Large Language Model Evaluation via Matrix Nuclear-Norm (2025)

Abstract: As large language models (LLMs) continue to evolve, efficient evaluation metrics are vital for assessing their ability to compress information and reduce redundancy. While traditional metrics like Matrix Entropy offer valuable insights, they are computationally intensive for large-scale models due to their O(n^3) time complexity with Singular Value Decomposition (SVD). To mitigate this issue, we introduce the Matrix Nuclear-Norm, which not only serves as a metric to quantify the data compression proficiency of LLM but also provides a convex approximation of matrix rank to capture both predictive discriminability and diversity. By employing the L_{1,2}\text{-norm} to further approximate the nuclear norm, we can effectively assess the model’s information compression capabilities. This approach reduces the time complexity to O(n^2) and eliminates the need for SVD computation. Consequently, the Matrix Nuclear-Norm achieves speeds 8 to 24 times faster than Matrix Entropy for the CEREBRAS-GPT model as sizes increase from 111M to 6.7B. This performance gap becomes more pronounced with larger models, as validated in tests with other models like Pythia. Additionally, evaluations on benchmarks and model responses confirm that our proposed Matrix Nuclear-Norm is a reliable, scalable, and efficient tool for assessing LLMs’ performance, striking a balance between accuracy and computational efficiency.

## introduction large language models (llms), such as gemini @xcite , deepseek @xcite , and @xcite , have shown exceptional performance in numerous natural language processing (nlp) tasks @xcite . these models are transforming the * corresponding author way we approach nlp tasks, providing unprecedented capabilities and solutions to complex problems. they are revolutionizing nlp @xcite @xcite and positively impacting computer vision @xcite and graph neural networks @xcite , achieving top results on leaderboards. despite these advancements, evaluating a model's ability to compress information remains a critical research challenge @xcite . this challenge is essential for improving the overall efficiency of these models. compression involves efficiently extracting essential information from large datasets while removing redundant data, highlighting a model's ability to understand the data's underlying structure @xcite . llms are expected to perform this compression during training @xcite . initially, after random initialization, the data representations are chaotic, but as training progresses, they become organized, allowing the model to filter out unnecessary information. thus, assessing an llm's compression capacity is vital for understanding its learning efficiency and representational power, which are crucial for practical applications and real-world deployment. current compression metrics like @xcite 's matrix entropy analyze output representations but face scalability limits due to o(n 3 ) svd complexity @xcite . to address this, we propose a novel metric called matrix nuclear-norm. this metric measures predictive discriminability and output diversity, serving as an upper bound for the frobenius norm and providing a convex approximation of the matrix rank. we enhance the matrix nuclear-norm by using the l 1,2 -norm to approximate the nuclear norm, improving stability across multiple classes. this approach efficiently assesses a model's compression capabilities and redundancy elimination, streamlining evaluation. the matrix nuclear-norm has a computational complexity of o(n 2 ), a significant improvement over matrix entropy's o(n 3 ). this optimization @xmath0. to validate the matrix nuclear-norm, we conducted preliminary experiments on two language models of different sizes. results showed a consistent decrease in matrix nuclear-norm values as model size increased, indicating enhanced compression capabilities. we also performed inference experiments on benchmark datasets, alpacaeval @xcite and chatbot arena @xcite , covering diverse language generation tasks. these benchmarks provide a comprehensive assessment of model inference performance. our findings confirm that the matrix nuclear-norm accurately measures model compression capabilities and ranks models based on performance, demonstrating its reliability and efficiency. our empirical investigations yield the following insights: ## preliminaries this section presents the fundamental concepts for model performance evaluation: discriminability, diversity, and nuclear norm. ## experiments of large language models the models and datasets used in this paper are thoroughly introduced in a.2. ## conclusion in conclusion, matrix nuclear-norm stands out as a promising evaluation metric for llms, offering significant advantages in assessing information compression and redundancy elimination. its key strengths include remarkable computational efficiency, greatly exceeding that of existing metrics like matrix entropy, along with exceptional stability across diverse datasets. matrix nuclear-norm's responsiveness to model performance under varying inputs emphasizes its ability to gauge not only performance but also the intricate adaptability of models. this metric marks a significant advancement in nlp, establishing a clear and effective framework for future research and development in the evaluation and optimization of language models.