arxiv: 2605.01627 · v2 · submitted 2026-05-02 · 💻 cs.LG

Recognition: unknown

Importance-Guided Basis Selection for Low-Rank Decomposition of Large Language Models

Daniel Agyei Asante, Ernie Chang, Yang Li

Authors on Pith no claims yet

Pith reviewed 2026-05-09 14:28 UTC · model grok-4.3

classification 💻 cs.LG

keywords low-rank decompositionlarge language modelsmodel compressionbasis selectionimportance scoringHessian estimationTaylor expansion

0 comments

The pith

A method ranks singular-vector bases in LLM low-rank compression by estimating how much each removal would increase task loss, using second-order loss curvature.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Basis Selection with Importance (BSI) to improve low-rank decomposition of large language models. Existing approaches prune bases based on re-learned magnitudes after adaptation, but this can misalign with actual task performance. BSI instead computes an importance score for each basis by approximating the expected increase in task loss if that basis is removed, derived from a second-order Taylor expansion of the loss with respect to singular values. An efficient estimator adapts the Hutchinson method to compute the necessary diagonal of the Hessian matrix. Experiments show BSI achieves better compression performance than prior methods, particularly when models are compressed deeply on mathematical reasoning tasks.

Core claim

BSI ranks and prunes singular-vector bases by directly estimating the loss increase from removal via a second-order Taylor expansion of the task loss, combined with an efficient randomized Hessian-diagonal estimator, leading to superior low-rank compression of LLMs compared to magnitude-based heuristics.

What carries the argument

The derivative-based importance score obtained from the second-order Taylor expansion of task loss with respect to singular values, estimated using a symmetric-perturbation adaptation of the Hutchinson method for the Hessian diagonal.

If this is right

Loss-increase bounds are derived that account for both pruning and the error in the Hessian-diagonal estimate.
High-probability sample-complexity guarantees are provided for reaching a target accuracy in the importance scores.
Variance of the estimator is characterized in terms of the spectrum of the Hessian.
Explicit guidance is given on choosing the intensity of the symmetric parameter perturbations.
Performance gains are largest under deep compression on mathematical reasoning benchmarks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same curvature-based ranking could be tested on other model families or non-reasoning tasks to check whether the advantage holds beyond the reported setting.
If the scores are reliable, they might allow compression steps to be inserted earlier in the training pipeline without separate adaptation phases.
The method supplies a concrete way to trade off estimation accuracy against compute by varying the number of Hutchinson probes.

Load-bearing premise

The second-order Taylor expansion of the task loss with respect to singular values provides an accurate estimate of the actual loss increase incurred by basis pruning in large language models.

What would settle it

Measure the actual task loss increase after removing one specific basis and compare it to the importance score that BSI assigns to that basis; large mismatches would show the ranking fails to predict real impact.

Figures

Figures reproduced from arXiv: 2605.01627 by Daniel Agyei Asante, Ernie Chang, Yang Li.

**Figure 1.** Figure 1: Accuracy and model size of Llama 2-7B compressed with various low-rank decomposition view at source ↗

**Figure 2.** Figure 2: Hessian eigenvalue spectrum in the reparameterized singular-value space for math-finetuned view at source ↗

**Figure 3.** Figure 3: Accuracy and model size of Llama 2-7B compressed using BSI and its variant without the view at source ↗

read the original abstract

Low-rank decomposition is a compelling approach for compressing large language models, but its effectiveness hinges on selecting which singular-vector bases to retain for a target task. Existing methods such as Basel adapt singular-value coefficients on downstream data and prune bases with small re-learned magnitudes, a heuristic that can be misaligned with task performance because it ignores the local geometry of the loss landscape. We present Basis Selection with Importance (BSI), a principled low-rank compression framework that ranks and prunes bases by directly estimating the expected loss increase incurred when each basis is removed. BSI derives a derivative-based importance score from a second-order Taylor expansion of the task loss with respect to singular values, combining first-order sensitivity and second-order curvature to quantify pruning impact. To make this criterion practical for LLMs, we develop an efficient Hessian-diagonal estimator by adapting the Hutchinson randomized-probing method to loss curvature with symmetric parameter perturbations. We provide a comprehensive theoretical analysis, including loss-increase bounds under basis pruning, explicit propagation of Hessian-diagonal estimation error into these bounds, variance characterization tied to the Hessian spectrum, high-probability sample-complexity guarantees for achieving a target estimation accuracy, and guidance on perturbation intensity. Extensive experiments on mathematical reasoning benchmarks demonstrate that BSI consistently outperforms state-of-the-art low-rank decomposition baselines, with especially strong improvements under deep compression.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

BSI uses a second-order Taylor expansion plus adapted Hutchinson estimator to rank bases by predicted loss increase, with reported gains on math benchmarks but the approximation's reliability under heavy pruning still needs direct checks.

read the letter

The main takeaway is that this paper gives a more direct way to pick which singular bases to keep during low-rank LLM compression. Instead of pruning by small re-learned magnitudes after adaptation, BSI scores each basis by the expected rise in task loss from a second-order Taylor expansion of the loss with respect to the singular values, then estimates the needed Hessian diagonal efficiently with a symmetric-perturbation version of Hutchinson's method. That is the actual new piece, and the derivation plus the error-propagation bounds are cleanly laid out. The experiments show consistent improvements over the cited baselines on mathematical reasoning tasks, and the advantage grows as compression gets deeper, which matches the claim that the loss-aware ranking helps most when many bases are removed. The theoretical sample-complexity guarantees and guidance on perturbation size are also useful additions for anyone trying to implement something similar. The soft spot is exactly the one the stress-test flags: the quadratic approximation is local, yet the strongest results are in the deep-compression regime where the perturbation is no longer small. The paper supplies bounds and variance characterizations, but it does not appear to report a direct correlation between the predicted loss deltas and the measured post-pruning loss (before or after any re-optimization). Without that check, it is hard to tell how much of the reported outperformance comes from better ranking versus other implementation details. All evaluation is on math-reasoning benchmarks, so generalization to other domains is still open. This is the kind of targeted, technically grounded compression paper that deserves a serious referee. The method is honest about its approximations and the authors have done the homework on the estimator, so an editor should send it out rather than desk-reject it. Reviewers can then verify the empirical validation and see whether the gains survive broader testing.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes Basis Selection with Importance (BSI) for low-rank decomposition of large language models. It derives importance scores for singular-vector bases from a second-order Taylor expansion of the task loss with respect to singular values, uses an adapted Hutchinson randomized-probing estimator for the Hessian diagonal, supplies theoretical loss-increase bounds with error propagation and sample-complexity guarantees, and reports experiments on mathematical reasoning benchmarks showing consistent outperformance over baselines, especially under deep compression.

Significance. If the importance scores prove to accurately predict pruning effects, BSI would replace heuristic magnitude-based selection with a loss-landscape-aware criterion, offering a more principled compression approach. The theoretical components (bounds, Hessian error propagation, variance characterization) add rigor, and the reported gains at high compression ratios on math tasks suggest practical value for deploying compressed LLMs.

major comments (2)

[Abstract and experimental evaluation] Abstract and experimental evaluation: The central claim that BSI outperforms SOTA baselines (with strongest gains in deep compression) depends on the Taylor-derived importance scores correctly ranking bases by actual downstream impact. The manuscript supplies loss-increase bounds and Hessian-diagonal error propagation but does not report a direct correlation between the estimated ΔL and measured task loss after basis removal (before re-optimization). This validation is load-bearing, as the local-quadratic assumption is violated when perturbation sizes are large.
[Theoretical analysis] Theoretical analysis (loss-increase bounds and estimator guarantees): While high-probability sample-complexity results and perturbation-intensity guidance are given, these do not automatically ensure that ranking errors from the Hutchinson estimator remain below the threshold needed to preserve the observed outperformance; an empirical check tying predicted rankings to actual performance degradation is required to confirm the bounds translate to practice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address the major comments point by point below. Where the comments identify missing empirical validations, we have incorporated the requested analyses into the revised manuscript.

read point-by-point responses

Referee: [Abstract and experimental evaluation] Abstract and experimental evaluation: The central claim that BSI outperforms SOTA baselines (with strongest gains in deep compression) depends on the Taylor-derived importance scores correctly ranking bases by actual downstream impact. The manuscript supplies loss-increase bounds and Hessian-diagonal error propagation but does not report a direct correlation between the estimated ΔL and measured task loss after basis removal (before re-optimization). This validation is load-bearing, as the local-quadratic assumption is violated when perturbation sizes are large.

Authors: We agree that directly reporting the correlation between the estimated ΔL (from the second-order Taylor expansion) and the measured task loss increase after basis removal—prior to any re-optimization—would strengthen the central claim, particularly given potential violations of the local quadratic assumption at larger perturbation scales. While the consistent outperformance on mathematical reasoning benchmarks provides indirect support, this explicit validation is a valuable addition. In the revised manuscript we have added a new experimental subsection containing scatter plots, Pearson and Spearman correlation coefficients, and quantitative analysis of predicted versus observed loss changes across multiple models, tasks, and compression ratios. We also include a brief discussion of how the observed correlations relate to the validity of the quadratic approximation. revision: yes
Referee: [Theoretical analysis] Theoretical analysis (loss-increase bounds and estimator guarantees): While high-probability sample-complexity results and perturbation-intensity guidance are given, these do not automatically ensure that ranking errors from the Hutchinson estimator remain below the threshold needed to preserve the observed outperformance; an empirical check tying predicted rankings to actual performance degradation is required to confirm the bounds translate to practice.

Authors: We thank the referee for emphasizing the importance of verifying that the theoretical guarantees on estimation accuracy translate into reliable ranking decisions that preserve the reported performance gains. The high-probability bounds and sample-complexity results characterize estimator variance, yet an empirical link to ranking fidelity and downstream degradation is indeed needed. The revised manuscript now includes an additional empirical study that measures the correlation between BSI-predicted base rankings and actual post-pruning performance degradation. This analysis reports ranking stability metrics, the effect of Hutchinson estimator variance on final task accuracy, and confirmation that ranking errors remain sufficiently small to maintain the observed advantages over baselines. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The core derivation of the BSI importance score proceeds directly from the standard second-order Taylor expansion of the task loss with respect to singular values, followed by application of the Hutchinson randomized-probing estimator to the Hessian diagonal. These steps rely on classical calculus and stochastic trace estimation; they do not reduce by the paper's own equations to any fitted parameter, self-citation, or quantity defined in terms of the target ranking. The supplied loss-increase bounds and error-propagation analysis are likewise obtained from the same expansion and estimator variance properties without circular closure. No load-bearing premise is justified solely by prior work of the same authors, and the method remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard mathematical tools rather than new postulates or fitted constants; the perturbation intensity receives guidance but is not described as a fitted free parameter.

axioms (2)

standard math A second-order Taylor expansion approximates the change in task loss when a singular-vector basis is removed.
Invoked to derive the derivative-based importance score combining first- and second-order terms.
standard math Hutchinson's randomized probing method yields a reliable estimate of the Hessian diagonal when applied to loss curvature with symmetric perturbations.
Used to make the curvature term computationally feasible for LLMs.

pith-pipeline@v0.9.0 · 5538 in / 1424 out tokens · 32204 ms · 2026-05-09T14:28:01.881060+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 8 canonical work pages · 3 internal anchors

[1]

Fluctuation-based Adaptive Structured Pruning for Large Language Models

Yongqi An, Xu Zhao, Tao Yu, Ming Tang, and Jinqiao Wang. Fluctuation-based Adaptive Structured Pruning for Large Language Models. InAAAI Conference on Artificial Intelligence (AAAI), 2024

2024
[2]

Baston and Yuji Nakatsukasa

Robert A. Baston and Yuji Nakatsukasa. Stochastic Diagonal Estimation: Probabilistic Bounds and an Improved Algorithm.arXiv preprint arXiv:2201.10684, 2022

work page arXiv 2022
[3]

An Estimator for the Diagonal of a Matrix.Applied Numerical Mathematics, 57(11-12):1214–1229, 2007

Costas Bekas, Effrosyni Kokiopoulou, and Yousef Saad. An Estimator for the Diagonal of a Matrix.Applied Numerical Mathematics, 57(11-12):1214–1229, 2007

2007
[4]

Training Verifiers to Solve Math Word Problems

Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training Verifiers to Solve Math Word Problems.arXiv preprint arXiv:2110.14168, 2021

work page internal anchor Pith review Pith/arXiv arXiv 2021
[5]

Exploiting Linear Structure within Convolutional Networks for Efficient Evaluation

Emily Denton, Wojciech Zaremba, Joan Bruna, Yann LeCun, and Rob Fergus. Exploiting Linear Structure within Convolutional Networks for Efficient Evaluation. InConference on Neural Information Processing Systems (NeurIPS), 2014

2014
[6]

LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale

Tim Dettmers, Mike Lewis, Younes Belkada, and Luke Zettlemoyer. LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale. InConference on Neural Information Processing Systems (NeurIPS), 2022

2022
[7]

GPTQ: Accurate Post- Training Quantization for Generative Pre-trained Transformers

Elias Frantar, Saleh Ashkboos, Torsten Hoefler, and Dan Alistarh. GPTQ: Accurate Post- Training Quantization for Generative Pre-trained Transformers. InInternational Conference on Learning Representations (ICLR), 2023

2023
[8]

Golub and Charles F

Gene H. Golub and Charles F. Van Loan.Matrix Computations. Johns Hopkins University Press, Baltimore and London, 1996

1996
[9]

Song Han, Jeff Pool, John Tran, and William J. Dally. Learning Both Weights and Connections for Efficient Neural Networks. InConference on Neural Information Processing Systems (NeurIPS), 2015

2015
[10]

Measuring Mathematical Problem Solving with the MATH Dataset

Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. Measuring Mathematical Problem Solving with the MATH Dataset. Conference on Neural Information Processing Systems (NeurIPS), 2021

2021
[11]

Rae, and Laurent Sifre

Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, Tom Hennigan, Eric Noland, Katie Millican, George van den Driessche, Bogdan Damoc, Aurelia Guy, Simon Osindero, Karen Simonyan, Erich Elsen, Oriol Vinyals, Jack W. Rae, and Laurent Sifre...

2022
[12]

Language Model Compression with Weighted Low-Rank Factorization

Yen-Chang Hsu, Ting Hua, Sung-En Chang, Qian Lou, Yilin Shen, and Hongxia Jin. Language Model Compression with Weighted Low-Rank Factorization. InInternational Conference on Learning Representations (ICLR), 2022

2022
[13]

Hutchinson

Michael F. Hutchinson. A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines.Communications in Statistics - Simulation and Computation, 19 (2):433–450, 1990

1990
[14]

Speeding up convo- lutional neural networks with low rank expansions,

Max Jaderberg, Andrea Vedaldi, and Andrew Zisserman. Speeding Up Convolutional Neural Networks with Low Rank Expansions.arXiv preprint arXiv:1405.3866, 2014

work page arXiv 2014
[15]

Scaling Laws for Neural Language Models

Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling Laws for Neural Language Models.arXiv preprint arXiv:2001.08361, 2020

work page internal anchor Pith review Pith/arXiv arXiv 2001
[16]

Streamlining Language Models via Semantic Basis Analysis.Transactions on Machine Learning Research (TMLR), 2025

Yang Li, Daniel Agyei Asante, Changsheng Zhao, Ernie Chang, Yangyang Shi, and Vikas Chandra. Streamlining Language Models via Semantic Basis Analysis.Transactions on Machine Learning Research (TMLR), 2025. 10

2025
[17]

Optimizing Neural Networks with Kronecker-factored Approximate Curvature

James Martens and Roger Grosse. Optimizing Neural Networks with Kronecker-factored Approximate Curvature. InInternational Conference on Machine Learning (ICML), 2015

2015
[18]

Compressing Pre-trained Language Models by Ma- trix Decomposition

Matan Ben Noach and Yoav Goldberg. Compressing Pre-trained Language Models by Ma- trix Decomposition. InConference of the Asia-Pacific Chapter of the Association for Com- putational Linguistics and International Joint Conference on Natural Language Processing (AACL-IJCNLP), 2020

2020
[19]

Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks

Daniel Povey, Gaofeng Cheng, Yiming Wang, Ke Li, Hainan Xu, Mahsa Yarmohammadi, and Sanjeev Khudanpur. Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks. InConference of International Speech Communication Association (INTERSPEECH), 2018

2018
[20]

Empirical Analysis of the Hessian of Over-Parametrized Neural Networks

Levent Sagun, Utku Evci, V Ugur Guney, Yann Dauphin, and Leon Bottou. Empirical Analysis of the Hessian of Over-Parametrized Neural Networks. InInternational Conference on Learning Representations (ICLR) Workshop Track, 2018

2018
[21]

Ash, and Dipendra Misra

Pratyusha Sharma, Jordan T. Ash, and Dipendra Misra. The Truth is in There: Improv- ing Reasoning in Language Models with Layer-Selective Rank Reduction.arXiv preprint arXiv:2312.13558, 2023

work page arXiv 2023
[22]

A Simple and Effective Pruning Approach for Large Language Models

Mingjie Sun, Zhuang Liu, Anna Bair, and J Zico Kolter. A Simple and Effective Pruning Approach for Large Language Models. InInternational Conference on Learning Representations (ICLR), 2024

2024
[23]

Investigating the Overlooked Hessian Structure: From CNNs to LLMs

Qian-Yuan Tang, Yufei Gu, Yunfeng Cai, Mingming Sun, Ping Li, zhou Xun, and Zeke Xie. Investigating the Overlooked Hessian Structure: From CNNs to LLMs. InInternational Conference on Machine Learning (ICLR), 2025

2025
[24]

Llama 2: Open Foundation and Fine-Tuned Chat Models

Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al. Llama 2: Open Foundation and Finetuned Chat Models.arXiv preprint arXiv:2307.09288, 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023
[25]

Restructuring of Deep Neural Network Acoustic Models with Singular Value Decomposition

Jian Xue, Jinyu Li, and Yifan Gong. Restructuring of Deep Neural Network Acoustic Models with Singular Value Decomposition. InConference of International Speech Communication Association (INTERSPEECH), 2013

2013
[26]

MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models

Longhui Yu, Weisen Jiang, Han Shi, Jincheng Yu, Zhengying Liu, Yu Zhang, James T Kwok, Zhenguo Li, Adrian Weller, and Weiyang Liu. MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models. InInternational Conference on Learning Representation (ICLR), 2024

2024
[27]

Asvd: Activation-aware singular value decomposition for compressing large language models,

Zhihang Yuan, Yuzhang Shang, Yue Song, Dawei Yang, Qiang Wu, Yan Yan, and Guangyu Sun. ASVD: Activation-aware Singular Value Decomposition for Compressing Large Language Models.arXiv preprint arXiv:2312.05821, 2023

work page arXiv 2023
[28]

Block-diagonal Hessian- free Optimization for Training Neural Networks.arXiv preprint arXiv:1712.07296, 2017

Huishuai Zhang, Caiming Xiong, James Bradbury, and Richard Socher. Block-diagonal Hessian- free Optimization for Training Neural Networks.arXiv preprint arXiv:1712.07296, 2017

work page arXiv 2017
[29]

Accelerating Very Deep Convolutional Networks for Classification and Detection.IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), 38(10):1943–1955, 2016

Xiangyu Zhang, Jianhua Zou, Kaiming He, and Jian Sun. Accelerating Very Deep Convolutional Networks for Classification and Detection.IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), 38(10):1943–1955, 2016. 11 Appendix Table of Contents Appendix A: Theoretical Analysis of Basis Selection with Importance A.1 Efficient Estimation of th...

1943