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Coarse-Graining Hidden Representations: Unsupervised Neuron Selection via Mapping Entropy
Margherita Mele, Andrea Castagna, Roberto Menichetti, Raffaello Potestio, Alessandro Ingrosso
TL;DR
Overparameterized networks contain more hidden units than their target computation may require, motivating the search for essential neurons using only the representation itself. The paper frames selection as hidden-layer coarse-graining, minimises mapping entropy using hidden-activation statistics, and reports structurally meaningful subsets that outperform equal-size random subsets under compression.
Problem
Overparameterized networks raise whether essential, task-relevant neurons can be identified from hidden representations without labels or gradients.
Method
The paper retains subsets of hidden neurons as coarse-grained representations and selects mappings that minimise mapping entropy computed from hidden-activation statistics.
Results
ME-selected subsets align with teacher-defined informative directions, select coherent functional classes in the nonlinear Gaussian-process task, and outperform equal-size random subsets across evaluated tasks.
Takeaways & Limitations
Mapping entropy provides an unsupervised proxy for pruning that preserves configurational distinguishability and is most advantageous under strong compression.
Takeaways & Limitations
The method does not identify the absolute optimal reduced network because it is task-blind and relies only on hidden activation statistics.
Abstract
from arXiv · showhide
Overparameterized neural networks carry far more hidden units than a task nominally requires, raising the question of which neurons are essential and whether that distinction is legible in the representation itself, without labels or gradients. We cast neuron selection as the problem of coarse-graining the hidden layer by retaining a subset of its neurons, and score each putative selection by the mapping entropy (ME). This quantity measures the loss of discriminatory power inherent in discarding part of the network neurons, and the selection that minimises the ME is taken as particularly informative. This criterion is fully unsupervised, in that it depends only on hidden-activation statistics. In teacher-student networks, ME optimisation recovers the minimal teacher-consistent representation and retains extra units in proportion to the hidden layer's residual variability; in a non-linear Gaussian process task, it selects coherent functional-class mappings whose preferred class shifts across training. On this task and on translation-augmented MNIST, ME-selected subnetworks outperform random subsets of equal size, most clearly under strong compression - linking configurational distinguishability to predictive performance.
I. INTRODUCTION
The paper asks whether task-relevant neurons can be identified from hidden representations without labels or gradients. It frames selection as unsupervised coarse-graining and evaluates mapping entropy as a criterion for retaining informative subsets.
- Motivation: Overparameterization raises whether task-relevant computation is concentrated in specific neurons or distributed across redundant hidden units.This question matters for both model compression and interpretability.
- Approach: The paper treats neuron selection as coarse-graining the hidden configuration space by retaining only a subset of hidden neurons.The reduced representation is obtained by mapping full hidden configurations onto lower-dimensional ones.
- Approach: Mapping entropy measures the loss of distinguishability induced by a coarse-graining and is minimised over candidate neuron subsets.The method is applied to hidden activation patterns rather than supervised loss changes.
- Approach: The criterion is fully unsupervised and depends only on hidden-activation statistics.The introduction presents this as a selection method for relevant subsets of hidden units.
- Evaluation: The study begins with controlled teacher-student settings and additionally uses a nonlinear Gaussian-process scenario to examine selection in richer representations.The teacher-student setup permits explicit monitoring of alignment between hidden units and teacher features.
II. METHODS
The method represents hidden activations as empirical binary configurations, then evaluates how much information is lost when a mapping retains only a subset of neurons. The preferred subset minimises mapping entropy at a fixed reduced size.
- Representation: A hidden layer of width K is converted into binary configurations by taking the sign of each postactivation.The binary configurations form the microscopic states used in the analysis.
- Coarse-graining: A decimation mapping retains ncg neurons, causing distinct microscopic configurations to become identical when they share the same reduced state.The reduced configuration is obtained by applying the mapping to each hidden configuration.
- Representation: Binarization discards activation magnitudes but avoids arbitrary binning and makes detected structure more robust.This is presented as a conservative modelling choice.
- Entropy: Mapping entropy is computed from the original empirical distribution and a back-mapped distribution formed by uniformly redistributing reduced-state probabilities.The loss is measured as a divergence between the original and reconstructed distributions.
- Optimisation: At fixed ncg, the selected subset is the mapping that minimises ME over all mappings retaining exactly ncg neurons.Smaller Smap indicates better preservation of statistical information.
- Optimisation: Small mapping spaces are searched exhaustively, whereas larger spaces use stochastic simulated annealing.Complete enumeration becomes computationally prohibitive for larger hidden layers.
A. Teacher-Student
The teacher-student experiments provide controlled hidden representations with more student units than teacher units. ME-selected subsets are analysed by their teacher-class composition and alignment with the teacher structure.
- Experimental setup: The controlled setup uses a teacher with M hidden units and an over-realised student with K > M hidden units.Both networks are soft committee machines with fixed second-layer weights.
- Experimental setup: Two teacher-student instances are studied: analytically replicated student units and trained students learned from teacher-generated examples.The replicated construction gives an explicit reference structure for interpreting selected subsets.
- Controlled representation: Student units are generated from teacher weights using a mismatch parameter η that controls their alignment with teacher directions.η = 0 gives exact replication, while η = 1 gives an orthogonal representation in the complementary subspace.
- Trained representation: For trained students, each neuron is assigned to the teacher class with which it has the largest final overlap, defining an effective mismatch parameter.Only first-layer weights are updated during online learning.
- Selection analysis: ME-selected mappings are characterised by the number of retained neurons associated with each teacher class and by a balance index Δ.Δ = 0 denotes a single-class selection, whereas Δ = 1 denotes perfect balance across M classes.
B. Non-Linear Gaussian Process
The NLGP benchmark uses a one-hidden-layer classifier for binary patterns distinguished by correlation length, and tracks ME-selected neuron subsets throughout training. Neurons are also classified by the structure of their incoming weights and associated biases.
- Task and network: The task generates two classes of N-dimensional patterns from Gaussian fields with distinct correlation lengths, ξ+ and ξ−.The fixed setup uses N = 50, ξ+ = 7.5, ξ− = 2, γ = 15, and α = 300.
- Task and network: The classifier has N input units, K = 30 hidden neurons, and one linear output unit.Only the first-layer weights and biases are trained; the output weights are fixed.
- Neuron organisation: Hidden neurons are partitioned into localised and oscillatory groups according to the structure of their incoming weight vectors.The inverse participation ratio distinguishes concentration of weights, while biases provide a complementary indicator.
- ME analysis: The ME minimisation is performed independently at each training checkpoint and retained-neuron count using multiple stochastic runs.Hidden representations are computed from all training patterns before selection, and selected mappings can be characterised by their localised-neuron fraction.
C. Wang-Landau Sampling of the Mapping Space
Wang–Landau sampling complements ME minimisation by characterising the global organisation of the mapping space through its density of states. The analysis constrains retained subsets by size and composition, then reconstructs the joint density using exact combinatorial normalisation.
- Purpose: ME minimisation finds low-information-loss mappings, whereas Wang–Landau sampling estimates the density of states across the mapping space.The density of states counts configurations associated with a given effective energy.
- Constrained mapping space: At fixed retained size ncg and localised-neuron count nloc, the mapping space contains subsets with exactly that composition.The equivalent composition variable is the fraction floc = nloc/ncg.
- Sampling design: The Wang–Landau analysis fixes ncg = 15 because the 30-neuron hidden layer contains 15 localised and 15 oscillatory units.Independent simulations are run for each admissible nloc, with moves preserving both ncg and nloc.
- Density reconstruction: The joint density of states is reconstructed by combining spectra sampled under each fixed-composition constraint.Known counts of admissible mappings provide the normalisation needed to assemble the restricted densities.
D. Performance Evaluation of Reduced Networks
The study evaluates whether ME-selected mappings produce effective reduced networks by comparing pruned networks against random subsets of equal size. The evaluation covers the NLGP classifier and translation-augmented MNIST.
- Evaluation protocol: For each retained-neuron count ncg, networks using ME-selected subsets are compared with networks using random subsets of the same size.The comparison tests whether minimum-entropy mappings identify effective reduced representations.
- NLGP evaluation: In the NLGP classifier, pruning is applied to the hidden layer after training and the output bias is re-optimised.The hidden neurons selected by the mapping are retained while evaluating the reduced network.
- MNIST evaluation: A second benchmark uses a one-hidden-layer network with K = 30 erf-activated neurons for binary classification of translated MNIST digits 1 and 7.Random translations make the classification problem translationally invariant and more challenging.
A. Structure of the Mappings Selected by Mapping Entropy Minimisation
The paper studies ME-selected subsets in two controlled settings where hidden-representation structure is interpretable: teacher-aligned units in an overparameterised regression model and functional neuron classes in NLGP classification.
- Controlled settings: The controlled experiments analyse ME-selected subsets where the organisation of the hidden representation is already understood.This design supports interpretation of selected neurons through known teacher alignments or functional classes.
- Teacher–student model: The teacher–student regression model interprets hidden units through their alignment with the teacher’s units.
- NLGP model: The NLGP classification problem provides a complementary setting in which neurons organise into distinct functional classes.
1. Teacher-Student system
In the teacher-student setting, mapping-entropy minimisation recovers a minimal teacher-consistent subset when replicas are exact, while retaining more neurons as hidden-representation variability increases. Trained networks follow the same relationship through their effective mismatch.
- Exact replicated limit: At η = 0, ME minimisation selects balanced subsets of ncg = 2 for M2-K10 and ncg = 5 for M5-K25, matching the smallest teacher-covering representations.The selected subset contains one representative degree of freedom for each teacher mode despite ME using only hidden-configuration statistics.
- Finite mismatch: As mismatch increases, the smallest balanced subset grows: M2-K10 requires ncg = 4 up to η ≈0.2, while M5-K25 first becomes fully balanced at ncg = 15.For M5-K25, ncg ≈10 remains only partially balanced with ∆≃0.875.
- Finite mismatch: ME responds to statistical distinguishability rather than teacher functional equivalence, so additional orthogonal variability shifts the optimal retained dimensionality upward.The increase in optimal ncg measures residual variability relative to the exact replicated limit.
- Trained networks: Training drives student-teacher overlaps toward one, suppresses orthogonal components, reduces generalisation error by several orders of magnitude, and lowers ηeff from approximately 1 to ηfinal ≃0.04.The trained student approaches but does not exactly reach the replicated limit.
- Trained networks: Trained networks lie on the controlled ME branches at their corresponding ηeff, with smallest balanced mappings ncg = 4 for M2-K10 and ncg = 15 for M5-K25.Thus the controlled and trained settings support the same interpretation of retained dimensionality as residual hidden-layer variability.
2. NLGP System
In the NLGP task, hidden neurons form two individually informative functional classes, and ME selects coherent rather than arbitrary mixtures. The preferred class changes with training stage and retained subset size as the ME landscape reorganises.
- Functional classes: Localised and oscillatory neuron populations each preserve separation between the two input classes when retained alone, representing alternative rather than necessarily complementary modes.The classes differ in weight structure: localised neurons concentrate on restricted coordinates, whereas oscillatory neurons have extended alternating-sign profiles.
- Training-dependent selection: At epoch 100, ME minimisation repeatedly selects the same band of localised neurons with pin ≈1, while oscillatory neurons are almost never included.Early optimal mappings are almost invariably localised across subset sizes.
- Training-dependent selection: Around epoch 300, the localised fraction decreases for intermediate and large ncg, signalling a competing oscillatory solution.The preference becomes strongly dependent on the number of retained neurons.
- Training-dependent selection: At epoch 1000, localised selections dominate for small ncg, but the localised fraction falls below 1/2 around ncg ≃6−7 and becomes essentially zero for ncg ≳10.The crossover reflects near-exclusive selection of one class or the other rather than gradual mixing.
- ME comparison: At ncg = 15, purely localised mappings have Smap ≈0.26 versus Smap ≈0.31 for oscillatory mappings at epoch 100, while the values become comparable at epoch 200 near Smap ≃0.32.At later epochs, the ME ordering reverses in favour of oscillatory selections.
- ME landscape: The mapping-space landscape shifts from a low-Smap localised basin at epoch 100 to a purely oscillatory low-Smap basin by epoch 1000, with a bimodal intermediate stage.ME minima remain near class-dominated edges rather than substantial mixed mappings, indicating a genuine landscape restructuring.
- Conclusion: Overall, ME selects coherent functional-class mappings whose preferred class depends on training stage and retained dimensionality, rather than favouring sparse or mixed subsets.Early training favours localised neurons; oscillatory mappings become competitive and eventually dominate for sufficiently large ncg.
B. Performance of the Reduced Network
ME-selected neuron subsets preserve task-relevant information after pruning, outperforming random subsets most clearly under strong compression. This pattern appears in both the NLGP classifier and translation-augmented MNIST, while the advantage weakens as more neurons are retained.
- NLGP classifier: ME-selected subsets typically achieve higher accuracy than most random subsets across NLGP training epochs.The comparison uses equal-cardinality random subsets and a full-network reference, with output bias re-optimised after pruning.
- NLGP classifier: At ncg = 8 and ncg = 12, ME-selected subsets consistently outperform typical random choices across all examined epochs.The separation remains at ncg = 15 and ncg = 18 but becomes smaller as more neurons are retained.
- Translation-augmented MNIST: ME-selected subsets also rank in the upper part of the random distribution on translation-augmented MNIST for 3 ≲ ncg ≲ 12.Except at ncg = 2, where all reduced networks perform poorly, selected subsets are often close to the random upper tail.
- Cross-task pattern: For larger retained subsets, the distinction between ME-selected and random configurations progressively weakens as admissible subsets become less heterogeneous.The benefit of structured selection is therefore most evident when pruning is strong and subset choice is non-trivial.
- Scope: ME minimisation provides an effective unsupervised pruning proxy but does not identify the globally optimal reduced network.The method uses hidden activation statistics alone and is not compared here with supervised pruning criteria such as weight magnitude, saliency, or gradient scores.
IV. CONCLUSIONS
The conclusions present MEOW as an unsupervised strategy for identifying structurally meaningful and functionally effective neuron subsets. Across controlled and diverse settings, ME-selected reductions preserve predictive performance better than random subsets, while broader applications remain future work.
- ME minimization provides a purely unsupervised method for selecting informative neurons and tracking functional heterogeneity in hidden layers.The method relies on the statistical structure of hidden configurations rather than supervised signals.
- In teacher-student settings, selected neurons align strongly with the teacher’s informative directions, while NLGP selections form coherent mappings from one representational class.The preferred functional class in the NLGP task changes during training.
- ME-selected subnetworks consistently outperform random subsets of equal size, with markedly smaller degradation under equal compression.This links preserved configurational distinguishability to better-preserved predictive performance.
- Future directions: The present study prunes a single hidden layer in one shot, motivating iterative ME-based pruning with repeated re-estimation and optional fine-tuning.Such an extension could support stronger compression in deep architectures.
- Future directions: Applying ME-based coarse-graining to convolutional maps, residual streams, attention heads, or transformer MLP neurons requires more general coarse-graining beyond individual-neuron selection.These extensions could connect ME-based selection with mechanistic interpretability.
- Future directions: Future studies could assess whether ME-critical neurons are disproportionately involved in sensitivity to adversarial perturbations and distribution shift.The proposed direction would examine how ME-guided pruning affects the accuracy–robustness trade-off.