# Pretraining Induces a Reusable Spectral Basis for Downstream Task Adaptation

arXiv: https://arxiv.org/abs/2605.07302 · HTML: https://mlsanity.com/paper/2605.07302/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 31 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 26 of 26)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [BitFit: Simple Parameter-efficient Fine-tuning for Transformer-based Masked Language-models](https://mlsanity.com/paper/2106.10199/) (2021)
- [How transferable are features in deep neural networks?](https://mlsanity.com/paper/1411.1792/) (2014)
- [Towards a Unified View of Parameter-Efficient Transfer Learning](https://mlsanity.com/paper/2110.04366/) (2021)
- [AdapterFusion: Non-Destructive Task Composition for Transfer Learning](https://mlsanity.com/paper/2005.00247/) (2020)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey](https://mlsanity.com/paper/2403.14608/) (2024)
- [Scaling Data-Constrained Language Models](https://mlsanity.com/paper/2305.16264/) (2023)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [What is being transferred in transfer learning?](https://mlsanity.com/paper/2008.11687/) (2020)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)
- [Large Brain Model for Learning Generic Representations with Tremendous EEG Data in BCI](https://mlsanity.com/paper/2405.18765/) (2024)
- [Scaling Laws for Transfer](https://mlsanity.com/paper/2102.01293/) (2021)
- [PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2404.02948/) (2024)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [Traditional and Heavy-Tailed Self Regularization in Neural Network Models](https://mlsanity.com/paper/1901.08276/) (2019)
- [Brain-JEPA: Brain Dynamics Foundation Model with Gradient Positioning and Spatiotemporal Masking](https://mlsanity.com/paper/2409.19407/) (2024)
- [SVFT: Parameter-Efficient Fine-Tuning with Singular Vectors](https://mlsanity.com/paper/2405.19597/) (2024)
- [Fine-tuning Happens in Tiny Subspaces: Exploring Intrinsic Task-specific Subspaces of Pre-trained Language Models](https://mlsanity.com/paper/2305.17446/) (2023)
- [Omni-fMRI: A Universal Atlas-Free fMRI Foundation Model](https://mlsanity.com/paper/2601.23090/) (2026)
- [Brain-DiT: A Universal Multi-state fMRI Foundation Model with Metadata-Conditioned Pretraining](https://mlsanity.com/paper/2604.12683/) (2026)
- [Few Dimensions are Enough: Fine-tuning BERT with Selected Dimensions Revealed Its Redundant Nature](https://mlsanity.com/paper/2504.04966/) (2025)

## Built on by (most cited first, 1 of 1)

- [Logic Before Language: Pre-pretraining on Formal Derivations Fosters Skill Acquisition and Compressibility](https://mlsanity.com/paper/2608.03930/) (2026)

## Abstract (arXiv)

Finetuning pretrained models occurs in a low-dimensional subspace of the full parameter space. Prior work has focused on characterizing this optimization subspace, but largely ignored the complementary question: why do certain directions remain unexplored during finetuning? Are these stable directions irrelevant to downstream tasks, or do they already encode task-relevant structure that requires no further adjustment? Answering this question is central to understanding how pretrained knowledge transfers. Through systematic spectral analysis across vision and language models, we show that the leading singular vectors of pretrained weight matrices remain highly stable under finetuning and are shared across unrelated downstream tasks, revealing that pretraining establishes a reusable spectral coordinate system. Models pretrained on larger datasets exhibit greater spectral stability under distribution shift or task change, directly linking pretraining scale to geometric transferability. Motivated by these findings, we propose a parameter-efficient method that freezes pretrained singular vectors and optimizes only leading spectral coefficients, achieving competitive performance on GLUE with 0.2% trainable parameters. Our results reveal that the stable directions encode transferable structure rather than irrelevant noise: successful pretraining discovers spectral bases that downstream tasks inherit and operate within.