# Small Singular Values Matter: A Random Matrix Analysis of Transformer Models

arXiv: https://arxiv.org/abs/2410.17770 · HTML: https://mlsanity.com/paper/2410.17770/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 54 references that are in the map.
- First posted on arXiv 23 Oct 2024.

## Builds on (most cited first, 30 of 36)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [XLNet: Generalized Autoregressive Pretraining for Language Understanding](https://mlsanity.com/paper/1906.08237/) (2019)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [BLOOM: A 176B-Parameter Open-Access Multilingual Language Model](https://mlsanity.com/paper/2211.05100/) (2022)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [RULER: What's the Real Context Size of Your Long-Context Language Models?](https://mlsanity.com/paper/2404.06654/) (2024)
- [Aligning Books and Movies: Towards Story-like Visual Explanations by Watching Movies and Reading Books](https://mlsanity.com/paper/1506.06724/) (2015)
- [Discovering Language Model Behaviors with Model-Written Evaluations](https://mlsanity.com/paper/2212.09251/) (2022)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [SliceGPT: Compress Large Language Models by Deleting Rows and Columns](https://mlsanity.com/paper/2401.15024/) (2024)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [SVD-LLM: Truncation-aware Singular Value Decomposition for Large Language Model Compression](https://mlsanity.com/paper/2403.07378/) (2024)
- [Extreme Compression of Large Language Models via Additive Quantization](https://mlsanity.com/paper/2401.06118/) (2024)
- [Visualizing and Measuring the Geometry of BERT](https://mlsanity.com/paper/1906.02715/) (2019)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [Transformer Dissection: A Unified Understanding of Transformer's Attention via the Lens of Kernel](https://mlsanity.com/paper/1908.11775/) (2019)
- [Language model compression with weighted low-rank factorization](https://mlsanity.com/paper/2207.00112/) (2022)
- [The Truth is in There: Improving Reasoning in Language Models with Layer-Selective Rank Reduction](https://mlsanity.com/paper/2312.13558/) (2023)
- [Predicting trends in the quality of state-of-the-art neural networks without access to training or testing data](https://mlsanity.com/paper/2002.06716/) (2020)
- [Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra](https://mlsanity.com/paper/2008.11865/) (2020)
- [Compressing Large Language Models using Low Rank and Low Precision Decomposition](https://mlsanity.com/paper/2405.18886/) (2024)
- [Random matrix analysis of deep neural network weight matrices](https://mlsanity.com/paper/2203.14661/) (2022)

## Built on by (most cited first, 21 of 21)

- [RL Fine-Tuning Heals OOD Forgetting in SFT](https://mlsanity.com/paper/2509.12235/) (2025)
- [RL Is Neither a Panacea Nor a Mirage: Understanding Supervised vs. Reinforcement Learning Fine-Tuning for LLMs](https://mlsanity.com/paper/2508.16546/) (2025)
- [Dimensional Collapse in Transformer Attention Outputs: A Challenge for Sparse Dictionary Learning](https://mlsanity.com/paper/2508.16929/) (2025)
- [Single-Head Attention in High Dimensions: A Theory of Generalization, Weights Spectra, and Scaling Laws](https://mlsanity.com/paper/2509.24914/) (2025)
- [Towards Worst-Case Guarantees with Scale-Aware Interpretability](https://mlsanity.com/paper/2602.05184/) (2026)
- [Algebraic Dead Directions in LayerNorm Transformers: A Forward-Pass-Only Diagnostic at LLM Scale](https://mlsanity.com/paper/2606.19491/) (2026)
- [Task complexity shapes internal representations and robustness in neural networks](https://mlsanity.com/paper/2508.05463/) (2025)
- [SMoA: Spectrum Modulation Adapter for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2605.21147/) (2026)
- [Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity](https://mlsanity.com/paper/2606.21158/) (2026)
- [Theory of Minimal Weight Perturbations in Deep Networks and its Applications for Low-Rank Activated Backdoor Attacks](https://mlsanity.com/paper/2601.16880/) (2026)
- [Quantifying the Uncertainty of Foundation Models with Singular Value Ensembles](https://mlsanity.com/paper/2601.22068/) (2026)
- [Eigenvectors of Experts are Training-free Non-collapsing Routers](https://mlsanity.com/paper/2605.30992/) (2026)
- [ResidualDroppath: Enhancing Feature Reuse over Residual Connections](https://mlsanity.com/paper/2411.09475/) (2024)
- [Evaluating Singular Value Thresholds for DNN Weight Matrices based on Random Matrix Theory](https://mlsanity.com/paper/2512.12911/) (2025)
- [Concepts Whisper: Spectral Anti-Concentration and the Dual Geometry of Transformer Representations](https://mlsanity.com/paper/2605.01609/) (2026)
- [Foundation-Preserving Optimization in Generalized Eigenspace](https://mlsanity.com/paper/2606.00132/) (2026)
- [Complexity-Guided Component-wise Initialization for Language Model Pretraining](https://mlsanity.com/paper/2607.09204/) (2026)
- [Spectral Tail Interventions in Decoder-Only Language Models: Reasoning-Sensitive Weight Structure from Controlled Surgery](https://mlsanity.com/paper/2609.26165/) (2026)
- [Reuse or Relearn? A Spectral View of Earth Observation Foundation Models](https://mlsanity.com/paper/2609.32756/) (2026)
- [Pre-registered tests of solid-state-physics-inspired LLM compression: a cluster-level negative result at small-language-model scale](https://mlsanity.com/paper/2609.34292/) (2026)
- [Surfing the Activation Wave: Analyzing the Information and Geometry of LLM Hidden States for Speculative Decoding](https://mlsanity.com/paper/2610.12939/) (2026)

## Abstract (arXiv)

This work analyzes singular-value spectra of weight matrices in pretrained transformer models to understand how information is stored at both ends of the spectrum. Using Random Matrix Theory (RMT) as a zero information hypothesis, we associate agreement with RMT as evidence of randomness and deviations as evidence for learning. Surprisingly, we observe pronounced departures from RMT not only among the largest singular values -- the usual outliers -- but also among the smallest ones. A comparison of the associated singular vectors with the eigenvectors of the activation covariance matrices shows that there is considerable overlap wherever RMT is violated. Thus, significant directions in the data are captured by small singular values and their vectors as well as by the large ones. We confirm this empirically: zeroing out the singular values that deviate from RMT raises language-model perplexity far more than removing values from the bulk, and after fine-tuning the smallest decile can be the third most influential part of the spectrum. To explain how vectors linked to small singular values can carry more information than those linked to larger values, we propose a linear random-matrix model. Our findings highlight the overlooked importance of the low end of the spectrum and provide theoretical and practical guidance for SVD-based pruning and compression of large language models.