Spectral Compact Training: Pre-Training Large Language Models via Permanent Truncated SVD and Stiefel QR Retraction

This study presents an ensemble technique, SPQ (SVD-Pruning-Quantization), for large language model (LLM) compression that combines variance-retained singular value decomposition (SVD), activation-based pruning, and post-training linear quantization. Each component targets a different source of inefficiency: i) pruning removes redundant neurons in MLP layers, ii) SVD reduces attention projections into compact low-rank factors, iii) and 8-bit quantization uniformly compresses all linear layers. At matched compression ratios, SPQ outperforms individual methods (SVD-only, pruning-only, or quantization-only) in perplexity, demonstrating the benefit of combining complementary techniques. Applied to LLaMA-2-7B, SPQ achieves up to 75% memory reduction while maintaining or improving perplexity (e.g., WikiText-2 5.47 to 4.91) and preserving accuracy on downstream benchmarks such as C4, TruthfulQA, and GSM8K. Compared to strong baselines like GPTQ and SparseGPT, SPQ offers competitive perplexity and accuracy while using less memory (6.86 GB vs. 7.16 GB for GPTQ). Moreover, SPQ improves inference throughput over GPTQ, achieving up to a 1.9x speedup, which further enhances its practicality for real-world deployment. The effectiveness of SPQ's robust compression through layer-aware and complementary compression techniques may provide practical deployment of LLMs in memory-constrained environments. Code is available at: https://github.com/JiaminYao/SPQ_LLM_Compression/

The deployment of Large Language Models is constrained by the memory and bandwidth demands of static weights and dynamic Key-Value cache. SVD-based compression provides a hardware-friendly solution to reduce these costs. However, existing methods suffer from two key limitations: some are suboptimal in reconstruction error, while others are theoretically optimal but practically inefficient. In this paper, we propose Swift-SVD, an activation-aware, closed-form compression framework that simultaneously guarantees theoretical optimum, practical efficiency and numerical stability. Swift-SVD incrementally aggregates covariance of output activations given a batch of inputs and performs a single eigenvalue decomposition after aggregation, enabling training-free, fast, and optimal layer-wise low-rank approximation. We employ effective rank to analyze local layer-wise compressibility and design a dynamic rank allocation strategy that jointly accounts for local reconstruction loss and end-to-end layer importance. Extensive experiments across six LLMs and eight datasets demonstrate that Swift-SVD outperforms state-of-the-art baselines, achieving optimal compression accuracy while delivering 3-70X speedups in end-to-end compression time. Our code is available at https://github.com/hiahei/Swift-SVD.

Large language models (LLMs) achieve remarkable performance across a wide range of tasks, but their deployment is constrained by substantial memory and compute requirements. Low-rank compression via singular value decomposition (SVD) is an effective remedy, but existing methods focus on how to factorize and which components to keep. We introduce SVD-Surgeon, a training-free method that brings the Optimal Brain Surgeon (OBS) framework to the singular-value basis. Treating each singular value as a parameter, it computes a closed-form update of the retained singular values that compensates, to second order in the model loss, for those removed by truncation. The same analysis yields a saliency for choosing which values to prune. As it operates directly on the singular-value factorization, SVD-Surgeon can be layered on top of existing SVD compressors. Applied to SVD-LLM, a leading SVD-based method, it improves the perplexity-compression trade-off on the OPT family and LLaMA 2-7B without any retraining.