Spectral Tempering for Embedding Compression in Dense Passage Retrieval

Dimensionality reduction is critical for deploying dense retrieval systems at scale, yet mainstream post-hoc methods face a fundamental trade-off: principal component analysis (PCA) preserves dominant variance but underutilizes representational capacity, while whitening enforces isotropy at the cost of amplifying noise in the heavy-tailed eigenspectrum of retrieval embeddings. Intermediate spectral scaling methods unify these extremes by reweighting dimensions with a power coefficient $\gamma$, but treat $\gamma$ as a fixed hyperparameter that requires task-specific tuning. We show that the optimal scaling strength $\gamma$ is not a global constant: it varies systematically with target dimensionality $k$ and is governed by the signal-to-noise ratio (SNR) of the retained subspace. Based on this insight, we propose Spectral Tempering (\textbf{SpecTemp}), a learning-free method that derives an adaptive $\gamma(k)$ directly from the corpus eigenspectrum using local SNR analysis and knee-point normalization, requiring no labeled data or validation-based search. Extensive experiments demonstrate that Spectral Tempering consistently achieves near-oracle performance relative to grid-searched $\gamma^*(k)$ while remaining fully learning-free and model-agnostic. Our code is publicly available at https://github.com/liyongkang123/SpecTemp.