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Understanding Representation of Deep Equilibrium Models from Neural Collapse Perspective
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Uses neural collapse and layer-peeled models to theoretically compare Deep Equilibrium Models (DEQ) with explicit networks under balanced and imbalanced class distributions, and validates on CIFAR-10/100 with ResNet18-based backbones.
Problem
Theoretical understanding of the learned representations of implicit (DEQ) networks is limited, and DEQ has not been analyzed under class-imbalanced settings through the neural collapse lens.
Approach
Applies the layer-peeled (UFM) model to DEQ by constraining the fixed-point output z* = f(H0; W_DEQ) and derives loss lower bounds for DEQ vs. explicit layers, showing NC1-NC3 in balanced settings and that DEQ features are closer to a simplex ETF and better aligned with classifier weights under mild imbalanced conditions.
Key results
- On CIFAR-10 with R=10, DEQ overall test accuracy was 73.84% vs 72.57% for explicit NN (balanced-ish comparison in the imbalanced table).
- On CIFAR-100 with R=100 (K_A=3), DEQ overall accuracy was 24.78% vs 23.43% for explicit NN, with minority accuracy 3.31% vs 1.99%.
- Theory claims DEQ's loss lower bound is -2C1 K/(K-1) sqrt(E_W E_H) versus -C1 K/(K-1) sqrt(E_W E_H) for explicit layers under balanced data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-10, R=10, overall | accuracy: 72.57$\pm$0.25 -> 73.84$\pm$0.72 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-10, R=50, overall | accuracy: 44.32$\pm$0.23 -> 46.08$\pm$1.06 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-10, R=100, overall | accuracy: 32.14$\pm$0.81 -> 34.18$\pm$1.28 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-100, R=10, overall | accuracy: 41.41$\pm$0.56 -> 43.72$\pm$0.60 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-100, R=50, overall | accuracy: 28.18$\pm$0.42 -> 30.46$\pm$1.27 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-100, R=100, overall | accuracy: 23.43$\pm$0.92 -> 24.78$\pm$1.93 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-10, R=10, minority | accuracy: 62.36$\pm$0.12 -> 64.06$\pm$0.66 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-10, R=50, minority | accuracy: 21.83$\pm$0.20 -> 24.42$\pm$1.32 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-10, R=100, minority | accuracy: 6.64$\pm$0.99 -> 8.83$\pm$1.08 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-100, R=10, minority | accuracy: 27.86$\pm$0.39 -> 30.67$\pm$0.53 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-100, R=50, minority | accuracy: 8.31$\pm$0.38 -> 11.96$\pm$1.66 | Helped |
| Use DEQ instead of Explicit NN | ResNet18 · CIFAR-100, R=100, minority | accuracy: 1.99$\pm$1.06 -> 3.31$\pm$2.45 | Helped |
Surprising findings
- Minority collapse persists for DEQ under strong imbalance: minority accuracy drops to 3.31% at R=100 on CIFAR-100 despite DEQ's advantage.
Limitations
- Theory restricted to linear DEQ and a simple layer-peeled (UFM) model with one or few layers
- Advantages of DEQ over explicit networks are marginal in experiments and depend on scaling coefficients E_H, E_W
- Training limited to 100 epochs due to DEQ instability, so terminal phase of training is not reached
Experimental setup
Datasets: CIFAR-10 CIFAR-100
Benchmarks: CIFAR-10 CIFAR-100
Metrics: test accuracy overall accuracy majority accuracy minority accuracy NC1 NC2 NC3
Baselines: Explicit NN (ResNet18)
Builds on: layer-peeled model deep equilibrium model neural collapse
Closest papers by meaning
- Separation and Bias of Deep Equilibrium Models on Expressivity and Learning Dynamics (NeurIPS 2024 · similarity 0.89)
- Wide Neural Networks Trained with Weight Decay Provably Exhibit Neural Collapse (ICLR 2025 · similarity 0.89)
- Neural Collapse under Gradient Flow on Shallow ReLU Networks for Orthogonally Separable Data (NeurIPS 2025 · similarity 0.89)
- Neural Collapse for Cross-entropy Class-Imbalanced Learning with Unconstrained ReLU Features Model (ICML 2024 · similarity 0.88)
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- Neural Collapse Beyond the Unconstrained Features Model: Landscape, Dynamics, and Generalization in the Mean-Field Regime (ICML 2025 · similarity 0.87)
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