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THEORY-288: In small CIFAR and MNIST autoencoders whose only basis-privileging primitive is the activation function, elementwise tanh and Leaky-ReLU concentrate latent representations into narrow beams arranged by the activation's discrete symmetry group, even in layers no activation precedes, while their orthogonally equivariant radial counterparts leave them spread, so axis-aligned discreteness there comes from the primitive, not the task

4 min read

Properties5
StatusProposed
Version2
Tagsrepresentation-learning, mathematics
SourceLIT-976 — Emergence of Quantised Representations Isolated to Anisotropic Functions
Bears onQUESTION-058 — When a trained network's representation carries a group's structure, how much of it comes from the task, the architecture, the training set's relations and the symmetry of the update, and how much can be read off the network without supplying the group?

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Source

George Bird (2025; v4 of 2026-01-26), LIT-976, read in NOTE-759: §§1–5 and Appendices A–E.

What was actually shown

The ablation. These are autoencoders with:

  • an affine encoder from the image to width n ∈ {12, 18, 24, 32};
  • L ≤ 3 hidden blocks, each an activation then an affine n → n map;
  • an activation and an affine decoder.

Weights are Xavier-normal, biases zero, training is momentum SGD on MSE. All of these commute with rotations of the hidden spaces. Only the activation varies:

  • elementwise tanh, Σᵢ tanh(x·êᵢ)êᵢ, equivariant to the signed permutations B_n, against isotropic tanh, tanh(‖x‖)x̂, equivariant to O(n);
  • elementwise Leaky-ReLU, equivariant to S_n, against a radial Leaky-ReLU;
  • two B_n Leaky-ReLUs.

The observation. Latent representations were projected onto all planes spanned by pairs of standard basis vectors and kept within an angular threshold. The plots show:

  • Elementwise tanh: narrow beams along and against the axes, growing through training (Figs. 1–2, 8–17). The count above threshold rises from about 2,000 to about 200,000.
  • Elementwise Leaky-ReLU: two antipodal beams on one diagonal at layer 0 (Figs. 3, 18–21). The ±1 orthant is the only one S_n treats specially.
  • B_n Leaky-ReLU: beams along the axes, like tanh and unlike S_n Leaky-ReLU (Figs. 22–27).
  • Radial versions: spread, continuous distributions. Deeper isotropic networks form some clusters, not aligned with the basis.
  • Layer 0, which no activation precedes, shows the effect. So it reaches backward through training, not only through the forward map’s geometry.
  • MNIST shows the same contrast (Figs. 28–30).

What could have come out the other way: elementwise networks could have given spread representations, with the basis showing only in parameters. Or radial networks could have formed the same beams through the data’s own pixel axes, which the encoder mixes at random.

A symmetry argument behind it. This is the record’s, from the setup, and since 2026-10-19 it is CLAIM-382, demonstrated and Active, with its proof. With a radial activation, every part of the network and its training is equivariant to rotations of each hidden space, and the initialisation is invariant in distribution. So the law of the learned representation is rotation-invariant, and no beam can prefer the standard axes. The argument needs an orthogonally equivariant optimiser. Bird’s momentum SGD is one, and Adam would not be. The non-trivial findings are three:

  • elementwise activations produce strong concentration, not a mild basis-dependence;
  • the concentration is on the directions the activation’s group singles out;
  • the radial networks stay spread, not clustered in random directions.

What this does not say

  • Not a measured effect size. There is no statistic, and the method has artefacts along the measured basis (App. A). The contrast is large in the plots, but it is visual.
  • Not “symmetry alone”. Every O(n)-equivariant map is radial, so each pair also contrasts coordinate-wise against radial nonlinearity. The B_n against S_n Leaky-ReLU comparison is the evidence for the group over the form, and it is qualitative.
  • Not large models. Widths up to 32 and depths up to 3, reconstruction only. Classifiers, convolutions, residual streams and attention are discussed, not run.
  • Not the downstream phenomena. That this is the origin of grandmother neurons, discrete superposition or neural collapse is the author’s conjecture.
  • Not that isotropic networks are better. The reconstruction results (App. D) are preliminary, and mixed without input standardisation.

Connections

  • QUESTION-058. Isolates the activation as a source of group structure in a representation: B_n or S_n, from the network, with no counterpart in the task. A discrete arrangement read off a trained network can belong to its primitive.
  • THEORY-184. The permutation-equivariant update preserves the symmetry it starts with. This adds that the primitive imprints its group on how examples are distributed.
  • THEORY-017. A basis is supplied from outside the space. Here the activation supplies it, and removing that removes the axes.
  • THEORY-022. There the group of the characters came from the task. Here a group comes from the network. “Not the network” does not generalise beyond that case.
  • THEORY-021. The packed geometries of superposition are found in toy models with ReLU, an S_n primitive. Bird conjectures that such geometries depend on the activation’s group. That is untested, and THEORY-021 does not depend on it.

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