Spectral Signatures of Large Language Models

The rapidly growing repository of publicly available large language models (LLMs) presents significant challenges for systematic management and quantification at scale, such as model lineage tracing, licensing, and evaluation. However, task-specific benchmarks are insufficient for this setting, as LLMs differ widely in architectures, scales, and training procedures. To address this challenge, we adopt spectral shape-based metrics for managing and quantifying LLMs based on Heavy-Tailed Self-Regularization theory. Our approach uses the shape information of the weight empirical spectral density as a compact spectral signature of each model. This signature captures intrinsic properties of pretrained models and remains robust during post-training, making it suitable for model-level analysis. In addition, this metric is data-free, computationally-efficient, and scale-invariant, enabling large-scale analysis in practice. Moreover, we curate a large and diverse model corpus consisting of major open-source LLM families, and use it to systematically benchmark spectral and non-spectral metrics across models and downstream tasks. We show that our spectral signature supports the tracking of the model lineage, the unsupervised clustering of similar models, and the quantification of the model performance. Overall, the proposed spectral signature provides a meaningful proxy for broad performance trends across LLMs, enabling efficient organization, comparison, and analysis of large model collections.

ResMerge: Residual-based Spectral Merging of Large Language Models

Model merging offers a training-free way to combine multiple post-trained expert models, but merging experts obtained through reinforcement learning (RL) remains challenging. Existing spectral merging methods often assume that leading singular directions contain the main task signal, while lower-energy residual components can be compressed, selected, or attenuated to reduce interference. We find that this assumption does not hold for RL task vectors: after decomposing each task vector into a leading spectral head and a residual component, both parts can independently recover substantial behavior knowledge, while exhibiting different merging properties. The head is highly concentrated and informative but more prone to sharp cross-expert conflicts, whereas the residual component is more dispersed and provides a more stable basis for aggregation. Based on this observation, we propose ResMerge, a residual-based spectral merging framework for RL experts. ResMerge first constructs a stable residual backbone with Spherical Residual Consensus Adaptation, which estimates a reliability-weighted consensus direction on the Frobenius sphere. It then reintroduces leading-head information through a Lightweight Head Correction module gated by positive cross-expert agreement. Experiments across multiple RL expert groups and capability domains show that ResMerge better preserves expert capabilities than representative task-vector and spectral merging baselines. The implementation of ResMerge is publicly available at https://github.com/sunyd0303-cpu/ResMerge-release.

SMoA: Spectrum Modulation Adapter for Parameter-Efficient Fine-Tuning

As the number of model parameters increases, parameter-efficient fine-tuning (PEFT) has become the go-to choice for tailoring pre-trained large language models. Low-rank Adaptation (LoRA) uses a low-rank update method to simulate full parameter fine-tuning, which is widely used to reduce resource requirements. However, decreasing the rank encounters challenges with limited representational capacity. Theory suggests that LoRA fine-tuning with rank r converges toward the top r singular values of the pre-trained weight matrix. As the rank increases, more principal singular directions are preserved, which generally improves the model's performance. However, a larger rank also introduces more trainable parameters, leading to higher computational cost. To overcome this dilemma, we propose SMoA, a \textbf{S}pectrum \textbf{Mo}dulation \textbf{A}dapter that enlarges the accessible family of spectrum-aware updates under a smaller parameter budget. SMoA partitions the layer into multiple aligned spectral blocks and applies one in-block Hadamard-modulated low-rank branch to each diagonal block, yielding broader coverage of pretrained spectral directions. We provide theoretical analysis and empirical results on multiple tasks. In our experiments, SMoA improves average performance in the current lower-budget setting over LoRA and competitive LoRA-style baselines.