Superposition

The fundamental mechanism where a model layer with n neurons can represent m > n independent features. This is possible because features in natural data are sparse—only a small fraction are active at any given time. The network leverages this sparsity to pack features into a compressed representation, analogous to how compressed sensing in signal processing reconstructs signals from fewer samples than Nyquist-Shannon requires. This compression is not lossless; interference between non-orthogonal features creates the polysemanticity observed in individual neurons.

Superposition is only computationally viable when features exhibit high sparsity. If most features were active simultaneously, the compressed representation would suffer catastrophic interference. Key properties of sparsity in this context:

• Feature sparsity: The probability that any given feature is active on a random input is low (often < 1%)
• Dimensionality expansion: A layer with 512 neurons might represent 100,000+ sparse features
• Interference floor: As more features are packed in, the noise floor rises, eventually degrading model performance
• Sparsity penalty: Models appear to learn a trade-off between representing more features and maintaining feature purity

The strongest empirical evidence for superposition comes from polysemantic neurons—individual neurons that activate for multiple unrelated concepts. For example, a single neuron in a vision model might fire for both cat faces and car fronts. Under the superposition hypothesis, this neuron participates in multiple almost-orthogonal feature directions in activation space. The neuron's activation is the projection of the input onto its learned direction, and when features are packed, each direction must serve multiple masters. This explains why monosemanticity is rare in dense models without explicit disentanglement pressure.

Superposition can be formalized using the sparse coding framework. Given a dictionary of features W (a matrix of feature directions) and a sparse activation vector h, the model's representation is x = Wh. When the number of features exceeds the dimensionality of x, the dictionary is overcomplete. The model learns to recover h from x by exploiting the fact that h is sparse. Key mathematical properties:

• Mutual coherence: The maximum absolute inner product between distinct feature directions; lower coherence enables cleaner recovery
• Restricted isometry property: A condition ensuring that sparse vectors can be uniquely recovered from compressed measurements
• L1 minimization: The optimization objective that promotes sparsity in the recovered feature activations

Researchers at Anthropic demonstrated superposition in controlled toy models—small ReLU networks trained on synthetic data with known ground-truth features. Key findings from these experiments:

• Superposition emerges naturally when features are sparse and the model benefits from representing more features than dimensions
• The model learns to pack features into antipodal pairs (opposing directions) and geometric structures like regular polytopes
• Feature importance drives allocation: more important features get more orthogonal, higher-magnitude directions
• The transition from monosemantic to polysemantic representations occurs as a phase change when the feature-to-dimension ratio crosses a critical threshold
• These toy models provide a tractable sandbox for developing dictionary learning and sparse autoencoder techniques later applied to production models

Superposition is the central obstacle that mechanistic interpretability must overcome. If features are not aligned with individual neurons, then simply reading neuron activations is misleading. This has driven the development of:

• Sparse autoencoders: Trained to decompose dense activations into a higher-dimensional, sparse set of interpretable features
• Dictionary learning: Unsupervised discovery of the overcomplete basis of features actually used by the model
• Causal interventions: Techniques like activation patching that test whether a direction causally mediates a behavior, even when it's distributed across neurons
• Linear representation hypothesis: The conjecture that features are linear directions, making them recoverable through linear methods despite superposition Understanding superposition is prerequisite to any serious attempt at reverse engineering neural networks.