# Frozen Factor or Spectral Band? Disentangling Two Choices in Low-Rank LoRA

arXiv: https://arxiv.org/abs/2610.06621 · HTML: https://mlsanity.com/paper/2610.06621/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 34 references that are in the map.
- First posted on arXiv 5 Oct 2026.

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

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [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)
- [VeRA: Vector-based Random Matrix Adaptation](https://mlsanity.com/paper/2310.11454/) (2023)
- [PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2404.02948/) (2024)
- [LoRA-GA: Low-Rank Adaptation with Gradient Approximation](https://mlsanity.com/paper/2407.05000/) (2024)
- [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)
- [MiLoRA: Harnessing Minor Singular Components for Parameter-Efficient LLM Finetuning](https://mlsanity.com/paper/2406.09044/) (2024)
- [Asymmetry in Low-Rank Adapters of Foundation Models](https://mlsanity.com/paper/2402.16842/) (2024)
- [LoRA-XS: Low-Rank Adaptation with Extremely Small Number of Parameters](https://mlsanity.com/paper/2405.17604/) (2024)
- [SVFT: Parameter-Efficient Fine-Tuning with Singular Vectors](https://mlsanity.com/paper/2405.19597/) (2024)
- [The Impact of Initialization on LoRA Finetuning Dynamics](https://mlsanity.com/paper/2406.08447/) (2024)
- [CorDA: Context-Oriented Decomposition Adaptation of Large Language Models for Task-Aware Parameter-Efficient Fine-tuning](https://mlsanity.com/paper/2406.05223/) (2024)
- [LoRA-One: One-Step Full Gradient Could Suffice for Fine-Tuning Large Language Models, Provably and Efficiently](https://mlsanity.com/paper/2502.01235/) (2025)
- [Initialization using Update Approximation is a Silver Bullet for Extremely Efficient Low-Rank Fine-Tuning](https://mlsanity.com/paper/2411.19557/) (2024)
- [Parameter Efficient Fine-tuning via Explained Variance Adaptation](https://mlsanity.com/paper/2410.07170/) (2024)
- [Sharp Generalization Bounds for Foundation Models with Asymmetric Randomized Low-Rank Adapters](https://mlsanity.com/paper/2506.14530/) (2025)
- [PiCa: Parameter-Efficient Fine-Tuning with Column Space Projection](https://mlsanity.com/paper/2505.20211/) (2025)
- [SSVD: Structured SVD for Parameter-Efficient Fine-Tuning and Benchmarking under Domain Shift in ASR](https://mlsanity.com/paper/2509.02830/) (2025)
- [Beware of the Batch Size: Hyperparameter Bias in Evaluating LoRA](https://mlsanity.com/paper/2602.09492/) (2026)
- [Least but not Last: Fine-tuning Intermediate Principal Components for Better Performance-Forgetting Trade-Offs](https://mlsanity.com/paper/2602.03493/) (2026)
- [MiCA Learns More Knowledge Than LoRA and Full Fine-Tuning](https://mlsanity.com/paper/2604.01694/) (2026)
- [Bilinear Optimization Divergence: Diagnosing Factor-Constrained LoRA Continual Learning](https://mlsanity.com/paper/2609.23594/) (2026)

## Abstract (arXiv)

Spectral variants of low-rank adaptation (LoRA) choose both a subspace and which factor to freeze. We separate these choices by freezing the input factor A or output factor B on the top or bottom singular directions of pretrained weights, with learning rates selected separately. At rank 2, the same-band advantage of freezing A is larger than either within-factor band difference on all four task-model pairs with complete comparisons. Freezing B also trails comparable-budget free LoRA by 8-18 percentage points on five pairs spanning a formatting task and OpenBookQA. The A-frozen advantage persists in a single-GPU-model replication and within individual MLP module groups, including controls with equal or greater trainable counts for B frozen, and when A is frozen on a random orthonormal basis. The factor contrast weakens with rank. On OpenBookQA / Qwen2.5-1.5B at rank 16, PEFT's MiCA implementation trails comparable-budget LoRA by 3.08 points under a shared training recipe transferred from the MiCA paper. A trained oracle output subspace largely removes the low-rank deficit; partial warm-up gains recur across three direction seeds. The factor-versus-band ordering is descriptive; an approximate multiplicity audit weakens several earlier significance claims. These results extend known factor asymmetry by showing how its magnitude depends on spectral placement, rank and training conditions.