Provably Extracting the Features from a General Superposition

It is widely believed that complex machine learning models generally encode features through linear representations. This is the foundational hypothesis behind a vast body of work on interpretability. A key challenge toward extracting interpretable features, however, is that they exist in superposition. In this work, we study the question of extracting features in superposition from a learning theoretic perspective. We start with the following fundamental setting: we are given query access to a function \[ f(x)=\sum_{i=1}^n \sigma_i(v_i^\top x), \] where each unit vector $v_i$ encodes a feature direction and $\sigma_i:\R\to\R$ is an arbitrary response function and our goal is to recover the $v_i$ and the function $f$.In learning-theoretic terms, superposition refers to the \emph{overcomplete regime}, when the number of features is larger than the underlying dimension (i.e. $n > d$), which has proven especially challenging for typical algorithmic approaches. Our main result is an efficient query algorithm that, from noisy oracle access to $f$, identifies all feature directions whose responses are non-degenerate and reconstructs the function $f$. Crucially, our algorithm works in a significantly more general setting than all related prior results. We allow for essentially arbitrary superpositions, only requiring that $v_i, v_j$ are not nearly identical for $i \neq j$, and allowing for general response functions $\sigma_i$. At a high level, our algorithm introduces an approach for searching in Fourier space by iteratively refining the search space to locate the hidden directions $v_i$.