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 Ψ . C Parameter-space compression Λ . 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 Λ , from the minimum for Ψ ).
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 Lt drive a fixed random reservoir with states Xt ; a trainable linear bottleneck maps Xt to Zt , whose top three principal components form the learned representation Vt ; 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 Λ 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, Λ=1 . B Λ with the Gaussian Barber–Agakov bound used in the main text and with a multinomial-logistic bound. C The numerator of Λ , I(θ;F(V)) with F(V)=I(V;L) , and the same bound with shuffled θ labels. Both choices shift the level of Λ but not its rise, and I(θ;F(V)) stays above its null, so the approach of Λ 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 ( σy=0.1 ). Λ 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.
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.
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
Representations continually change as a network learns new tasks. We ask whether early representational changes naturally form a geometric structure that continues to shape later learning. We identify a low-dimensional subspace of early representation drift, which we call a scaffold, and test whether it is reused across subsequent tasks. Across four pretrained visual encoders and two datasets, later representational changes consistently favor this early-defined subspace over matched random alternatives. This reuse is history-dependent: when networks experience different early tasks but identical later training inputs, each network preferentially reuses the scaffold induced by its own learning history. The same preference appears in individual optimizer updates, even though the network's dominant local response directions shift away from the original scaffold. Finally, constraining motion within the scaffold slows new-task acquisition more than matched random constraints, while effects on old-task retention are less consistent. In summary, these results suggest that early experience leaves a persistent geometric imprint on how neural networks adapt to future tasks.Code is available at https://github.com/YuantaoDeng/latent-scaffold.
Information-theoretic and geometric compression have been widely studied as possible explanations for generalization. Yet their temporal dynamics in recurrent networks remain largely unexplored. Using discretized LSTMs, whose finite embedding space enables direct estimates of information quantities, we observe pronounced compression across the final timesteps of many datasets. We explain this effect through three competing mechanisms: (i) class separability permits stronger compression at the final timestep; (ii) shared recurrent dynamics propagate this pressure across timesteps with similar activation states, making its extent depend on temporal stationarity; and (iii) limited representational capacity opposes prolonged compression because it restricts information available for subsequent computation. We systematically test these mechanisms through controlled synthetic interventions and real-world sequence tasks. Our results identify class separability, temporal stationarity, and network capacity as key determinants of temporal compression, and suggest that compression acts as a regularizer-potentially beneficial when excess capacity is available, but harmful to generalization when it consumes memory needed for the task.
Nikolaus Kopp, Franz Pernkopf, Bernhard Geiger
Graz University of Technology, Graz, Austria · Graz University of Technology & Know Center Research GmbH, Graz, Austria