Spectral Superposition: A Theory of Feature Geometry

TL;DR

This paper introduces a spectral theory for understanding how neural network features are geometrically organized in high-dimensional space, revealing global interactions and feature localization through eigenvalue analysis.

Contribution

It develops a spectral framework using the frame operator to analyze feature geometry, capturing global interactions and feature localization beyond pairwise methods.

Findings

Spectral localization occurs under capacity saturation in toy models.

Features organize into tight frames and discrete classifications.

Spectral measures diagnose feature localization in arbitrary matrices.

Abstract

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.