Representational Stability in Learning Systems

• Representational stability is defined as the invariance of internal feature spaces during model adaptations and perturbations.
• It is measured using methods like linear probing, CCA, and alignment errors to track the fidelity of feature spaces in neural networks and dynamic systems.
• This concept underpins robust design in AI and mathematics, balancing model expressivity with resistance to representational collapse.

Representational stability refers to the preservation of internal feature spaces or population codes—often under reparameterization, finetuning, or long-term adaptation—such that the essential semantic, functional, or statistical properties of model outputs or behavioral patterns remain invariant. In machine learning, neuroscience, and mathematics, this concept characterizes the robustness of representations to perturbations in networks, tasks, model classes, or training signals. Across domains, representational stability is both a target of algorithmic design and a diagnostic for the integrity of learned or evolved information-processing systems.

1. Formal Definitions and Theoretical Foundations

In the context of neural networks and representation learning, representational stability is operationalized by measuring the invariance of internal feature distributions, similarity structures, or decision boundaries under interventions such as finetuning, retraining, parameter perturbation, or task redefinition. Conceptually, stability is often cast as the converse of representational collapse—the excessive distortion or loss of generalizable structure acquired during pretraining when adapting to specific downstream objectives.

A canonical definition emerges in the fine-tuning literature as follows: Let $f: X \to \mathbb{R}^p$ denote a pre-trained encoder and $g: \mathbb{R}^p \to \mathbb{R}^q$ a downstream task head; their composition $g\circ f$ maps inputs to outputs. Representational stability requires that the shift in the distribution $p(f(\cdot;\theta_f))$ under task adaptation $\theta_f \mapsto \theta_f + \Delta\theta_f$ remains bounded. The trust-region framework formalizes this via: $$\min_{\Delta\theta}\ L(\theta+\Delta\theta)\quad \text{subject to}\quad \mathrm{KL}(p(f(\cdot;\theta_f))\,\|\,p(f(\cdot;\theta_f+\Delta\theta_f))) = \epsilon,$$ for loss $L(\theta)$ and a small constraint parameter $\epsilon$ (Aghajanyan et al., 2020).

Measurement and diagnostic proxies include:

• Linear probing accuracy: Train a probe on frozen intermediate features. Drop in performance after further finetuning indicates collapse, while retention signals stability.
• Similarity and alignment metrics: Canonical Correlation Analysis (CCA), Centered Kernel Alignment (CKA), cosine similarity, and subspace overlap are used to gauge the geometric similarity of feature clouds across reparameterizations or model instances (Nikooroo et al., 5 Aug 2025).
• Decision-boundary robustness: In LLMs, representational stability is quantified by the resilience of truth-separating hyperplanes to relabeling or perturbation of semantic categories (Dies et al., 24 Nov 2025).

In mathematical contexts (e.g., algebraic topology and representation theory), representation stability characterizes convergence properties and multiplicity stabilization in sequences of representations indexed by size or rank (e.g., $S_n$, configuration spaces, diagram algebras) (Khomenko et al., 2016).

2. Representational Stability in Deep and Probabilistic Models

2.1 Neural Networks and LLMs

Empirical and computational approaches to stability emphasize controlling representation drift during task-specific adaptation. In LLMs and transformers, naively fine-tuning on small or narrow datasets induces collapse, where task-specialized layers overwrite the broad features learned through pretraining. To mitigate this, regularization-based and trust-region techniques are developed:

• Parametric Noise (R3F): Fine-tuning with symmetric KL divergence penalties between outputs of the same sample under small random perturbations ($z \sim \mathcal{N}(0,\sigma^2 I)$ or $z \sim \mathcal{U}(-\sigma, \sigma)$), thereby enforcing proximity of feature spaces pre- and post-update (Aghajanyan et al., 2020).
• Spectral normalization (R4F): Impose a 1-Lipschitz constraint on the task head to tightly bound representational change, further enhancing stability at the potential cost of slight accuracy reduction.
• Probing and chain/cyclic evaluation: Probing performance remains higher under R3F/R4F protocols versus standard or adversarial (SMART) approaches, demonstrating superior retention of generalizable internal states.

2.2 Dynamical Embedding Systems

Dynamic networks and time-varying graph embeddings present unique stability challenges due to inherent indeterminacies under distance-preserving transformations (rotation, translation, scaling). The alignment protocol formalizes explicit metrics:

• Alignment errors ($\xi_{\mathrm{tr}}$, $\xi_{\mathrm{rot}}$, $\xi_{\mathrm{sc}}$): Quantify spurious changes due to global transformations.
• Stability error ($\xi_{\mathrm{st}}$): Residual normed change after explicit alignment, interpreted as true representational drift (Gürsoy et al., 2021).

Empirical findings show that mitigating misalignment directly improves downstream prediction accuracy, while $\xi_{\mathrm{st}}$ robustly tracks genuine system evolution.

3. Geometry, Singularities, and Desingularization in Embedding Spaces

Recent advances have identified geometric pathologies in LLM token embedding spaces, notably the failure of the manifold hypothesis around polysemous or rare tokens. Instabilities arise from singularities—points where multiple semantic directions (branches) coalesce, yielding non-manifold structure and brittleness in model predictions.

Algebraic-geometric resolution via blow-up: The TokenBlowUp framework formalizes representational desingularization by applying the scheme-theoretic blow-up to each singular embedding point, replacing it with a projective bundle of contextual meanings—the exceptional divisor $E_s \cong \mathbb{P}^{n-1}_{\mathbb{R}}$ (Zhao, 26 Jul 2025). This procedure regularizes local dimension, ensuring that context disambiguation selects a resolved direction and that the new, augmented embedding space is locally smooth and stable. The approach suggests architectural modifications where static lookup tables are replaced, for singular tokens, by context-aware dynamic heads selecting the appropriate semantic fiber.

4. Stability in Reinforcement and Relational Learning

In reinforcement learning (RL) with function approximation, stability under off-policy training is nontrivial. Canonically, a linear value function approximation $Q_0(s,a) = \Phi(s,a)^{\top}\theta$ is subject to divergence unless the feature matrix $\Phi$ is "stable":

• Matrix criterion: $\Phi$ is stable if $A = \Phi^T D (I-\gamma P_\pi)\Phi$ is positive definite; equivalently, the projected Bellman operator $\Pi T_\pi$ is a contraction (Ghosh et al., 2020).
• Schur basis and Krylov subspaces: Constructing $\Phi$ from invariant subspaces of $P_\pi$ or from orthonormal Krylov subspaces (for fixed rewards) guarantees contractive updates and thus stability.

In molecular relational learning, stability under out-of-distribution shift (e.g., functional group or scaffold changes) is attained by chemically-informed dynamic substructure alignment and by imposing subgraph information bottlenecks to suppress confounding factors, maximizing core-structure alignment and minimizing spurious coupling. ReAlignFit exemplifies this paradigm and achieves superior robustness in MRL under distribution shifts (Zhang et al., 7 Feb 2025).

5. Representational Stability in Algebraic and Topological Structures

The classical notion of representation stability, originating in the work of Church, Farb, and collaborators (Khomenko et al., 2016), characterizes the eventual stabilization (injectivity, surjectivity, multiplicity constancy) of sequences of group representations indexed by natural parameters—often the symmetric group $S_n$, or more generally, in diagram algebras, braid groups, and mapping class groups.

• FI-modules and polynomial functors: The category-theoretic framework asserts that finitely generated FI-modules correspond to uniformly stable representation sequences.
• Stability theorems for configuration spaces: For configuration spaces $\mathrm{Conf}_n(X)$ of $n$ ordered points in a space $X$, and associated $S_n$-actions, the cohomology $H^k(\mathrm{Conf}_n(X))$ stabilizes separately for each $k$, with explicit stable ranges determined by the dimension and topology of $X$ (Hersh et al., 2015, Lütgehetmann, 2017, Fedah et al., 8 May 2025).
• Combinatorial and diagrammatic categories: Diagram algebras (Temperley–Lieb, Brauer, partition algebras) exhibit stabilization of simple-module multiplicities in finitely presented modules over corresponding stability categories (Patzt, 2020).

6. Trade-Offs, Practical Implications, and Open Directions

Stability is often in tension with representational capacity. In GNNs, greater degrees of freedom enable richer function classes but amplify instability to topological perturbations due to eigenvector misalignment; leaner, shift-invariant architectures maximize stability constants and robustness at the expense of expressivity (Gao et al., 2023).

Cross-network and cross-architecture transferability is enhanced by structural priors (e.g., linear shaping operators). When shared, these priors guarantee low alignment error, increasing the stability and interoperability of internal representations—critical for distillation and modular system design (Nikooroo et al., 5 Aug 2025). In VLMs, principal eigenvector dominance in attention layers underpins representational stability, enabling robust and interpretable manipulation of high-level concepts (Tian et al., 25 Mar 2025).

Empirical and theoretical work aligns in establishing that representational stability is a multi-scale, multi-domain property, vital for robustness, transfer, and interpretability in contemporary AI and mathematical structures. Future research will likely focus on new dynamical models of drift and stabilization (as in cortical circuits (Morales et al., 18 Dec 2024)), deepening algebraic formalization, and extending stabilization guarantees to broader, non-semisimple and non-manifold regimes.