# From Low Rank Gradient Subspace Stabilization to Low-Rank Weights: Observations, Theories, and Applications

arXiv: https://arxiv.org/abs/2407.11239 · HTML: https://mlsanity.com/paper/2407.11239/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 46 references that are in the map.
- First posted on arXiv 15 Jul 2024.

## Builds on (most cited first, 30 of 38)

- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [SparseGPT: Massive Language Models Can Be Accurately Pruned in One-Shot](https://mlsanity.com/paper/2301.00774/) (2023)
- [A Simple and Effective Pruning Approach for Large Language Models](https://mlsanity.com/paper/2306.11695/) (2023)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [LoRA+: Efficient Low Rank Adaptation of Large Models](https://mlsanity.com/paper/2402.12354/) (2024)
- [LoRA Learns Less and Forgets Less](https://mlsanity.com/paper/2405.09673/) (2024)
- [Factual Probing Is [MASK]: Learning vs. Learning to Recall](https://mlsanity.com/paper/2104.05240/) (2021)
- [SVD-LLM: Truncation-aware Singular Value Decomposition for Large Language Model Compression](https://mlsanity.com/paper/2403.07378/) (2024)
- [WARP: Word-level Adversarial ReProgramming](https://mlsanity.com/paper/2101.00121/) (2021)
- [Language model compression with weighted low-rank factorization](https://mlsanity.com/paper/2207.00112/) (2022)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [ASVD: Activation-aware Singular Value Decomposition for Compressing Large Language Models](https://mlsanity.com/paper/2312.05821/) (2023)
- [Outlier Weighed Layerwise Sparsity (OWL): A Missing Secret Sauce for Pruning LLMs to High Sparsity](https://mlsanity.com/paper/2310.05175/) (2023)
- [S-LoRA: Serving Thousands of Concurrent LoRA Adapters](https://mlsanity.com/paper/2311.03285/) (2023)
- [LoRA-FA: Efficient and Effective Low Rank Representation Fine-tuning](https://mlsanity.com/paper/2308.03303/) (2023)
- [Flora: Low-Rank Adapters Are Secretly Gradient Compressors](https://mlsanity.com/paper/2402.03293/) (2024)
- [LoSparse: Structured Compression of Large Language Models based on Low-Rank and Sparse Approximation](https://mlsanity.com/paper/2306.11222/) (2023)
- [Chain of LoRA: Efficient Fine-tuning of Language Models via Residual Learning](https://mlsanity.com/paper/2401.04151/) (2024)
- [A Thorough Examination of the CNN/Daily Mail Reading Comprehension Task](https://mlsanity.com/paper/1606.02858/) (2016)
- [Black-box Prompt Learning for Pre-trained Language Models](https://mlsanity.com/paper/2201.08531/) (2022)
- [Tied-Lora: Enhancing parameter efficiency of LoRA with weight tying](https://mlsanity.com/paper/2311.09578/) (2023)
- [Understanding Self-supervised Learning with Dual Deep Networks](https://mlsanity.com/paper/2010.00578/) (2020)
- [Compressing LLMs: The Truth is Rarely Pure and Never Simple](https://mlsanity.com/paper/2310.01382/) (2023)
- [FreezeOut: Accelerate Training by Progressively Freezing Layers](https://mlsanity.com/paper/1706.04983/) (2017)
- [Revealing the Structure of Deep Neural Networks via Convex Duality](https://mlsanity.com/paper/2002.09773/) (2020)
- [Implicit Regularization Towards Rank Minimization in ReLU Networks](https://mlsanity.com/paper/2201.12760/) (2022)
- [AutoFreeze: Automatically Freezing Model Blocks to Accelerate Fine-tuning](https://mlsanity.com/paper/2102.01386/) (2021)
- [Multi-Dimensional Evaluation of Text Summarization with In-Context Learning](https://mlsanity.com/paper/2306.01200/) (2023)

## Built on by (most cited first, 25 of 25)

- [LDAdam: Adaptive Optimization from Low-Dimensional Gradient Statistics](https://mlsanity.com/paper/2410.16103/) (2024)
- [Flow caching for autoregressive video generation](https://mlsanity.com/paper/2602.10825/) (2026)
- [R-Sparse: Rank-Aware Activation Sparsity for Efficient LLM Inference](https://mlsanity.com/paper/2504.19449/) (2025)
- [Finding Fantastic Experts in MoEs: A Unified Study for Expert Dropping Strategies and Observations](https://mlsanity.com/paper/2504.05586/) (2025)
- [GaLore 2: Large-Scale LLM Pre-Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2504.20437/) (2025)
- [LIFT the Veil for the Truth: Principal Weights Emerge after Rank Reduction for Reasoning-Focused Supervised Fine-Tuning](https://mlsanity.com/paper/2506.00772/) (2025)
- [LittleBit: Ultra Low-Bit Quantization via Latent Factorization](https://mlsanity.com/paper/2506.13771/) (2025)
- [An Overview of Low-Rank Structures in the Training and Adaptation of Large Models](https://mlsanity.com/paper/2503.19859/) (2025)
- [CE-LoRA: Computation-Efficient LoRA Fine-Tuning for Language Models](https://mlsanity.com/paper/2502.01378/) (2025)
- [Position: Weight Space Should Be a First-Class Generative AI Modality](https://mlsanity.com/paper/2605.18632/) (2026)
- [Deep Hierarchical Learning with Nested Subspace Networks for Large Language Models](https://mlsanity.com/paper/2509.17874/) (2025)
- [MemoryLLM: Plug-n-Play Interpretable Feed-Forward Memory for Transformers](https://mlsanity.com/paper/2602.00398/) (2026)
- [POET-X: Memory-efficient LLM Training by Scaling Orthogonal Transformation](https://mlsanity.com/paper/2603.05500/) (2026)
- [Memory-Efficient Fine-Tuning via Low-Rank Activation Compression](https://mlsanity.com/paper/2509.23472/) (2025)
- [From Memorization to Reasoning in the Spectrum of Loss Curvature](https://mlsanity.com/paper/2510.24256/) (2025)
- [Memory-Efficient LLM Training by Various-Grained Low-Rank Projection of Gradients](https://mlsanity.com/paper/2505.01744/) (2025)
- [AdamO: A Collapse-Suppressed Optimizer for Offline RL](https://mlsanity.com/paper/2605.01968/) (2026)
- [Adaptive Selection of LoRA Components in Privacy-Preserving Federated Learning](https://mlsanity.com/paper/2605.05769/) (2026)
- [Pro-KLShampoo: Projected KL-Shampoo with Whitening Recovered by Orthogonalization](https://mlsanity.com/paper/2605.06316/) (2026)
- [SLORR: Simple and Efficient In-Training Low-Rank Regularization](https://mlsanity.com/paper/2607.08754/) (2026)
- [Emergent Low-Rank Training Dynamics in MLPs with Smooth Activations](https://mlsanity.com/paper/2602.06208/) (2026)
- [From O(mn) to O(r2): Two-Sided Low-Rank Communication for Adam in Distributed Training with Memory Efficiency](https://mlsanity.com/paper/2602.08007/) (2026)
- [Understanding the Subspace Stabilization of the Hessian and Gradient Covariance Matrix](https://mlsanity.com/paper/2609.31983/) (2026)
- [AIM-ZO: Activation-Informed Subspace Maintenance for Zeroth-Order LLM Fine-Tuning](https://mlsanity.com/paper/2609.35257/) (2026)
- [DyRA: Dynamic Residual Approximation for Efficient Matrix Multiplication in DNNs](https://mlsanity.com/paper/2610.02882/) (2026)

## Abstract (arXiv)

Large Language Models' (LLMs) weight matrices can often be expressed in low-rank form with potential to relax memory and compute resource requirements. Unlike prior efforts that focus on developing novel matrix decompositions, in this work we study the non-uniform low-rank properties of weight matrices in LLMs through the lens of stabilizing gradient subspace. First, we provide a theoretical framework to understand the stabilization of gradient subspaces through Hessian analysis. Second, we empirically establish an important relationship between gradient dynamics and low-rank expressiveness of weight matrices. Our findings reveal that different LLM components exhibit varying levels of converged low-rank structures, necessitating variable rank reduction across them to minimize drop in performance due to compression. Drawing on this result, we present Weight Low-Rank Projection(WeLore) that unifies weight compression and memory-efficient fine-tuning into one, in a data-agnostic and one-shot manner. When used as a compression technique, WeLore categorizes weight matrices into Low-rank Components (LRCs) and Non-Low-rank Components (N-LRCs) and suitably encodes them for minimum performance loss. Our gradient dynamics perspective illustrates that LRCs tend to have better fine-tuning capabilities and their standalone fine-tuning can closely mimic and sometimes outperform the training loss trajectory and performance of full fine-tuning with notable memory and compute footprint reduction. Codes are available at https://github.com/VITA-Group/WeLore.