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From Density Matrices to Phase Transitions in Deep Learning: Spectral Early Warnings and Interpretability

NeurIPS 2026 · Learning theory and algorithms › Matrix and tensor methods · approach: Geometry of low-rank and SPD structure · 0 citations (OpenAlex)

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Introduces the 2-datapoint reduced density matrix (per-sample loss covariance over a probe set) and derives spectral heat capacity and participation ratio as online early-warning and interpretability signals for training phase transitions.

Problem

Detecting and predicting phase transitions (emergent capabilities, grokking, induction heads, emergent misalignment) during neural network training in an online, computationally cheap way, with interpretability of what changes.

Approach

Treat the covariance of per-sample losses over a probe set (under a Gaussian or trajectory-window distribution over parameters) as a 2-datapoint reduced density matrix; use the variance of its eigenvalues (spectral heat capacity) as an early warning signal via critical slowing down, the participation ratio as a dimensionality measure, and the top eigenvectors for mechanistic interpretation.

Key results

• In deep linear networks, the SHC spike precedes or coincides with silent alignment; average lag computed over 30 random seeds.
• In emergent misalignment on Qwen2.5-7B-Instruct with LoRA, the variance increase is concentrated in alignment probe groups (e.g., mean aligned 0.00-0.25 in templated/JSON/free-form subgroups vs 1.00 for benign groups at step 5).
• Grokking analysis reports SHC on test-sample block spikes before test accuracy increases, and top eigenvector power concentrates in few Fourier modes.

Surprising findings

• The participation ratio of the dynamical 2RDM detects a non-equilibrium steady state in grokking where weight decay and gradient update balance.
• Emergent misalignment is detected at very early steps (5-10) with only ~20% behavioral shift in a 7B model.

Limitations

• Detection only possible for transitions resolved by the probe set (basis-set analogy).
• Emergent misalignment effect is small in scale and fluctuations are very small, needing normalization.
• Proofs rely on linearization and Gaussian/rank-one assumptions; the early-warning proposition is stated with assumptions such as unbounded variance growth.

Experimental setup

Datasets: emergent misalignment probe questions bad medical advice data

Benchmarks: induction head formation grokking modular division emergent misalignment probes

Metrics: spectral heat capacity participation ratio induction head score alignment score train/test accuracy lag between SHC spike and alignment

Baselines: local learning coefficient gradient norm

Builds on: cumulant expansion silent alignment in deep linear networks local learning coefficient loss kernel critical slowing down

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