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Linear Algebra Foundations of Efficient Attention: A Phase Reversal in Rank Collapse Under SVD Compression

arXiv 2609.06341 ↗First posted 6 Sep 2026cs.LG cs.AI cs.NEcited by 0

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Abstract (arXiv)

Linear algebra provides the framework of concepts (matrix rank, singular value decomposition (SVD), and eigendecomposition) that modern artificial intelligence employs to encode, compress, and propagate information through neural networks. This paper unifies fourteen separate peer-reviewed works analyzing the usage of these techniques in the context of transformer-based foundation model research, focusing on three areas of the topic: derivations and properties of self-attention matrices' output rank, compression methods that purposefully utilize this phenomenon, and the low-rank key-value (KV) cache projection and its semiseparable-matrix duality to linear attention and state-space structured models. We were motivated to conduct this work after observing an open problem in this literature: the interplay of the mentioned compression methods with natural rank collapse of the network. With this paper, we report an original finding that using SVD compression of attention projections actually has the opposite effect on the rank collapse of the network: while it strongly suppresses it at initialization, it accelerates on pretrained models (for GPT-2 124M, GPT-2 Medium 355M, and Pythia-160M) with minimal risk of object aliasing artifacts appearing (verified on all compression ratios) and is consistent across four rank estimation methods. A controlled causal decomposition of the effect in both settings showed that the reason for this behavior can be explained by the choice of the subspace SVD makes when compressing the matrix better than the reduction of the operator norm it achieves, explaining roughly 76% of the effect at initialization and 83% on the pretrained weights, providing a refinement to the calibration-aware compression viewpoint and an explanation of why it outperformed naive SVD truncation.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 16 of its 19 references that are in the map. First posted on arXiv 6 Sep 2026.

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References · 16

  1. 2017Attention Is All You Needcited by 60,361
  2. 2021LoRA: Low-Rank Adaptation of Large Language Modelscited by 14,380
  3. 2024Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Dualitycited by 1,336
  4. 2024GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projectioncited by 463
  5. 2021Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depthcited by 419
  6. 2022Matryoshka Representation Learningcited by 282
  7. 2024SVD-LLM: Truncation-aware Singular Value Decomposition for Large Language Model Compressioncited by 243
  8. 2018Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learningcited by 215
  9. 2022Language model compression with weighted low-rank factorizationcited by 195
  10. 2022Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapsecited by 141
  11. 2023LoSparse: Structured Compression of Large Language Models based on Low-Rank and Sparse Approximationcited by 104
  12. 2024Loki: Low-rank Keys for Efficient Sparse Attentioncited by 89
4 more
  1. 2024Eigen Attention: Attention in Low-Rank Space for KV Cache Compressioncited by 72
  2. 2024Palu: Compressing KV-Cache with Low-Rank Projectioncited by 64
  3. 2024On the Role of Attention Masks and LayerNorm in Transformerscited by 48
  4. 2025Eigenspectrum Analysis of Neural Networks without Aspect Ratio Biascited by 10

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