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Representation-Weight-Gradient Alignment (Canonical Representation Hypothesis)

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A trained network's latent representation matrix R, weight matrix W, and neuron-gradient matrix G become mutually aligned through a set of covariance/correlation relations, so the resulting representation is invariant to task-irrelevant input transformations; when this alignment is instead broken, R, W, G settle into reciprocal power-law relations rather than tight alignment, with the balance of gradient noise and regularization determining which regime holds.

Replicationcomputed from the corpus — never hand-assigned
1 paper3 architecture classes · one study3 domains · one study3 model families · one study
Filled = two or more values reported by papers that share no author — replication. Outlined = two or more values, but all from a single study — breadth, not replication. Grey = a single value. Derived from paper authorship and each model's architecture class, domain and family; it updates itself when a paper is added.

Definition

For a network layer with representation matrix RR (activations over a data batch), weight matrix WW, and neuron-gradient matrix GG (gradients of the loss with respect to each neuron’s pre-activation), the canonical regime is the configuration in which RR, WW, GG satisfy a set of six covariance/correlation-based alignment relations simultaneously: RR correlates with GG, WW aligns with both RR and GG, and the composite effect is that the learned representation becomes invariant to input transformations that do not affect the task. When this three-way alignment does not hold, RR, WW, GG instead exhibit reciprocal power-law relations (the Polynomial Alignment Hypothesis regime) rather than tight alignment.

Intuition

Several previously separate empirical observations about trained networks — Neural Collapse (last-layer class means and classifier weights converging to a shared symmetric structure) and the Neural Feature Ansatz (weight matrices aligning with the covariance of their own gradients) — can be read as special cases of one underlying phenomenon: training dynamics push a network’s own representations, weights, and gradients into mutual alignment, and this alignment (not the training objective alone) is what produces compact, invariant representations as a byproduct.

Properties

  • A measured configuration, not an assumption. The alignment relations are covariance/correlation quantities computed directly from a real trained network’s own RR, WW, GG, not imposed by construction.
  • A regime, not a universal law. The canonical (tightly-aligned) configuration requires a specific balance between gradient noise and regularization strength; when that balance is absent, the same three matrices instead settle into the weaker, reciprocal-power-law Polynomial Alignment configuration.
  • Unifying scope. The alignment configuration is framed as unifying Neural Collapse and the Neural Feature Ansatz as two consequences of the same underlying RR-WW-GG relationship, rather than as an independent, narrower claim.

Key evidence

Ziyin, Chuang, Galanti & Poggio (2024) propose six alignment relations between the covariances of a fully connected layer’s representations RR, weights WW and neuron gradients GG and measure them via Pearson-correlation alignment in from-scratch fully connected nets on synthetic teacher data (all six hold at large weight decay), ResNet-18 on CIFAR-10 (alignment above 0.7 early in training), a self-supervised ResNet-18 on CIFAR-10/100 (strong for the fully connected layers, weaker for the convolutional ones), and a six-layer transformer on OpenWebText (measured on attention-head outputs; only the representation-gradient alignment and eigenvalue power laws are reported); a minimal-assumption theory ties the aligned versus power-law (Polynomial Alignment) regimes to the balance of gradient noise and regularization, and the CRH is proved equivalent to neural collapse under an interpolation assumption. See ziyin-etal-2024-representations-weights-and-gradients-become-mutually-aligned-during-training.

Found in (1 observation · 3 families)

Custom research feedforward MLP

Formation of Representations in Neural Networks (2024)measured

Latent representations, weights, and neuron gradients become mutually aligned during training, unifying Neural Collapse and the Neural Feature Ansatz

Details

- Ziyin, Chuang, Galanti & Poggio propose the Canonical Representation Hypothesis (CRH), six alignment relations (representation-gradient, representation-weight, gradient-weight; forward and backward) between the covariances of a fully connected layer's latent representations (R), weights (W) and neuron gradients (G), and measure alignment with a Pearson-correlation metric in several from-scratch settings: fully connected nets on synthetic teacher data, ResNet-18 on CIFAR-10, ResNet-18 self-supervised on CIFAR-10/100, and a six-layer eight-head transformer on OpenWebText. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - All six relations are shown to hold strongly in six-layer fully connected nets at large weight decay (at least one forward and one backward relation at small weight decay); in ResNet-18 on CIFAR-10 alignment above 0.7 is reached early in training; in the self-supervised ResNet-18 the representation-gradient alignment holds well for the fully connected layers but less strongly for the convolutional layers; in the 100M-parameter six-layer transformer (measured on the output of each attention head) only the representation-gradient alignment and eigenvalue power-law relations are reported. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - When alignment breaks, eigenvalues of the representation and gradient covariances follow reciprocal power laws (the Polynomial Alignment Hypothesis; exponents observed in [1/3, 3], e.g. exponents 3, 2 and 1 across transformer layers); a minimal-assumption theory (Theorem 2) attributes the regime to the balance between gradient noise and regularization. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - The CRH is proved equivalent to neural collapse for a quasi-interpolating classifier with identity loss-gradient covariance (Theorem 3) and the gradient-weight alignment equals the neural feature ansatz when B is proportional to I; the authors present it as a generalization of neural collapse and an "equivariant NFA", and state it applies only to fully connected layers. [ziyin-etal-2024-formation-of-representations-in-neural-networks]

models: Ziyin et al. 2024 fully connected nets on synthetic two-layer-teacher data (fc1/fc2) · method:

Custom Research ResNet (constant-width residual variant, purpose-built for SSL studies)

Formation of Representations in Neural Networks (2024)measured

Latent representations, weights, and neuron gradients become mutually aligned during training, unifying Neural Collapse and the Neural Feature Ansatz

Details

- Ziyin, Chuang, Galanti & Poggio propose the Canonical Representation Hypothesis (CRH), six alignment relations (representation-gradient, representation-weight, gradient-weight; forward and backward) between the covariances of a fully connected layer's latent representations (R), weights (W) and neuron gradients (G), and measure alignment with a Pearson-correlation metric in several from-scratch settings: fully connected nets on synthetic teacher data, ResNet-18 on CIFAR-10, ResNet-18 self-supervised on CIFAR-10/100, and a six-layer eight-head transformer on OpenWebText. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - All six relations are shown to hold strongly in six-layer fully connected nets at large weight decay (at least one forward and one backward relation at small weight decay); in ResNet-18 on CIFAR-10 alignment above 0.7 is reached early in training; in the self-supervised ResNet-18 the representation-gradient alignment holds well for the fully connected layers but less strongly for the convolutional layers; in the 100M-parameter six-layer transformer (measured on the output of each attention head) only the representation-gradient alignment and eigenvalue power-law relations are reported. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - When alignment breaks, eigenvalues of the representation and gradient covariances follow reciprocal power laws (the Polynomial Alignment Hypothesis; exponents observed in [1/3, 3], e.g. exponents 3, 2 and 1 across transformer layers); a minimal-assumption theory (Theorem 2) attributes the regime to the balance between gradient noise and regularization. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - The CRH is proved equivalent to neural collapse for a quasi-interpolating classifier with identity loss-gradient covariance (Theorem 3) and the gradient-weight alignment equals the neural feature ansatz when B is proportional to I; the authors present it as a generalization of neural collapse and an "equivariant NFA", and state it applies only to fully connected layers. [ziyin-etal-2024-formation-of-representations-in-neural-networks]

models: Ziyin et al. 2024 ResNet-18 (CIFAR-10, from scratch), Ziyin et al. 2024 ResNet-18 self-supervised (CIFAR-10/100, from scratch) · method:

Custom Research Transformer (small, purpose-built for interpretability studies)

Formation of Representations in Neural Networks (2024)measured

Latent representations, weights, and neuron gradients become mutually aligned during training, unifying Neural Collapse and the Neural Feature Ansatz

Details

- Ziyin, Chuang, Galanti & Poggio propose the Canonical Representation Hypothesis (CRH), six alignment relations (representation-gradient, representation-weight, gradient-weight; forward and backward) between the covariances of a fully connected layer's latent representations (R), weights (W) and neuron gradients (G), and measure alignment with a Pearson-correlation metric in several from-scratch settings: fully connected nets on synthetic teacher data, ResNet-18 on CIFAR-10, ResNet-18 self-supervised on CIFAR-10/100, and a six-layer eight-head transformer on OpenWebText. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - All six relations are shown to hold strongly in six-layer fully connected nets at large weight decay (at least one forward and one backward relation at small weight decay); in ResNet-18 on CIFAR-10 alignment above 0.7 is reached early in training; in the self-supervised ResNet-18 the representation-gradient alignment holds well for the fully connected layers but less strongly for the convolutional layers; in the 100M-parameter six-layer transformer (measured on the output of each attention head) only the representation-gradient alignment and eigenvalue power-law relations are reported. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - When alignment breaks, eigenvalues of the representation and gradient covariances follow reciprocal power laws (the Polynomial Alignment Hypothesis; exponents observed in [1/3, 3], e.g. exponents 3, 2 and 1 across transformer layers); a minimal-assumption theory (Theorem 2) attributes the regime to the balance between gradient noise and regularization. [ziyin-etal-2024-formation-of-representations-in-neural-networks] - The CRH is proved equivalent to neural collapse for a quasi-interpolating classifier with identity loss-gradient covariance (Theorem 3) and the gradient-weight alignment equals the neural feature ansatz when B is proportional to I; the authors present it as a generalization of neural collapse and an "equivariant NFA", and state it applies only to fully connected layers. [ziyin-etal-2024-formation-of-representations-in-neural-networks]

models: Ziyin et al. 2024 6-Layer Transformer (OpenWebText, from scratch) · method: