Continuous Function

Foundation reference

Superposition, Sparse Features & Monosemanticity

Superposition is the hypothesis that a network stores more features than it has dimensions as overlapping directions in activation space. It is one proposed explanation for polysemanticity, where a neuron responds to several unrelated features (Elhage et al., 2022).

Concept 11 of 100RepresentationsStudy phase 5
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Key equation
min⁡A,{si}∑i(∥hi−Asi∥22+λ∥si∥1),∥A:,j∥2=1\min_{A, \{s_i\}} \sum_i \big(\|h_i - A s_i\|_2^2 + \lambda \|s_i\|_1\big),\quad \|A_{:,j}\|_2=1
Reading map and next steps

Study guide

Where to start

Concept 11 of 100SuperpositionRepresentations · Study phase 5: Representation & interpretability
Question

Why is superposition useful to a network, and how does the trade-off between capacity and interference decide how many features it stores?

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CheckCheck against the math and the papers.

Compare what the demo shows with the key equation and with the papers cited below.

GeneralizeName what stays the same.

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ContinueFollow the connections.

Use the prerequisites and related concepts at the end of the page to fill the next gap.

Why It Matters for Modern Models

  • Superposition is the hypothesis that a network stores more features than it has dimensions as overlapping directions in activation space. It is one proposed explanation for polysemanticity, where a neuron responds to several unrelated features (Elhage et al., 2022).
  • Sparse autoencoders have since been scaled to production language models, for example Claude 3 Sonnet (Templeton et al., 2024), as a tool for interpretability and safety research.

Questions to Think About

  • How does a sparse autoencoder divide activation space into feature directions that may overlap?

Interactive Demo

Core Math

If you prefer intuition first, try the demo above, then come back for the math.

Key Equation
min⁡A,{si}∑i(∥hi−Asi∥22+λ∥si∥1),∥A:,j∥2=1\min_{A, \{s_i\}} \sum_i \big(\|h_i - A s_i\|_2^2 + \lambda \|s_i\|_1\big),\quad \|A_{:,j}\|_2=1

The superposition hypothesis: a network represents more features than it has neurons by storing them as nearly orthogonal directions in activation space, which works when each feature is active only rarely (Elhage et al., 2022). Recovering those directions is a dictionary learning problem:

min⁡A,{si}∑i(∥hi−Asi∥22+λ∥si∥1)\min_{A, \{s_i\}} \sum_i \big(\|h_i - A s_i\|_2^2 + \lambda \|s_i\|_1\big)

where hih_i are activations, columns of AA are unit-norm features (a constraint in the minimization), and sis_i are sparse codes.

Bricken et al. (2023) trained sparse autoencoders on the MLP activations of a one-layer transformer and found features that are much more interpretable ("monosemantic") than individual neurons.

Key Papers

Toy Models of Superposition

Elhage et al.2022Transformer Circuits Thread (Anthropic)
Read paper →

Towards Monosemanticity: Decomposing Language Models with Dictionary Learning

Bricken et al.2023Transformer Circuits Thread (Anthropic)
Read paper →

Connections

Prerequisites

Embeddings

Leads to

CircuitsSparse Autoencoders

Where to go next

Circuits builds directly on Superposition. The other ideas that build on this page are listed under Leads to, above.

Circuits
Other connections (6)

Which limitations does it address, or leave to a later idea?

  • Feature dictionaries: a limitation of Superposition that Sparse Autoencoders addresses. Open Sparse Autoencoders

    Superposition is one explanation for why neurons are polysemantic and hard to interpret (Elhage et al., 2022). Sparse autoencoders were developed to learn an overcomplete dictionary that recovers sparse, more interpretable features from dense activations (Bricken et al., 2023).

  • Disentangle features: a limitation of Sparse Autoencoders that Superposition addresses. Open Sparse Autoencoders

    Sparse autoencoders do not remove superposition from the model; they read features out of it. They learn more directions than the layer has neurons and explain each activation with a few of them, so the learned features are more often monosemantic than the neurons are (Bricken et al., 2023).

Where does it break down?

  • Features overlap: where Superposition breaks down, taken up in Probing. Open Probing

    Under superposition a layer stores more features than it has dimensions, as directions that are only nearly orthogonal. A linear probe for one feature then also picks up small contributions from the others that overlap with it, so its accuracy is limited by this interference, and the direction it finds need not be a single feature the model uses.

  • Non-causal probes: where Probing breaks down, taken up on this page. Open Probing

    With superposition, probes can "find" information in activations that isn't a clean, causal feature the model uses. High probe accuracy can overstate mechanistic understanding when features are entangled.

Where else does the same idea appear?

  • Sparse features: an analogy between Superposition and MoE. Open MoE

    Superposition lets a layer represent more features than it has dimensions, as overlapping directions, which works because only a few features are active at once (Elhage et al., 2022). A mixture-of-experts layer likewise relies on each token needing only a few experts. Both exploit sparsity.

  • Compression tradeoff: an analogy between Superposition and Tokenization. Open Tokenization

    Tokenizers compress frequent patterns into single tokens but fragment rare strings into many pieces. Superposition compresses many features into limited dimensions but creates interference—same capacity vs ambiguity tradeoff.