structurally similar latent representations by modeling internal world models as transducers. It introduces approximate structural comparisons and proves that convergence is stable for linear and predictive transducers but can fail for standard transducers.
- Motivation: The work asks whether transducers implementing similar behavior also share structure when their interfaces differ slightly because models train on different data.This extends convergence questions from exact to noisy and approximate settings.
- Contributions: Approximate homomorphisms capture local structural similarity, compose robustly, and preserve represented dynamics with discounted-metric error O(ε).Discounted metrics exponentially downweight differences at longer horizons.
- Contributions: Standard transducers can implement simple interfaces while remaining structurally far apart, so approximate homomorphisms do not generally restore a common minimal representation.The counterexample also applies to nearby interfaces and discounted distance measures.
- Contributions: For finite-rank interfaces, all minimal linear implementations of sufficiently close interfaces map approximately to a common minimal linear transducer, with error Γ_Iε.The result follows from the canonical Hankel-matrix construction.
- Contributions: An analogous stability result holds for predictive transducers under a residual metric, with assumptions concerning indistinguishability of belief states.The predictive result is restricted to settings where the residual distance is meaningful.
- Implications: Together, the results provide theoretical support for convergent latent structure when neural-network world models are represented by linear or predictive transducers.Predictive-state structure has been observed empirically, while linear transducers are proposed as a future direction for residual-stream analysis.
2 Types of transducers
Transducers are controlled stochastic systems whose hidden states mediate actions and outputs, inducing interfaces that describe observable sequence distributions. The paper compares standard, linear, and predictive forms through reductions and their minimal representations.
- Standard transducers: A transducer consists of states, actions, outputs, a Markov kernel, and an initial state distribution.The kernel specifies conditional distributions over next states and outputs.
- Standard transducers: Transducers represent world models by assigning output-sequence distributions to action sequences while using hidden states to track prior events.The interface records these conditional probabilities for finite traces and action sequences.
- Homomorphisms and reductions: A homomorphism maps states, actions, and outputs while preserving coarse-grained one-step dynamics and the initial distribution; reductions additionally keep action and output types fixed.Reductions use identity action and output maps and a surjective state map.
- Minimality: Different implementations of one interface form a reduction poset, but standard transducers may lack a unique minimum.Linear and predictive transducers instead have unique minima for every interface.
- Linear transducers: Linear transducers embed states in vector spaces and generate formal series through linear transition maps, with size measured by the dimension of the representing space.Every standard transducer can be converted into a linear one whose dimension equals its number of states.
- Linear transducers: The Hankel construction yields a canonical minimal linear transducer to which every linear implementation of the same interface reduces.Minimal linear transducers are unique up to invertible linear coordinate changes.
- Predictive transducers: The predictive construction yields a minimal predictive transducer, and every predictive implementation reduces to it.This minimum is identified with the epsilon-machine representation of the dynamics.
3 Approximate homomorphisms and the space of interfaces
The paper introduces approximate homomorphisms and interface metrics to compare locally similar transducers and their induced dynamics. Supremum interface distance can remain maximal despite arbitrarily small local errors, whereas discounted metrics support composability and continuity under suitable structural restrictions.
- Approximate homomorphisms: Approximate homomorphisms retain the state, action, and output maps while allowing the coarse-grained one-step dynamics to differ by ε.They generalize exact homomorphisms, which require the pushed-forward one-step mechanism to agree exactly.
- Approximate homomorphisms: Approximate reductions require mapped states to have locally similar one-step output and transition dynamics.For approximate reductions, the action and output maps are identities and the state map is surjective.
- Metrics on interfaces: Interface metrics aggregate distributional distances over all finite action sequences, including supremum and discounted variants.Interface distances also induce pseudometrics on transducers because distinct transducers can implement the same interface.
- Metrics on interfaces: For every ε > 0, the Figure 5 transducers can have interface supremum distance 1 despite an ε-reduction.Thus, small one-step errors need not control long-horizon discrepancies, which can become arbitrarily large in total variation.
- Stability properties: Discounted interface metrics are preserved under approximate homomorphisms, and approximate homomorphisms compose with additive error.The composability result relies on total variation for comparing one-step dynamics.
- Linear transducers: For finite standard transducers, an ε-reduction induces a 2ε-linear reduction between their ℓ1-norm linear implementations.Linear transducers must be contractive to prevent one-step errors from being amplified arbitrarily over subsequent steps; canonical Hankel representations satisfy this condition.
- Linear transducers: The linear stability bound is weaker because approximate linear reduction bounds each error independently, introducing a factor of |O|.Changing the definition could remove this factor but would exclude canonical Hankel representations from the contractive class used later.
4 Approximate reductions between implementations of similar interfaces
The section asks when transducers implementing nearby interfaces share approximate structural representations. It finds failure for standard transducers but stability for finite-rank linear and suitably separated predictive transducers.
- Setup: The section defines ε-approximate transducers and seeks δ-minima whose reductions ideally satisfy δ = O(ε).The goal is to identify interface classes whose nearby implementations share common structure.
- Standard transducers: Standard transducers can lack convergent structure: some interfaces have no δ-minima, even among approximate implementations.The obstruction persists for every ε ≥ 0 under the stated reduction conditions.
- Linear transducers: Finite-rank interfaces admit canonical linear representations for nearby interfaces, with approximation error scaling linearly in ε.The construction uses Hankel matrices and atomic norms; the canonical realization is a 2ΓIε-minimum for the restricted lattice of minimal linear implementations.
- Linear transducers: The linear result requires sufficiently small perturbations to ensure surjective reductions and is stated for minimal implementations because unrestricted atomic-norm reductions lack a uniform bound.Alternative norms may avoid the latter issue but are described as unnatural.
- Predictive transducers: Predictive transducers admit approximate reductions to the minimal implementation when residual distance is sufficiently small relative to the interface separation ΔI.The result applies under the condition ΔI > 0 and yields δ-minima for predictive transducers.
- Predictive transducers: The predictive stability result is restricted to settings where residual distance is appropriate and can overlook highly unlikely histories with sharply different conditioned interfaces.The discounted metric shares this limitation; a distance valuing every conditioning independently can avoid it.
5 Conclusion
The paper develops approximate homomorphisms to study whether transducer implementations of nearby dynamics share convergent structure. Convergence fails for standard transducers but holds under stated conditions for linear and predictive transducers, while the abstractions remain limited.
- Implications: The theory supports investigating structural convergence in neural representations, especially when latent dynamics are modeled by transducers.The authors connect these results to empirical observations about internal representations and residual streams.
- Contributions: The paper introduces approximate homomorphisms for standard and linear transducers, showing they preserve discounted dynamics and compose.The linear notion directly extends the standard one.
- Convergence results: Standard transducers can fail to exhibit convergent structure even when implementing simple dynamics.The paper identifies interfaces for which implementations remain structurally far apart.
- Convergence results: Finite-rank interfaces admit convergence among minimal linear implementations of ε-close dynamics.The paper presents this as a positive result for linear transducers.
- Convergence results: A suitable metric yields shared structural properties among predictive transducers implementing sufficiently close dynamics.This is the predictive-transducer convergence theorem.
- Limitations and future work: The framework remains incomplete because approximate convergence has multiple possible formalizations and linear transducers require a norm on their vector spaces.The authors also identify broader abstraction limitations and propose nonlinear and globally tolerant extensions.
A.1 Comparison of notions of exact homomorphisms
The appendix compares the paper’s exact homomorphism definition with the formulation from [44]. They coincide when the output map is injective, while non-injective maps can make the formulations differ.
- Original definition: The original homomorphism uses state, input, and output maps satisfying specified kernel conditions.The maps are written as ⟨ϕ, f, g⟩ between the corresponding transducer components.
- Injective output maps: The two formulations coincide when the output map g is injective.The appendix establishes this equivalence as Proposition 8.
- Proof of equivalence: The equivalence proof derives the marginal condition by summing the joint-kernel condition over target states.Injectivity then identifies each relevant source output with a unique target output.
- Proof of equivalence: The kernel condition remains valid both when the relevant source-output probability is positive and when it is zero.Nonnegativity forces both sides to vanish in the zero-probability case.
- Non-injective output maps: With non-injective g, the original definition permits coarse-graining only for outputs sharing statewise output laws, whereas the paper’s formulation averages over fibers.The distinction is considered irrelevant when focusing on reductions.
A.2 Total variation
This appendix collects total-variation tools used in the perturbation analysis. Push-forwards and Markov-kernel operations compose or contract total variation, enabling the stated perturbation bound.
- Definitions: Total variation is defined for probability distributions on a measurable space of finite or countable support.The appendix introduces the notation before stating the auxiliary properties.
- Definitions: Push-forward distributions transport probability measures through maps, while Markov kernels produce distributions by averaging conditional transitions.The appendix establishes notation for both operations.
- Basic properties: Push-forwards compose according to (r ◦ h)_∗µ = r_∗(h_∗µ).This identity follows directly from the definition of push-forward.
- Basic properties: Push-forward and kernel operations contract total variation, including under marginalization.These contractions are among the properties collected in Lemma 4.
- Perturbation bound: The perturbation bound follows by combining kernel contraction, the triangle inequality, and convexity.The proof bounds the two resulting terms separately before concluding.
A.3 Deferred proofs
The deferred proofs establish composability and stability bounds for approximate reductions, construct predictive minimal implementations, and support the standard-transducer counterexample. They also verify that the interface example admits both a minimal three-state implementation and a larger implementation.
- Predictive transducers: The canonical predictive implementation is predictive because its state after a history determines the residual future law.Every predictive transducer has an exact reduction to this implementation, and composing it with the stability theorem yields the desired result.
- Approximate reductions: The proof compares output-prefix and coarse-grained-state laws inductively, increasing the error by at most ε at each step.The final interface-distribution bound follows because marginalization contracts total variation.
- Composition: Approximate homomorphisms compose, with the resulting error bounded by ε1 + ε2.If both component maps are reductions, their composition remains a reduction.
- Linear transducers: The linearization maps states through a surjective linear map and yields a 2ε-linear reduction.The factor 2 arises from the normalization in total variation.
- Standard transducers: The standard-transducer interface has a three-state implementation, and no implementation with fewer than three states exists.The interface emits $ first, then repeats one fair-coin bit; its behavior is independent of actions.
- Standard transducers: A larger transducer also implements the same interface by uniformly choosing a pair after $, reading one coordinate, and then repeating the resulting bit.This construction verifies equivalence at the interface level despite the different internal state structure.