Title: Spectral Conditioning of Attention Improves Transformer Performance
Venue: NeurIPS 2025
Accepted: yes
Keywords: spectral conditioning

attention
Abstract: We present a theoretical analysis of the Jacobian of a attention block within a transformer, showing that it is governed by the query, key, and value projections that define the attention mechanism. Leveraging this insight, we introduce a method that systematically alters the spectral properties of each attention layer to reduce the Jacobian’s condition number, thereby improving the overall conditioning of the attention layers within a transformer network. We empirically show that this improved Jacobian conditioning translates to enhanced performance in practice. Our approach is simple, broadly applicable, and can be easily integrated as a drop-in replacement for a wide range of existing attention mechanisms. We validate its effectiveness across diverse transformer architectures and tasks, demonstrating consistent improvements in performance.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a theoretical analysis of the Jacobian matrix of self-attention layers in Transformers, revealing its dependence on query, key, and value projection matrices. Based on this insight, the authors propose a method to systematically adjust the spectral properties of self-attention layers to improve the overall conditioning of self-attention layers within Transformer networks. The method demonstrates consistent performance improvements across various architectures.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n**Solid theoretical foundation**: Establishes the mathematical relationship between the condition number of self-attention layer Jacobian matrices and the condition numbers of WQ, WK, WV matrices, transforming Transformer optimization into a matrix condition number optimization problem.\n\n**Simple and practical method**: The proposed spectral conditioning method serves as a drop-in replacement, requires no additional parameters, and is applicable to various attention mechanisms with good generalizability.\n\n**Comprehensive experimental validation**: Validates the method's effectiveness across multiple tasks including image classification, object detection, instance segmentation, and nlp.\n\n## Weaknesses:\n**Insufficient motivation**: While the authors mention the effectiveness of Jacobian conditioning optimization in feedforward networks [1], they do not adequately demonstrate whether similar problems exist in Transformers or the necessity of condition number optimization.\n\n**Lack of hyperparameter guidance**: The $\\lambda$ parameter relies on grid search, lacking theoretical guiding principles.\n\n[1] Weight conditioning for smooth optimization of neural networks. ECCV 2025.\n\nQUESTIONS:\n1. Suggest experimental analysis to verify whether attention layers in Transformers exhibit similar condition number problems as feedforward networks.\n\n2. Recommend extending the analysis to feedforward components to comprehensively validate the applicability of this analytical framework to Transformer architectures.\n\n3. Suggest providing theoretical selection criteria for the $\\lambda$ parameter to enhance the method's practicality in large-scale models.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on analyzing the conditioning of the self-attention mechanism in Transformers to enable more stable training. An upper bound on the condition number of the Jacobian of the self-attention function is derived, providing a principled way to control Jacobian conditioning during training. Specifically, the paper introduces a technique called spectral conditioning, which can be applied not only to the classical attention mechanism but also to more advanced variants such as shifted-window attention (e.g., Swin Transformer) and low-rank approximations (e.g., Nyströmformer). Empirical results demonstrate that spectral conditioning improves the conditioning of attention modules during training. Experiments across diverse learning tasks confirm the enhanced performance brought by this technique.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: Conditioning analysis in Transformer models remains relatively under-explored compared to other network architectures. Hence, this paper could inspire conditioning-aware design that improves the numerical stability of attention-based models.\n\nWeakness: A discussion and comparison with low-rank attention mechanisms may be necessary.\n\nQUESTIONS:\nWhile I do appreciate the work, there are some related concerns:\n1. Low-rank approximation is closely related to the conditioning of the attention mechanism. Both Theorem 3.3 and Theorem 3.4 involve the term $\\text{softmax}(XW_QW_K^\\top X^\\top)$, which is a core target of many low-rank Transformer variants, such as Nyströmformer, Linformer [1], Performer [2]. First, to what extent do the theoretical bounds provided in Theorems 3.3 and 3.4 generalize to these low-rank attention variants? From my perspective, these theorem may be applicable to these methods since some spectral components can be controlled under the low-rank condition. Second, is it sufficient to control only the weight matrices $(W_Q, W_K, W_V)$ in these low-rank attentions in order to reduce their Jacobian's condition number? A broader discussion and numerical comparison across different low-rank self-attention mechanisms is necessary, as it could reveal how spectral conditioning complements or overlaps with structural approximations in improving training stability and performance.\n2. To better evaluate the impact of Jacobian conditioning, I feel it is necessary to add experiments that directly reflect numerical stability during training, such as a tracking on gradient norm, losses, etc.\n3. There are still some unclear notations and typos in the manuscript. Eq (4), $J$ should be removed from the partial derivation; Table 1, it is not immediately clear whether “Spec. cons.” refers to “Spec. cons. + the corresponding attention mechanism” and also the subsequent experiments. Caption of Figure 14, ViT-B should be corrected to Nyströmformer. Please double check the typos and expressions.\n\nI would like to consider raising the score if these concerns are resolved.\n\nReferences\n\n[1] Wang et al., Linformer: Self-Attention with Linear Complexity, arXiv 2020\n\n[2] Choromanski et al., Rethinking Attention with Performers. ICLR 2021\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n- The paper presents a theoretical analysis of the conditioning of the Jacobian of a self-attention layer of transformers.\n- Citing previous work, the authors argue that poorly conditioned Jacobians can hinder performance and improving Jacobian conditioning can lead to better optimization and generalization performance.\n- A theoretical analysis is performed, showing how the conditioning of the Jacobian of a self-attention layer is influenced by the conditioning of the query (Q), key (K), and value (V) matrices.\n- Based on this analysis, the authors propose spectral conditioned self-attention, which adds correction terms to the Q, K, V matrices during self-attention.\n- Since computing the correction terms using SVD at each iteration is prohibitive, the paper proposes an implementation-friendly form of the correction terms as $\\lambda \\mathbf{I}$, for some scalar hyperparameter $\\lambda \\geq 2$ (in the experiments, $\\lambda=10$ is chosen).\n- The authors empirically show improved conditioning and generalization on ImageNet classification, COCO object and instance segmentation, LRA text classification, and the GLUE benchmark.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n- Quality\n    - The proposed method is simple and technically sound.\n    - Experiments are performed across multiple domains and tasks.\n    - Assumption of full rank in the theorems is empirically validated.\n- Clarity\n    - The paper is clear and easy to follow.\n- Significance\n    - Despite being simple, the method leads to generalization improvements across diverse settings.\n- Originality\n    - The proposed spectral conditioned self-attention layer is novel.\n\n### Weaknesses\n\n- Quality\n    - The method only aims to minimize the upper bound of the condition number, and the practical implementation further loosens the upper bound being minimized.\n    - Theorem 3.5 is a trivial result that is unnecessary to derive the final practical method (Theorem 3.8 is all you need).\n    - Self-attention is a special case of attention, and it is not clear why the analysis does not extend to cases like cross-attention.\n- Significance\n    - The paper evaluates relatively small models and the generalization improvements are relatively small. There is no clear signal that the generalization improvements will persist in larger-scale settings.\n- Originality\n    - The idea behind the final practical method (Theorem 3.8), adding values to the diagonal to improve condition number, is not novel and is a fairly standard approach in linear algebra.\n\nQUESTIONS:\nIs it possible to generalize the paper's analysis and method beyond self-attention (e.g., cross-attention)?\n\nLIMITATIONS:\nYes, except the limitation to the self-attention case of attention.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a **theoretically grounded** and **computationally efficient** method to enhance Transformer models by improving the *conditioning* of their self-attention layers. The authors show that the condition number of the Jacobian of the self-attention block—an important factor affecting optimization stability—depends on the *spectral properties of the query, key, and value projection matrices* (WQ, WK, WV). To mitigate ill-conditioning, they propose spectral conditioning, which adds **fixed correction terms** to these matrices, significantly reducing their condition numbers and, in turn, improving the conditioning of the self-attention Jacobian.\n\nThe method is simple to implement and compatible with a wide range of attention variants and Transformer architectures. It does not introduce additional trainable parameters and incurs minimal computational overhead. The authors validate their approach across diverse tasks including **image classification** (ViT, Swin, XCiT, etc.), **object detection** and **segmentation** (COCO dataset), **long-range sequence modeling** (LRA benchmark with Nyströmformer), and **language modeling** (Crammed BERT on GLUE). In all cases, spectral conditioning consistently improves performance and leads to more stable training dynamics.\n\nWhile the proposed method optimizes an *upper bound* on the Jacobian condition number rather than the *true value* itself, the empirical gains support the theoretical motivation. The technique’s **ease of integration** and **general applicability** position it as a valuable tool for improving Transformer robustness without modifying their architecture or training procedure.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n- This paper is **well-written** and **accessible**, offering a clear presentation of both theoretical and empirical contributions. The mathematical analysis is rigorous and well-motivated, with theorems and proofs presented in a concise and structured manner. The accompanying figures, especially those analyzing singular values and condition numbers, are clean and effectively support the theoretical claims. Overall, the exposition is strong and reinforces the soundness of the proposed method.\n\n- Despite the simplicity of the idea, the core contribution is **conceptually elegant** and **broadly applicable**. The method—spectral conditioning of self-attention—targets a foundational property of Transformers (the conditioning of the Jacobian), and can be applied as a drop-in modification to virtually any self-attention mechanism. This generality strengthens its practical impact. In terms of novelty, the work stands out by focusing specifically on self-attention layers, where previous conditioning methods (e.g., reference [21]) have typically targeted fully connected layers. Thus, I would characterize the level of novelty as medium to high.\n\n- The experimental section covers both vision and language domains, evaluating the method across diverse tasks. In vision, the authors explore not only image classification but also object detection and instance segmentation. The inclusion of results on long-range sequence modeling (via the LRA benchmark) and language modeling (Crammed BERT on GLUE) further supports the generality of the approach.\n\n**Weaknesses and Suggestions**\n\n- While the experiments are diverse in terms of task types, the breadth of datasets and model variants could be improved to strengthen the empirical evidence. For image classification, all experiments are limited to ImageNet-1k. Given current standards, it would be beneficial to include evaluations on additional datasets, such as Places365, iNaturalist, or fine-grained benchmarks like CUB-200 or CARS-196. These additions would test the robustness and generalization of spectral conditioning beyond ImageNet and increase the confidence in its effectiveness across visual domains.\n\n- Similarly, for detection and segmentation tasks, results are only provided for XCiT within the Mask R-CNN framework. To better demonstrate the generality of the method across vision backbones, I would suggest incorporating one or two additional models in this setup.\n\n- A potential omission is the lack of evaluation on vision-language transformer models, such as CLIP or SigLIP. These models also rely on self-attention, and a simple adaptation of spectral conditioning could be tested on tasks like zero-shot image recognition. Given the popularity and widespread use of such models, this would be a valuable addition and highlight the method’s potential in multimodal settings.\n\n- Lastly, while the focus is on self-attention, it would be helpful for the authors to briefly discuss whether the proposed conditioning method could extend to cross-attention mechanisms as well. Even a short theoretical or empirical remark on this could broaden the relevance of the work and preempt reader questions.\n\nQUESTIONS:\n**Questions and Suggestions for the Authors**\n\n1. Additional Classification Datasets.\n2. More Detection/Segmentation Models.\n3. Vision-Language Models.\n4. Discussion on Applicability to Cross-Attention.\n\nMy current rating is **borderline accept**, and *I lean towards accepting the paper* due to its solid theoretical contributions and promising empirical results. Addressing the questions above—particularly expanding the experimental validation—could strengthen my recommendation.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-03"}