Invariance before structure: two modes of emergent compression in representation learning

Hardik Rajpal; Dan Goodman

cs.LGNov 5, 2026

Invariance before structure: two modes of emergent compression in representation learning

Authors: Hardik Rajpal, Dan Goodman

Organizations: I-X Centre for AI in Science, Department of Electrical and Electronics Engineering, Imperial College London, London, UK, SW7 2AZ

Abstract

Learned representations show striking invariance across biological and artificial neural networks, with different substrates and variations driven by drift, heterogeneity and initialisation. To understand this invariance, we propose Emergent Adaptive Representation Learning (EARL), a framework that distinguishes two modes of emergence in learned representations: state-space compression, where a learned representation gains predictive power over its own future beyond what the network's units can predict individually, and parameter-space compression, where networks with different micro-level parameters converge to the same representation. We formalise both as Fisher sensitivities and introduce ensemble-level information-theoretic measures for each. In a recurrent reservoir with a trained bottleneck, parameter-space compression rises within the first few epochs of training and partly independently of the task, whereas state-space compression rises about 30 epochs later and only when the network begins to learn the temporal structure of the task. These results suggest that invariance to the substrate and task-specific emergent structure are acquired in distinct phases of learning.

Figures & tables

Figure 1: The two modes of compression. A State space: a stable macro configuration emerges as microscopic fluctuations are compressed away. B Parameter space: the same configuration emerges from different models of microscopic interactions.

Figure 2: Learning dynamics on noise-free input. A One-step prediction error. B State-space compression $\Psi$ . C Parameter-space compression $\Lambda$ . Solid: mean of 20 replicate ensembles (band: 10th–90th percentile) on training attractors, unseen rotations of the training attractors and held-out attractors; dashed/dotted: time-permuted and cross-sequence target controls (10 matched replicates each). Triangles: half-change epochs (from epoch 0 for $\Lambda$ , from the minimum for $\Psi$ ).

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Reservoir–bottleneck architecture, adapted from our previous work [ 31 ] . Random orthogonal embeddings of the attractor latent $L_{t}$ drive a fixed random reservoir with states $X_{t}$ ; a trainable linear bottleneck maps $X_{t}$ to $Z_{t}$ , whose top three principal components form the learned representation $V_{t}$ ; a ridge readout predicts the next input. The bottleneck is the only trained component.

Figure 4: Robustness of parameter-space compression (noise-free input, training sequences). A $\Lambda$ with the micro feature $F(X)$ built from $k$ random projections of the reservoir states ( $k=24$ in the main text); dashed line: shuffled-label null, $\Lambda=1$ . B $\Lambda$ with the Gaussian Barber–Agakov bound used in the main text and with a multinomial-logistic bound. C The numerator of $\Lambda$ , $I(\theta;F(V))$ with $F(V)=I(V;L)$ , and the same bound with shuffled $\theta$ labels. Both choices shift the level of $\Lambda$ but not its rise, and $I(\theta;F(V))$ stays above its null, so the approach of $\Lambda$ towards 1 is not an estimator floor.

Figure 5: Performance and alignment for noise-free input, with lines and bands as in Fig. 2 . A One-step prediction error (as Fig. 2 A). B Alignment of the learned with the true latent, $I(V;L)$ .

Figure 6: As Fig. 2 , for models trained and evaluated on noisy input ( $\sigma_{y}=0.1$ ). $\Lambda$ approaches 1 in trained runs and plateaus near 0.95 in controls; the ordering of the two modes and the differences from the controls are preserved.

Invariant Representation Learning

Invariant representation learning seeks features stable to irrelevant changes while preserving information needed for prediction and transfer.

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Emergent Generalization by Representation Learning in Artificial Neural Networks

Hardik Rajpal, Dan Goodman

I-X Centre for AI in Science, Imperial College London, W12 0BZ, UK · Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, UK

Neural Network Generalization Representation Learning Information Bottleneck Recurrent Neural Networks Neural Encoding

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Dimensionality reduction has proven powerful for identifying neural manifolds, which are low-dimensional structures underlying high-dimensional neural activity. These low-dimensional representations have improved the interpretability of population-level coding. Yet whether such low-dimensional representations are biologically relevant and confer functional advantages in learning systems, or merely reflect neuron-level activity, remains contested in neuroscience. We show that an explicit information bottleneck forcing a recurrent neural network to learn a low-dimensional representation is necessary for rotational and out-of-distribution generalisation in a time-series prediction task. Using information-theoretic measures of causal emergence, we characterise the dynamics of this representation across the memorisation-to-generalisation transition, finding a non-monotonic trajectory which shows an initial decrease, a minimum, and a subsequent rise to a maximum, even as prediction loss falls monotonically. This trajectory scales with task complexity, and the magnitude of emergent structure reliably predicts generalisation performance. Analysis of hippocampal CA1 and medial prefrontal activity in rats learning an alternating maze task reveals analogous non-monotonic emergence dynamics that track behavioural performance. Together, these findings indicate that the ability of neural networks to learn compact, distributed and emergent representations confers a functional advantage for generalisation, supporting a causal role for learned representations in cognition.

Abstract

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