HiSVD: Principled Low-Rank Approximation of LLMs via Hierarchical Modeling of Information Capacity and Spectral Structure

Singular Value Decomposition (SVD) enables hardware-agnostic LLM compression via lowrank approximation, yet optimal rank allocation remains a bottleneck. Existing methods predominantly derive layer importance from performance-oriented proxies. Yet, these metrics fail to distinguish between representational importance and structural compressibility, consequently obscuring the fine-grained influence of spectral distribution shape. We demonstrate this disconnect through spectral analysis, revealing that layers with similar information capacity can exhibit markedly different singular value decay behaviors, corresponding to varying degrees of redundancy in the spectral tail. This imperfect coupling implies that allocation strategies driven solely by importance leave significant compression opportunities underexploited. To address this gap, we propose HiSVD, a hierarchical rank allocation framework with two stages: (1) Capacity-Anchored Baseline Allocation, which preserves representational stability by aligning rank budgets with information capacity; and (2) Redundancy-Aware Refinement, which modulates this baseline using tail redundancy to penalize structural excess. Experiments on LLMs demonstrate that HiSVD achieves superior compression efficiency, significantly outperforming state-ofthe-art baselines by effectively exploiting this spectral heterogeneity.