Spectral Superposition: A Theory of Feature Geometry
This provides a theoretical foundation for interpreting feature interactions in neural networks, though it is incremental as it builds on prior work on superposition.
The paper tackles the problem of understanding feature geometry in neural networks by developing a spectral theory that analyzes weight matrices to study how features allocate norm across eigenspaces, proving in toy models that capacity saturation forces features to collapse onto single eigenspaces and organize into tight frames.
Neural networks represent more features than they have dimensions via superposition, forcing features to share representational space. Current methods decompose activations into sparse linear features but discard geometric structure. We develop a theory for studying the geometric structre of features by analyzing the spectra (eigenvalues, eigenspaces, etc.) of weight derived matrices. In particular, we introduce the frame operator $F = WW^\top$, which gives us a spectral measure that describes how each feature allocates norm across eigenspaces. While previous tools could describe the pairwise interactions between features, spectral methods capture the global geometry (``how do all features interact?''). In toy models of superposition, we use this theory to prove that capacity saturation forces spectral localization: features collapse onto single eigenspaces, organize into tight frames, and admit discrete classification via association schemes, classifying all geometries from prior work (simplices, polygons, antiprisms). The spectral measure formalism applies to arbitrary weight matrices, enabling diagnosis of feature localization beyond toy settings. These results point toward a broader program: applying operator theory to interpretability.