# PoLAR: Polar-Decomposed Low-Rank Adapter Representation

arXiv: https://arxiv.org/abs/2506.03133 · HTML: https://mlsanity.com/paper/2506.03133/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 74 of its 101 references that are in the map.
- First posted on arXiv 3 Jun 2025.

## Builds on (most cited first, 30 of 74)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [The Power of Scale for Parameter-Efficient Prompt Tuning](https://mlsanity.com/paper/2104.08691/) (2021)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference](https://mlsanity.com/paper/1704.05426/) (2017)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [DeBERTa: Decoding-enhanced BERT with Disentangled Attention](https://mlsanity.com/paper/2006.03654/) (2020)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Know What You Don't Know: Unanswerable Questions for SQuAD](https://mlsanity.com/paper/1806.03822/) (2018)
- [SemEval-2017 Task 1: Semantic Textual Similarity - Multilingual and Cross-lingual Focused Evaluation](https://mlsanity.com/paper/1708.00055/) (2017)
- [Neural Network Acceptability Judgments](https://mlsanity.com/paper/1805.12471/) (2018)
- [Few-Shot Parameter-Efficient Fine-Tuning is Better and Cheaper than In-Context Learning](https://mlsanity.com/paper/2205.05638/) (2022)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [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)
- [PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2404.02948/) (2024)
- [Mix-of-Show: Decentralized Low-Rank Adaptation for Multi-Concept Customization of Diffusion Models](https://mlsanity.com/paper/2305.18292/) (2023)
- [A Rank Stabilization Scaling Factor for Fine-Tuning with LoRA](https://mlsanity.com/paper/2312.03732/) (2023)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)

## Built on by (most cited first, 20 of 20)

- [StelLA: Subspace Learning in Low-rank Adaptation using Stiefel Manifold](https://mlsanity.com/paper/2510.01938/) (2025)
- [Powering Up Zeroth-Order Training via Subspace Gradient Orthogonalization](https://mlsanity.com/paper/2602.17155/) (2026)
- [ANCRe: Adaptive Neural Connection Reassignment for Efficient Depth Scaling](https://mlsanity.com/paper/2602.09009/) (2026)
- [Calibrating and Rotating: A Unified Framework for Weight Conditioning in PEFT](https://mlsanity.com/paper/2511.00051/) (2025)
- [On the Benefits of Weight Normalization for Overparameterized Matrix Sensing](https://mlsanity.com/paper/2510.01175/) (2025)
- [Low-Rank Adaptation Redux for Large Models](https://mlsanity.com/paper/2604.21905/) (2026)
- [Balanced LoRA: Removing Parameter Invariance to Accelerate Convergence](https://mlsanity.com/paper/2605.31484/) (2026)
- [Improving Neural Network Training by Decoupling the Magnitude and Direction of Weight Vectors](https://mlsanity.com/paper/2606.25971/) (2026)
- [Scalable Variational Bayesian Fine-Tuning of LLMs via Orthogonalized Low-Rank Adapters](https://mlsanity.com/paper/2604.03388/) (2026)
- [Same Architecture, Different Capacity: Optimizer-Induced Spectral Scaling Laws](https://mlsanity.com/paper/2605.21803/) (2026)
- [ScaLoRA: Optimally Scaled Low-Rank Adaptation for Efficient High-Rank Fine-Tuning](https://mlsanity.com/paper/2510.23818/) (2025)
- [SALAAD: Sparse And Low-Rank Adaptation via ADMM for Large Language Model Inference](https://mlsanity.com/paper/2602.00942/) (2026)
- [SDS-LoRA: Overcoming Anisotropic Gradient Scaling in Low-Rank Adaptation](https://mlsanity.com/paper/2606.16454/) (2026)
- [Automatic Rank Determination for Low-Rank Adaptation via Submodular Function Maximization](https://mlsanity.com/paper/2507.01841/) (2025)
- [Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics](https://mlsanity.com/paper/2606.31390/) (2026)
- [Muon on the Stiefel Manifold Admits an Exact Closed-Form Update](https://mlsanity.com/paper/2608.06218/) (2026)
- [Behaviorally Effective LoRA Writes Are Sparse and Structured](https://mlsanity.com/paper/2609.01374/) (2026)
- [Geometry-Aware Bayesian Parameter-Efficient Fine-Tuning on the Stiefel Manifold via Stein Variational Gradient Descent](https://mlsanity.com/paper/2609.08354/) (2026)
- [Behavior Quotient Learning for Low-Rank Adaptation of LLM Agents](https://mlsanity.com/paper/2609.12896/) (2026)
- [LoRA's Second Descent Extends Beyond Parameter Parity](https://mlsanity.com/paper/2610.04507/) (2026)

## Abstract (arXiv)

We show that low-rank adaptation of large-scale models suffers from a low stable rank that is well below the linear algebraic rank of the subspace, degrading fine-tuning performance. To mitigate the underutilization of the allocated subspace, we propose PoLAR, a parameterization inspired by the polar decomposition that factorizes the low-rank update into two direction matrices constrained to Stiefel manifolds and an unconstrained scale matrix. Our theory shows that PoLAR yields an exponentially faster convergence rate on a canonical low-rank adaptation problem. Pairing the parameterization with Riemannian optimization leads to consistent gains on three different benchmarks testing general language understanding, commonsense reasoning, and mathematical problem solving with base model sizes ranging from 350M to 27B.