Superposition and sparse autoencoders

The last lesson ended on the feature problem: units respond to unrelated things. The explanation is one of the most important ideas in modern interpretability: superposition — a network can represent more features than it has dimensions by storing them as not-quite-orthogonal directions and paying in a little noise.

The geometry #

Feel the compression:

The toy model behind this (Anthropic's "Toy Models of Superposition"): a network trained to reconstruct $m$ features into $n < m$ dimensions, under a sparsity prior, converges to the geometry you just watched — first the densest orthogonal set, then increasingly tight angular packing with interference as the price. Sparsity is the budget: rare features can share space because they rarely collide; frequent features earn their own private dimensions (which is why the most important features — induction heads' "current token" copies, etc. — are often clean even while rarer ones superpose).

The formal picture, in one line: activations are $x = \sum_i f_i \, w_i + \text{noise}$ with more features $f_i$ than basis vectors $w_i$ — the reconstruction is lossy only when multiple active features are near-parallel, which sparsity makes rare.

Sparse autoencoders: un-mixing the signal #

If superposition is the disease, sparsity is also the cure. A sparse autoencoder (SAE) takes the model's activations and learns to rewrite them as a wider representation with one dimension per recovered feature:

$$f = \mathrm{ReLU}(W_{\text{enc}} x + b_{\text{enc}}), \qquad \hat{x} = W_{\text{dec}} f + b_{\text{dec}}, \qquad \mathcal{L} = \lVert x - \hat{x} \rVert_2^2 + \lambda \lVert f \rVert_1$$

Read the loss: reconstruct the activations faithfully ($L_2$ term) while keeping the feature vector $f$ mostly zeros ($L_1$ term). The $L_1$ penalty forces sparsity, which forces each learned dimension to be a single feature — dictionary learning. Turn the toggle on in the artifact above: the SAE's wide space has one clean, nameable direction per feature, and interference drops to zero by construction of the objective, not by magic.

What this bought in real models: named, steerable features at scale — deception-related features, sycophancy features, code-vulnerability features, "San Francisco" features with their neighborhoods — in production LLMs, not toys.

Next: once features are clean and nameable, you can do more than look — you can push.

This lesson has exercises attached — predicting interference levels and diagnosing an SAE trained with λ too low or too high — once the exercises layer ships.