Title: Shortcut Features as Top Eigenfunctions of NTK: A Linear Neural Network Case and More
Venue: NeurIPS 2025
Accepted: yes
Keywords: shortcut learning

neural tangent kernel
Abstract: One of the chronic problems of deep-learning models is shortcut learning. In a case where the majority of training data are dominated by a certain feature, neural networks prefer to learn such a feature even if the feature is not generalizable outside the training set. Based on the framework of Neural Tangent Kernel (NTK), we analyzed the case of linear neural networks to derive some important properties of shortcut learning. We defined a “feature” of a neural network as an eigenfunction of NTK. Then, we found that shortcut features correspond to features with larger eigenvalues when the shortcuts stem from the imbalanced number of samples in the clustered distribution. We also showed that the features with larger eigenvalues still have a large influence on the neural network output even after training, due to data variances in the clusters. Such a preference for certain features remains even when a margin of a neural network output is controlled, which shows that the max-margin bias is not the only major reason for shortcut learning. These properties of linear neural networks are empirically extended for more complex neural networks as a two-layer ReLU FC network and a ResNet-18.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates shortcut learning in deep learning models, where networks overly rely on dominant but non-generalizable features in the training data. Using the Neural Tangent Kernel (NTK) framework, the authors analyze linear neural networks and define a “feature” as an eigenfunction of the NTK. \nThey find that shortcut features align with large NTK eigenvalues, particularly in settings with imbalanced or clustered data distributions. These dominant features significantly affect the model output even after training, suggesting that shortcut learning isn't solely due to max-margin biases. The insights are also shown to extend to nonlinear networks, like two-layer ReLU networks and ResNet-18.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The paper presents a thorough theoretical analysis using the Neural Tangent Kernel (NTK) framework to dissect shortcut learning, starting from linear models and extending the insights to more complex architectures. It provides formal theorems and proofs (e.g., Theorem 3.1 and 3.2) that explain how shortcut features arise due to spectral properties of NTK and data distribution characteristics.\n\n2. Introduces a clean interpretation of features as NTK eigenfunctions and relates shortcut features to eigenvalues, thereby linking shortcut learning to fundamental learning dynamics like spectral bias.\n\n3. The paper empirically shows that findings from linear settings extend to two-layer ReLU networks and ResNet-18, increasing the real-world relevance of the theoretical insights.\n\nWeaknesses:\n\n1. Core theoretical results are derived under simplifying assumptions: linear networks, Gaussian mixture models, infinite-width limit. While extensions to complex models are attempted empirically, theoretical generality is limited.\n\n2. The paper does not include error bars or statistical tests for empirical results. This undermines confidence in the generality of findings, especially on small or noisy datasets.\n\nQUESTIONS:\n1. The theoretical analysis relies heavily on the NTK framework (infinite-width, lazy training). In practice, finite-width networks often exhibit feature learning.  Is there any empirical observations or theoretical intuitions about how the findings generalize to feature-learning regimes?\n\n2. The shortcut labels in real-world datasets are manually constructed based on model predictions. How consistent are these across runs or architectures? Could a data-driven or automated method for shortcut label discovery be developed using availability scores?\n\nLIMITATIONS:\nTheoretical results rely on idealized settings (linear networks, infinite width, Gaussian mixture data), which limits applicability.\n\nThe empirical results do not provide statistical significance analysis.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes an analysis of shortcut learning resulting from the dominance of biased attributes in the data for the case of linear networks using NTK theory, by tying the network features to the eigenfunctions of the NTK and measuring the influence of each eigenfunction to the neural network outputs.\n\nThe central parts of the theoretical argument are two theorems, formulated for the case of linear neural networks and datasets generated from mixtures of Gaussians with isotropic covariances:\n\n**Theorem 3.1** quantitatively inspects the spectral bias by relating the convergence speed of each feature direction to the eigenvalues​ of the between-means covariance matrix M. It states that the eigenfunctions of the integral operator defined for the inner product kernel are linear projections on the eigenvectors of M and the eigenvalues decompose in the sum of average within-cluster variances, which is a constant term, and the eigenvalues associated with the eigenvectors of M. The theorem provides insight on why shortcut features (which, in the considered scenario are aligned with large cluster mixing probabilities), corresponding to larger M eigenvalues, are fitted faster: their kernel eigenvalues are large. In contrast, discriminative features that correspond to either small or similar clusters show a slower learning speed.\n\n**Theorem 3.2** further shows that, under training with MSE loss, eigenfunctions of the kernel operator corresponding to clusters with larger mixing weights have a higher influence on the network output. Coupled with theorem 3.1, it states that shortcut features (which correspond to larger kernel eigenvalues) are fitted early and dominate the network predictions after convergence. One important note from the authors is that the impact of bias feature after convergence depend not only on the cluster mixing weights but also on the variances of the samples, highlighted empirically on a toy example.\n\nThe theoretical results and experiments on toy data additionally challenge the idea that maximal margin control (for example via SD) alone can eliminate the shortcut bias, by stating that the SD decision boundary converges to the MSE decision boundary in the infinite-limit of the regularization coefficient, which is affected by shortcut learning. Further, the authors provide metrics for the concepts of predictability (how well a feature is aligned with the ground truth) and availability (the alignment of the feature with the top eigenspace of NTK) and empirically apply them on real datasets.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n* The authors take an interesting perspective on shortcut features through the lens of NTK theory and advances the theoretical understanding\n* The theoretical argument (restricted to the case of linear networks and GMM data) provides explicit forms for the eigenfunctions and weights as functions of the dataset mixing probabilities\n* The results improve understanding on debiasing, by first challenging the idea that maximal margin alone cannot eliminate shortcut biases and by additionally highlighting the impact of within-cluster variance on both spectral bias and feature weights\n* The experimental setup is extended beyond the limited regime considered in the theoretical argument (linear networks, GMM data) to real data and networks and empirically validates the spectral bias properties through newly introduced metrics for predictability and availability\n* The results might potentially provide insights applicable for the study of interpretability, as shown in Figure 3 where saliency maps for directions with high and low eigenvalues are tied to human-interpretable features\n\n**Weaknesses**:\n* In the follow up analysis of the theorem 3.1, it is implied that eigenvalues depend primarily on the cluster mixing weights (\"Since eigenvectors close to a cluster with a larger weight pk have larger eigenvalues, an inner product with a vector from a larger cluster has a larger eigenvalue and converges faster due to the spectral bias\"). However, it follows from Theorem 3.1 that eigenvalues scale with both mixing weights and the magnitude of the mean vectors (concretely, with pk * ||uk||). Therefore, in scenarios where a cluster with a high mixing weight has a small-norm mean, its direction might not dominate the spectral bias anymore. I recommend clarifying this aspect in the analysis, as it potentially restricts the theoretical setup even further to datasets where the mean vectors have approximately equal norms, further drifting away from practice.\n* Some of the limitations already pointed out by the authors: highly limiting regime in the theoretical analysis, with respect to data generation and class of models, as well as limitation of availability due to multiple shortcuts. While the empirical results show agreement with theory, the gap between the two is still considerable.\n* Theory relies on static NTK assumption that only holds in the infinite-width limit\n\nQUESTIONS:\n* Given the spectral bias analysis, I am curious whether you have thought about how the new spectral bias insights (and the metrics for availability and predictability) might inform designing further debiasing strategies?\n* Since the theory assumes a static NTK (which holds for infinite-width limit), have you observed a particular width threshold beyond which the spectral bias observations tend to emerge in the experiments?\n* As mentioned in the Weaknesses section, for the toy experiments highlighting theorems 3.1 and 3.2, has the norm of the cluster means been controlled in some way in order to observe its effect on the eigenvalue scales?\n* Can some aspects from the theoretical argument be directly applied from linear NTK to other types of kernels?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIntroduces an NTK-based theory explaining why shortcut (spurious) features dominate training: clusters with higher sample weight π acquire larger NTK eigen-values, are fitted fastest, and retain the largest post-training influence. \n\nProves (Theorems 3.1–3.2) these properties for linear networks; then shows empirically that two-layer ReLU nets and ResNet-18 exhibit the same behaviour on Patched-/Colored-MNIST, Waterbirds, CelebA, Dogs vs Cats.\n\nDefines predictability (alignment to ground-truth) and availability (alignment to top-NTK eigen-vectors) and demonstrates that shortcut labels have low predictability but high availability across five biased benchmarks. \n\nArgues, both theoretically (Cor. 3.3) and with 2-D toy data, that margin-control methods (SD / Marg-Ctrl) cannot fully remove shortcut bias.\n\nSTRENGTHS AND WEAKNESSES:\n**Quality**\nStrength\n1. Solid linear-kernel proofs, clearly laid out in Appendix B.\n2. Comprehensive experiments (5 real, 2 synthetic datasets) validate theory.\n\nWeakness\n\n3. All curves come from single runs; no confidence intervals reported.\n\n4. Manual construction of shortcut labels may inject subjectivity.\n\n**Clarity**\n\nStrength \n\n1 Helpful visuals (Fig. 3 saliency, Fig. 4-5 decision boundaries) illuminate ideas.\n\nWeakness\n\n2 Main text is heavy with notation; many key plots relegated to appendix.\n\n**Significance**\n\nStrength\n\n1. Bridges spectral bias and shortcut learning, highlighting cluster variance as a new factor.\n\nWeakness\n\n2. Results remain in infinite-width/lazy-training regime; impact on feature-learning settings uncertain.\n\n**Originality**\n\nStrength\n\n1. Novel availability metric linking shortcut strength to NTK spectrum; challenges margin-bias narrative.\n\nWeakness\n\n2. Core insight “large eigen-values ⇒ faster learning” is rooted in prior spectral-bias work, limiting novelty. \n3. Core contribution also overlaps with Learnability in the Context of NTK: because mathematically Tr K = ∑ λᵢ, their large-k(x,x) “easy” samples are exactly authors large-λ “shortcuts”.\n\nQUESTIONS:\nFinite-width drift – Could you run a width-sweep (e.g. 256-2 k channels) on Patched-MNIST to show when NTK approximation breaks?\n\nStatistical confidence – Please repeat each benchmark over ≥5 seeds and report mean ± s.d.; how stable is the availability ranking?\n\nShortcut-label sensitivity – How do availability curves change if shortcut labels are derived automatically (e.g. k-means on top eigen-projections) instead of manual inspection?\n\nVariance term in Thm 3.2 – Provide a concrete bound showing when a small but high-variance cluster can outweigh a large low-variance one.\n\nLate-epoch NTK – Have you measured availability using the empirical NTK after feature learning (e.g. epoch 50 ResNet-18) to see if shortcut alignment grows?\n\nLIMITATIONS:\nAuthors acknowledge linear-kernel and infinite-width assumptions but societal-impact section is brief; suggest expanding on potential mis-use of availability for data filtering.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nPropose a toy model where the problem of shortcut features can be investigated, using a mixture of two Gaussians with biased clusters (unbalanced probabilities) and considerable variance. Demonstrate, using an NTK setting in a linear network, that in this setup, shortcut features get learned earlier and influence the learned predictions. Furthermore, it demonstrates that a previously suggested measure to avoid shortcut features, controlling the classification margin, does not seem to alleviate the problem. Propose measures of availability and predictability to identify shortcut features in real-world datasets and demonstrate that they exhibit similar properties.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths \n 1. A toy model where the problem of shortcut features can be reproduced and against which possible solutions can be tested.\n 2. Use an existing theoretic framework (NTK) in a simplified manner to better understand the learned features which correspond to shortcut features.\n 3. The predictability and availability metrics seem reasonable and useful.\n 4. Test to see if a previously proposed mitigation for shortcut features work on the model, and found it does not.\n\n## Weaknesses\n 1. Unclear logical structure. The manuscript would benefit from a more layered structured, which makes clearer claims. Specifically, it is unclear which results depend on the Gaussian mixture model, which on the NTK analysis, and which do not. I think the structure of the logical argument is:  \n   ** Propose a model of biased Gaussian mixture to test shortcut features (with equations).  \n   ** Show when the model demonstrates shortcut features and when it does not (e.g., yes when biased larger than something and variance larger than something, show figure).  \n   **Apply a previously suggested method to understand the learned features in this model, demonstrating that they exist, are learned faster, and contribute to task performance.  \n   ** Show that a previously suggested solution for shortcut features, controlling prediction margin, is not helpful in this model.\n   **Based on the analysis, suggest measures for identifying shortcut features, availability, and predictability.    \n   ** Replicate the results of real-world datasets, without the model's assumptions.   \n 2. The predictability and availability metrics seem reasonable, to the point that I would guess they already have a name or previous usage.\n 3. The NTK analysis is relevant only in the lazy regime, which is not what happens when neural networks get trained on real-world samples. The derived results provide only an intuitive understanding that needs to be tested.\n4. While it is note that \"the data variance of each cluster is not really small\", the contribution of this phenomenon is not well investigated. The point of Figure 4 is unclear.\n\nQUESTIONS:\n1. Can you show that the Gaussian mixture model exhibits shortcut features without using the NTK analysis?\n2. Can you quantitatively show the parameters of the GMM under which shortcut feature do or do not show, based on the NTK analysis?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}