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. The code is available at https://github.com/MLGroupJLU/MatrixNuclearNorm.

Deploying large language models (LLMs) is challenging due to their high memory and computational demands, especially during long-context inference. While keyvalue (KV) caching accelerates inference by reusing previously computed keys and values, it also introduces significant memory overhead. Existing KV cache compression methods--such as eviction and merging--typically compress the KV cache after it is generated and overlook the hidden states, failing to improve the speed of the prefilling stage. Additionally, applying a uniform compression rate across different attention heads can harm crucial retrieval heads in needle-in-ahaystack tasks due to excessive compression. In this paper, we propose UNComp, an uncertainty-aware compression scheme that leverages matrix entropy to estimate model uncertainty across layers and heads at the token sequence level. By grouping layers and heads based on their uncertainty, UNComp adaptively compresses both the hidden states and the KV cache. Our method achieves a 1.6 speedup in the prefilling stage and reduces the KV cache to 4.74% of its original size, resulting in a 6.4 increase in throughput and a 1.4 speedup in inference with only a 1.41% performance loss. Remarkably, in needle-in-a-haystack tasks, UNComp outperforms the full-size KV cache even when compressed to 9.38% of its original size. Our approach offers an efficient, training-free Grouped-Query Attention paradigm that can be seamlessly integrated into existing KV cache schemes. The proliferation of large language models (LLMs) has led to unprecedented advancements in natural language processing (Achiam et al., 2023; Kaplan et al., 2020), enabling capabilities ranging from simple text generation to complex reasoning and dialogue.

In this paper, we utilize information-theoretic metrics like matrix entropy and mutual information to analyze supervised learning. We explore the information content of data representations and classification head weights and their information interplay during supervised training. Experiments show that matrix entropy cannot solely describe the interaction of the information content of data representation and classification head weights but it can effectively reflect the similarity and clustering behavior of the data. Inspired by this, we propose a cross-modal alignment loss to improve the alignment between the representations of the same class from different modalities. Moreover, in order to assess the interaction of the information content of data representation and classification head weights more accurately, we utilize new metrics like matrix mutual information ratio (MIR) and matrix information entropy difference ratio (HDR). Through theory and experiment, we show that HDR and MIR can not only effectively describe the information interplay of supervised training but also improve the performance of supervised and semi-supervised learning.

Large language models (LLMs) have revolutionized the field of natural language processing, extending their strong capabilities into multi-modal domains. Thus, it is vital to define proper and diversified metrics for the evaluation of LLMs. In this paper, we introduce matrix entropy, a novel metric rooted in information theory and geometry principles to quantify the data compression proficiency in LLMs. It reflects the model's ability to extract relevant information and eliminate unnecessary elements, thereby providing insight into the language model's intrinsic capability. Specifically, we demonstrate its applicability in both single-modal (language) and multi-modal settings. For language models, our findings reveal that the matrix entropy of representations follows a scaling law type reduction when the model scales up, serving as a complement to the traditional loss scaling law. For the multi-modal setting, we also propose an evaluation method based on matrix entropy for assessing alignment quality and we find that modern large multi-modal models exhibit great alignment performance.