A General Error-Theoretical Analysis Framework for Constructing Compression Strategies

The exponential growth in parameter size and computational complexity of deep models poses significant challenges for efficient deployment. The core problem of existing compression methods is that different layers of the model have significant differences in their tolerance to compression levels. For instance, the first layer of a model can typically sustain a higher compression level compared to the last layer without compromising performance. Thus, the key challenge lies in how to allocate compression levels across layers in a way that minimizes performance loss while maximizing parameter reduction. To address this challenge, we propose a Compression Error Theory (CET) framework, designed to determine the optimal compression level for each layer. Taking quantization as an example, CET leverages differential expansion and algebraic geometry to reconstruct the quadratic form of quantization error as ellipsoids and hyperbolic paraboloids, and utilizes their geometric structures to define an error subspace. To identify the error subspace with minimal performance loss, by performing orthogonal decomposition of the geometric space, CET transforms the optimization process of the error subspace into a complementary problem. The final theoretical analysis shows that constructing the quantization subspace along the major axis results in minimal performance degradation. Through experimental verification of the theory, CET can greatly retain performance while compressing. Specifically, on the ResNet-34 model, CET achieves nearly 11$\times$ parameter compression while even surpassing performance comparable to the original model.

Lossless Model Compression via Joint Low-Rank Factorization Optimization

Low-rank factorization is a popular model compression technique that minimizes the error $\delta$ between approximated and original weight matrices. Despite achieving performances close to the original models when $\delta$ is optimized, a performance discrepancy remains due to the separate optimization processes for low-rank factorization and model performance, resulting in unavoidable losses. We address this issue by introducing a novel joint optimization strategy for lossless low-rank weight factorization, which, for the first time, enhances the model's performance beyond the original. Our approach begins with a theoretical analysis of the relationship between low-rank factorization and model optimization objectives, establishing a precise perturbation range for matrix factorization errors on model performance. This challenge is then reformulated as a numerical rank deficiency problem with inequality constraints and develop a joint objective that simultaneously addresses factorization error and model performance. Based on the above analysis, we propose two optimization algorithms: \textbf{a lossless optimization algorithm} that maximizes model accuracy while ensuring compression, and \textbf{a compact optimization algorithm} that minimizes model size while preserving performance. These algorithms do not require fine-tuning and can directly compress numerous deep models to achieve lossless results. Our methods demonstrate robust efficacy across various vision and language tasks. For example, the compressed model reduced by 70\% on ResNext50 outperforms the original. Our code will be made public.