Convergence of Spectral Principal Paths: How Deep Networks Distill Linear Representations from Noisy Inputs

High-level representations have become a central focus in enhancing AI transparency and control, shifting attention from individual neurons or circuits to structured semantic directions that align with human-interpretable concepts. Motivated by the Linear Representation Hypothesis (LRH), we propose the Input-Space Linearity Hypothesis (ISLH), which posits that concept-aligned directions originate in the input space and are selectively amplified with increasing depth. We then introduce the Spectral Principal Path (SPP) framework, which formalizes how deep networks progressively distill linear representations along a small set of dominant spectral directions. Building on this framework, we further demonstrate the multimodal robustness of these representations in Vision-Language Models (VLMs). By bridging theoretical insights with empirical validation, this work advances a structured theory of representation formation in deep networks, paving the way for improving AI robustness, fairness, and transparency.