Superposition
Sparse Autoencoders Find Highly Interpretable Features in Language Models — inherited
A hypothesized cause of polysemanticity (Elhage et al., 2022b) in which a network represents more features than it has dimensions by assigning them to an overcomplete set of non-orthogonal directions, relying on activation sparsity to limit interference. The paper's stated goal is to resolve superposition in language models by recovering these directions via sparse dictionary learning.
Packing thousands of distinct ideas into a network with only a few hundred dimensions to hold them sounds impossible, and the explanation isn't that most of those ideas are illusions. It's that the network squeezes more directions into the space than the space technically has room for, betting that they rarely interfere with each other.
Set out next is the hypothesis in the form it gets adopted, how it dictates concrete choices in the shape and training of a dictionary meant to undo it, and the basic geometry of packing more directions into a space than that space can hold without any of them staying fully independent.
A hypothesis that turns a symptom into a structural claim
Superposition is the hypothesis, proposed by Elhage et al. (2022b) in the Anthropic 'Toy Models of Superposition' line of work, that a network represents more distinct features than it has neurons or activation dimensions by packing those features into an overcomplete set of non-orthogonal directions rather than giving each one its own coordinate. It is offered as the structural cause of Polysemanticity: when several features share a direction, the neuron reading out that direction fires for all of them. This paper adopts the hypothesis directly, describing superposition as a 'hypothesised cause of polysemanticity' and setting its own explicit goal as showing 'it is possible to resolve superposition in language models using a scalable, unsupervised method' (sparse-autoencoders, §"ABSTRACT", p. 1).
Superposition mandates the sparse-autoencoder's shape
Because superposition is a specific structural claim, more features than dimensions, sharing directions, rather than a vague call to 'improve interpretability,' it dictates concrete design choices in the paper's Sparse autoencoders (SAEs). The hidden layer must be able to represent an overcomplete basis, so its size is set to $d_{hid} = R \cdot d_{in}$ with $R$ a tunable expansion factor (sparse-autoencoders, §"2 TAKING FEATURES OUT OF SUPERPOSITION WITH SPARSE DICTIONARY LEARNING", p. 2), and recovering directions that were only ever sparsely active requires an $\ell_1$ penalty on the hidden coefficients rather than on the weights. The paper states plainly that 'without high sparsity, interference between non-orthogonal features prevents any performance gain from superposition' (sparse-autoencoders, §"1 INTRODUCTION", p. 1), which is why activation sparsity, not just the dimension count, is the assumption connecting the hypothesis to the method built on top of it.
Packing more features than there are dimensions
Geometrically, an orthogonal basis for a $d$-dimensional vector space can hold at most $d$ mutually non-interfering directions. Superposition proposes that a network sidesteps this limit by writing each activation vector as a sparse linear combination of many more directions than $d$: $$x_i = \sum_j a_{i,j} g_j$$ where the ground-truth features $g_j$ outnumber the dimensions of $x_i$ and the coefficients $a_i$ are mostly zero for any given input. Because the $g_j$ cannot all be mutually orthogonal once there are more of them than dimensions, two features sharing a component of the same direction interfere whenever both are active at once, which is precisely why sparsity is required for the scheme to pay off. The paper's entire method (Section 2) is aimed at inverting this equation: recovering the $g_j$ given only samples of $x_i$.