Research map / Architectures and efficiency / RNNs and normalization
Formation of Representations in Neural Networks
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes the Canonical Representation Hypothesis (six alignment relations among representations, gradients and weights in fully connected layers), proves it under noise-regularization balance, and derives the Polynomial Alignment Hypothesis for broken-CRH phases, with experiments on ResNet, transformers and MLPs.
Problem
How complex, structured and transferable neural representations form during training, and what universal relations govern hidden-layer representations, weights and neuron gradients.
Approach
Within a fully connected layer, representation second moments, neuron-gradient second moments and weight Gram matrices become mutually aligned after training because SGD gradient noise balances weight decay; when this alignment breaks, the matrices become reciprocally polynomially related (PAH), giving distinct phases.
Key results
- In ResNet-18 on CIFAR-10, alignment alpha > 0.7 is reached early in training and keeps improving in later stages.
- In a six-layer transformer, eigenvalue power-law exponents observed were 3, 2 and 1 for successive layers, within the predicted range [1/3, 3].
- Theory shows NC1-NC4 is equivalent to the CRH for quasi-interpolating classifiers under a loss-covariance-proportional-to-identity assumption.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| increase weight decay | fully connected networks (fc1) · fc1 synthetic dataset | alignment (α) | Helped |
| decrease batch size and increase weight decay | ResNet-18 self-supervised training · CIFAR-10/100 (res2) | alignment (α) | Helped |
| increase batch size | fully connected networks (fc1) · fc1 synthetic dataset, earlier layers | alignment (α) | Hurt |
| increase batch size | fully connected networks (fc1) · fc1 synthetic dataset, later layers | alignment (α) | Helped |
| increase network width with Adam | fully connected networks (fc1) · fc1 synthetic dataset, early versus later layers | alignment (α) | Mixed |
| normalize activations and gradients and subtract their means | alignment measurement · experiments across settings | alignment (α) | Helped |
| compare representation-gradient alignment against random-projection alignment | six-layer transformer · OWT | alignment (α): 0.14 | Helped |
| use a later fully connected layer rather than a convolutional layer | ResNet-18 self-supervised training · CIFAR-10/100 (res2) | representation-gradient alignment | Helped |
| move to deeper layers | fully connected networks (fc1) · fc1 synthetic dataset | alignment (α) | Helped |
Surprising findings
- Almost all observed positive power-law exponents fall within the predicted range [1/3, 3].
- Representation-gradient alignment holds better in later layers than earlier layers, and the alignment is stronger at larger weight decay.
Limitations
- Results and theory apply only to fully connected layers; extension to other layer types is left open.
- Focuses on the final stage of representation formation rather than full dynamics.
- Alignment is weaker in convolutional layers in self-supervised settings and CRH breaking is not fully characterized.
Experimental setup
Datasets: CIFAR-10 CIFAR-100 OpenWebText synthetic data from two-layer teacher network
Benchmarks: CIFAR-10 CIFAR-100 OpenWebText
Metrics: Pearson-correlation alignment alpha between covariance matri power-law exponent between eigenvalues
Baselines: random projection baseline initial-vs-final feature alignment baseline cross-head alignment baseline
Builds on: neural collapse neural feature ansatz neural tangent kernel feature learning
Closest papers by meaning
- Not all solutions are created equal: An analytical dissociation of functional and representational similarity in deep linear neural networks (ICML 2025 · similarity 0.88)
- When Representations Align: Universality in Representation Learning Dynamics (ICML 2024 · similarity 0.88)
- Parameter symmetries determine representational geometry in overparameterized nonlinear networks (NeurIPS 2026 · similarity 0.88)
- Bridging Critical Gaps in Convergent Learning: How Representational Alignment Evolves Across Layers, Training, and Distribution Shifts (NeurIPS 2025 · similarity 0.88)
- How are linear representations learned? Exact solutions to the dynamics of abstraction (NeurIPS 2026 · similarity 0.87)
- From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks (ICLR 2025 · similarity 0.87)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.