Legacy Concept Lab

Superposition, Sparse Features & Monosemanticity

Frontier LMs heavily rely on superposition: neurons implement many overlapping features

Concept 11 of 100RepresentationsPhase 5

Open Demo All Foundations

#11SuperpositionRepresentations

key equation

\min_{A, s_i} \sum_i \|h_i - A s_i\|_2^2 + \lambda \|s_i\|_1

Phase 5: Representation & interpretabilityConcept 11 of 100

Why It Matters for Modern Models

Frontier LMs heavily rely on superposition: neurons implement many overlapping features
Monosemantic dictionaries being applied to Claude-class models for interpretability & safety

What Tutorials Skip

What is still poorly explained in textbooks and papers:

Simple geometric story for why superposition is useful (capacity vs interference trade-offs)
Interactive views of how sparse autoencoders carve up activation space into overlapping feature directions

Interactive Visualization

Core Math (Optional Deep Dive)

If you want intuition first, start with the key equation and the visualization. Come back here for the full walkthrough.

Key Equation

\min_{A, s_i} \sum_i \|h_i - A s_i\|_2^2 + \lambda \|s_i\|_1

Features are represented not by one neuron each, but as sparse directions in activation space. Formalized via dictionary learning:

\min_{A, s_i} \sum_i \|h_i - A s_i\|_2^2 + \lambda \|s_i\|_1

where $h_i$ are activations, columns of $A$ are features, and $s_i$ are sparse codes.

Sparse autoencoders applied to transformer MLP activations recover relatively interpretable, "monosemantic" features.

Canonical Papers

Toy Models of Superposition

Elhage et al.2022Anthropic

Towards Monosemanticity: Decomposing Language Models with Dictionary Learning

Bricken et al.2023Anthropic

Connections

Prerequisites

Enables

⊛Circuits 🔍Sparse Autoencoders

Next Moves

Explore this concept from different angles — like a mathematician would.

Semantic Connections

≈ Analogy

🔀Sparse features→MoE 🔤Compression tradeoff→Tokens

🔧 Invented to Fix

🔍Feature dictionaries→Sparse Autoencoders 🔍Disentangle features→Sparse Autoencoders

⚠️ Breaks When

⚲Features overlap→Probing ⚲Non-causal probes→Probing