# A geometric framework for momentum-based optimizers for low-rank training

arXiv: https://arxiv.org/abs/2506.17475 · HTML: https://mlsanity.com/paper/2506.17475/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 44 references that are in the map.
- First posted on arXiv 20 Jun 2025.

## Builds on (most cited first, 28 of 28)

- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [DeBERTaV3: Improving DeBERTa using ELECTRA-Style Pre-Training with Gradient-Disentangled Embedding Sharing](https://mlsanity.com/paper/2111.09543/) (2021)
- [BitFit: Simple Parameter-efficient Fine-tuning for Transformer-based Masked Language-models](https://mlsanity.com/paper/2106.10199/) (2021)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [AdapterFusion: Non-Destructive Task Composition for Transfer Learning](https://mlsanity.com/paper/2005.00247/) (2020)
- [Channel Pruning for Accelerating Very Deep Neural Networks](https://mlsanity.com/paper/1707.06168/) (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)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)
- [Dynamic Network Surgery for Efficient DNNs](https://mlsanity.com/paper/1608.04493/) (2016)
- [DyLoRA: Parameter Efficient Tuning of Pre-trained Models using Dynamic Search-Free Low-Rank Adaptation](https://mlsanity.com/paper/2210.07558/) (2022)
- [Quantized Convolutional Neural Networks for Mobile Devices](https://mlsanity.com/paper/1512.06473/) (2015)
- [ReLoRA: High-Rank Training Through Low-Rank Updates](https://mlsanity.com/paper/2307.05695/) (2023)
- [Pufferfish: Communication-efficient Models At No Extra Cost](https://mlsanity.com/paper/2103.03936/) (2021)
- [Initialization and Regularization of Factorized Neural Layers](https://mlsanity.com/paper/2105.01029/) (2021)
- [Analytic Insights into Structure and Rank of Neural Network Hessian Maps](https://mlsanity.com/paper/2106.16225/) (2021)
- [Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations](https://mlsanity.com/paper/2205.13571/) (2022)
- [DoRA: Enhancing Parameter-Efficient Fine-Tuning with Dynamic Rank Distribution](https://mlsanity.com/paper/2405.17357/) (2024)
- [Robust low-rank training via approximate orthonormal constraints](https://mlsanity.com/paper/2306.01485/) (2023)
- [Geometry-aware training of factorized layers in tensor Tucker format](https://mlsanity.com/paper/2305.19059/) (2023)
- [Federated Dynamical Low-Rank Training with Global Loss Convergence Guarantees](https://mlsanity.com/paper/2406.17887/) (2024)
- [Dynamical Low-Rank Compression of Neural Networks with Robustness under Adversarial Attacks](https://mlsanity.com/paper/2505.08022/) (2025)
- [GeoLoRA: Geometric integration for parameter efficient fine-tuning](https://mlsanity.com/paper/2410.18720/) (2024)
- [Harnessing Orthogonality to Train Low-Rank Neural Networks](https://mlsanity.com/paper/2401.08505/) (2024)
- [An Augmented Backward-Corrected Projector Splitting Integrator for Dynamical Low-Rank Training](https://mlsanity.com/paper/2502.03006/) (2025)

## Built on by (most cited first, 5 of 5)

- [Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds](https://mlsanity.com/paper/2605.09238/) (2026)
- [Mitigating Subject Dependency in EEG Decoding with Subject-Specific Low-Rank Adapters](https://mlsanity.com/paper/2510.08059/) (2025)
- [Compressing Vision Transformers in Geospatial Transfer Learning with Manifold-Constrained Optimization](https://mlsanity.com/paper/2601.08882/) (2026)
- [Stiefel-AdamW: Geometry-Aware AdamW for Linear Factorization Blocks](https://mlsanity.com/paper/2609.21039/) (2026)
- [RECLAIM: Can Agents Reproduce the Claims of Machine Learning Papers?](https://mlsanity.com/paper/2609.28850/) (2026)

## Abstract (arXiv)

Low-rank pre-training and fine-tuning have recently emerged as promising techniques for reducing the computational and storage costs of large neural networks. Training low-rank parameterizations typically relies on conventional optimizers such as heavy ball momentum methods or Adam. In this work, we identify and analyze potential difficulties that these training methods encounter when used to train low-rank parameterizations of weights. In particular, we show that classical momentum methods can struggle to converge to a local optimum due to the geometry of the underlying optimization landscape. To address this, we introduce novel training strategies derived from dynamical low-rank approximation, which explicitly account for the underlying geometric structure. Our approach leverages and combines tools from dynamical low-rank approximation and momentum-based optimization to design optimizers that respect the intrinsic geometry of the parameter space. We validate our methods through numerical experiments, demonstrating faster convergence, and stronger validation metrics at given parameter budgets.