Habish Dhakal | Searcharxiv

arXiv2026

Concepts Whisper: Spectral Anti-Concentration and the Dual Geometry of Transformer Representations

We find that transformer concept representations systematically anti-concentrate in the spectral tail of the unembedding covariance, encoding word-level concepts in low-variance directions across a 17-model core suite and an expanded set of 22 semantic concept categories, with convergent replications from three independent extraction methods. Residual-stream difference-of-means vectors anti-concentrate in all 17 models (model-level one-sample t-test, p = 3.8e-9), and remain more tail-aligned than norm-matched random directions in 13 of 17; convergent support comes from sparse autoencoder (SAE) features (p = 4.5e-19 across concepts within a model) and linear probes on Llama and Qwen. We identify a dual geometry: activation-space concept directions anti-concentrate while static unembedding-row contrasts concentrate in high-variance directions (p < 10^-4). This investigation arose from testing whether the causal inner product of Park et al. (2024) aids cross-lingual concept transport; a matched-spectrum randomization across 17 models and four language pairs finds no evidence that Whitened Causal Alignment improves over spectral regularization alone (p = 0.95). Split-injection interventions, restricted to steering strengths at which both arms remain interpretable, show the predicted interference asymmetry in four of five models (paired Cohen's d_z up to 1.19) with no significant reversal inside that regime, and POS-tag probing across eight models shows syntax preferentially encoded in the high-variance subspace in six of eight architectures, with a significant reversal in the Qwen 2.5 family. These results suggest transformers rotate semantic content into spectrally quiet regions during contextualized processing, where, in some architectures, interventions may reduce grammatical disruption relative to high-variance steering.

cs.LG↗

arXiv2026

Grokking as a Variance-Limited Phase Transition: Spectral Gating and the Epsilon-Stability Threshold

Standard optimization theories struggle to explain grokking, where generalization occurs long after training convergence. While geometric studies attribute this to slow drift, they often overlook the interaction between the optimizer's noise structure and landscape curvature. This work analyzes AdamW dynamics on modular arithmetic tasks, revealing a ``Spectral Gating'' mechanism that regulates the transition from memorization to generalization. We find that AdamW operates as a variance-gated stochastic system. Grokking is constrained by a stability condition: the generalizing solution resides in a sharp basin ($λ_{max}^H$) initially inaccessible under low-variance regimes. The ``delayed'' phase represents the accumulation of gradient variance required to lift the effective stability ceiling, permitting entry into this sharp manifold. Our ablation studies identify three complexity regimes: (1) \textbf{Capacity Collapse} ($P < 23$), where rank-deficiency prevents structural learning; (2) \textbf{The Variance-Limited Regime} ($P \approx 41$), where generalization waits for the spectral gate to open; and (3) \textbf{Stability Override} ($P > 67$), where memorization becomes dimensionally unstable. Furthermore, we challenge the "Flat Minima" hypothesis for algorithmic tasks, showing that isotropic noise injection fails to induce grokking. Generalization requires the \textit{anisotropic rectification} unique to adaptive optimizers, which directs noise into the tangent space of the solution manifold.

cs.LG↗