Title: How Classifier Features Transfer to Downstream: An Asymptotic Analysis in a Two-Layer Model
Venue: NeurIPS 2025
Accepted: yes
Keywords: open-set clustering

feature transfer

two-layer neural network

feature learning

metric learning

retrieval
Abstract: Neural networks learn effective feature representations, which can be transferred to new tasks without additional training.
While larger datasets are known to improve feature transfer, the theoretical conditions for the success of such transfer remain unclear.
This work investigates feature transfer in networks trained for classification to identify the conditions that enable effective clustering in unseen classes.
We first reveal that higher similarity between training and unseen distributions leads to improved Cohesion and Separability.
We then show that feature expressiveness is enhanced when inputs are similar to the training classes, while the features of irrelevant inputs remain indistinguishable.
We validate our analysis on synthetic and benchmark datasets, including CAR, CUB, SOP, ISC, and ImageNet.
Our analysis highlights the importance of the similarity between training classes and the input distribution for successful feature transfer.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work establishes a theoretical framework to justify that the source representations referred to as spike direction can serve to indicate the cohesion and separability in the sense of alignment when applying downstream unseen data. The paper treats multi-classification as multiple two-class classification problems, leading to the Eq. (2) represented hidden-layer gradient in terms of MSE loss of a two-layer neural classifier as formulated in Eq. (1). Then, Eq. (4) formulates the feature undergoing update into two parts: The original one of random initialization, and the incremental part by applying a bounded proxy of gradient defined in Eq. (3), in the framework of Hermite decomposition, where the latter one is termed as Dominant Feature or Spike Component. Then, the train-unseen similarity is deifined as the alignment between the mean unseen pattern and the spike direction to allow computing the so-called Cohesion and Separability (definition 4.3), which leads to proposition 4.4 and proposition 4.5 in Eq. (6) and Eq. (7), respectively. Eq. (6) and Eq. (7) reveal that the cohesion increases with the alignment of unseen pattern to the spike direction, and the separability increases when the unseen patterns are aligned oppositely in reference to the spike direction. Figure 3(c) illustrates some cases in which the two patterns are distinguishable/indistinguishable. Numerical experiments are included.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: Domain adaptation is studied from a theoretical perspective.\nWeakness: The conclusion is not surprising, and not informative to indicate the possible trend.\n\nQUESTIONS:\nQ1: In fact, we concern manifolds more than the mean representations of two classes in measuring the separability. If Figure 3(c) illustrates the distribution of two classes, it is possible that only a few examples are indistinguishable when projecting to the spike direction, subject to the manifolds of distributions. Besides, the practical case is more complex in that the convolution or self-attention operations have modified the representations prior to feeding them to the classifier and the end-to-end learning can thus avoid the ad hoc cases shown in Figure 3(c).\n\nQ2: I cannot understand why recall@1 can manifest the alignment of unseen pattern to the spike direction in Eq. (6) and Eq. (7), which is subject to only the mean representation. \n\nQ3: A possible bug is: The recall@1 concerns the nearest neighbor only but omits the overall distribution of the test data.\n\nQ4: Does the bias term omitted in Eq. (1) affect the generality of the neural approximation?\n\nLIMITATIONS:\nThe experiments are not solid.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper theoretically investigates the feature transfer abilities of two-linear neural network classifiers. \n\nThe authors consider the setting of a two-layer neural network with specific assumptions on the activation function and initialization of parameters and uncentered sub-gaussian data distribution. In a 1-vs-1 K-class classification setting with MSE loss, one gradient descent (GD) step is asymptotically shown to be approximated by a specific rank-1 structure. This yields a feature decomposition result that essentially decomposes the feature mapping function realized by the network after one GD step into two components: random initialization features and a so-called Spike Component, which is effectively responsible for feature learning from the training data.\n\nGiven this, the authors derive closed-form expressions for the two clustering performance criteria for unseen data, Cohesion and Separability, w.r.t. the Spike Component (Propositions 4.4 and 4.5) and relate them to the so-called Train-Unseen Similarity.\nNamely, *the better the alignment between training and testing data, the more effective the clustering of test data in the feature space*.\n\nExperimental results on synthetic and image data support theory.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n\nThis paper proposes a profound theoretical analysis of feature transferring capabilities in two-layer neural networks.\nThe presented results and proof techniques are novel and extend the assumptions and scope of the prior work.\nTheoretical analysis is well supported by both illustrative examples (Fig. 1-3) and actual experiments on synthetic (Fig. 4-5) and real data (Fig. 6-7).\n\n# Weaknesses\n\nThere is a significant gap between the specific setup considered in this study for theoretical derivations and real practice.\nNot to mention the general questionability of applying the two-layer analysis to explain feature learning in actual deep neural networks, I hardly find the parallels between the common practice and the provided results for 1-vs-1 perfectly class-balanced K-classification in the limit of data dimension $d \\to \\infty$ and one gradient descent step with learning rate $\\eta = 1$ under additional assumptions.  \nTo me, the main contribution of this work &mdash; proving in a very limited context an intuitive thesis that *clustering performance in feature space depends on the degree of alignment between training and testing distributions* &mdash; is marginal and does not provide any significant novel insights.\n\nThe paper is not very clearly written, it contains a large amount of inaccuracies, undefined values, and notation abuse that hinders comprehension of the paper content.\n\nRelated Work: I am not an expert in this specific domain, but it seems that some related works on transfer learning theory (in two-layer models specifically) might be missing here.\n\nQUESTIONS:\n1. Can the current results obtained for the non-conventional setting of 1-vs-1 K-class classification be transferred to the more standard K-way multi-class classification (i.e., when last layer is an $N \\times K$ matrix)?\n2. What if the classes are not perfectly balanced?\n3. Can the requirement of $d \\to \\infty$ be avoided in the analysis? It seems very unnatural.\n4. Line 111: if $W_0[i]$ denotes the $i$-th row of matrix $W_0 \\in \\mathbb{R}^{d \\times N}$, how can it be initialized from the $(d-1)$-dimensional sphere? Is that a typo?\n5. Can the sphere initialization be relaxed with the Gaussian initialization?\n6. Line 114: $P \\to K$.\n7. Does Shifted ReLU actually follow Assumption 2.1? It seems that it is not bounded by some $\\lambda_\\sigma$ a.s.\n8. How exactly are $H_k(z)$ defined?\n9. Doesn't Theorem 3.1 imply the degenerate limit of $F_0(x) \\to 0$ as $n \\to \\infty$? \n10. There is a discrepancy between the definition of $\\beta$ (without indices) and the way it is used in the text $\\beta_{ij}$ (with indices).\n11. Line 162: Definition 4.3.\n12. Line 165: What is $\\mathcal{n}(\\mu_i, I_d)$?\n13. Line 166: What is $\\rho_{k,k',r,r'}$?\n14. What are $C(s_L)$ and $S(s_L)$ used in Propositions 4.4, 4.5?\n15. Recall@1 definition: is the data sampled from the test or train distribution?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a theoretical and empirical analysis of how features learned by a two-layer classifier generalize to unseen classes in clustering tasks. The authors use a simplified setting — one gradient descent step and Hermite polynomial expansions — to decompose features into components and study how similarity between training and unseen data affects clustering quality. Experiments on both synthetic and real-world data support their theoretical findings.\n\nSTRENGTHS AND WEAKNESSES:\n**Quality:**\n- (+) The theoretical analysis is detailed and mathematically sound.\n- (+) The experiments are well-designed to support the theoretical claims.\n- (-) A two-layer network trained with just one step is far from how real models are used.\n- (-) The main result (features transfer better when unseen inputs are similar to training data) is well-known.\n\n**Clarity:**\n- (-) The introduction section is very hard to read. I recommend a profesional proofreading to make it more accessible to other researchers.\n- (-) The English writing is often overly complex. It is very hard to follow. \n- (-) I suggest the authors to separate simpler explanations from hard demonstrations.\n\n**Significance:**\n- (+) Understanding when and how features generalize to unseen classes is an important question in the machine learning community.\n- (+) The topic is relevant for researchers working on transfer learning, clustering, and metric learning.\n- (-) The contribution is **mostly theoretical formalization** of ideas that are already intuitively known (e.g., aligned inputs transfer better).\n- (-) The impact on practice is limited, as the analysis is far from current Machine Learning architectures. It would be interesting to know is the assumptions correlate to bigger networks.\n\n**Originality:**\n- (+) The authors' approach seems creative and novel.\n- (-) It is not a negative per se, but it is dissappointing that the theory confirms known behavior rather than discovering surprising or counterintuitive results\n\nQUESTIONS:\n- Is it possible to perform a similar analysis of deeper/multi-step models?\n- Did you find any unexpected behavior that could led to novel approeaches on the topic?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper theoretically analyzes how features learned by a classifier perform in clustering unseen classes. The study uses a simple model where a two-layer nonlinear network is trained with a single gradient descent step. As a result, it is shown that the learned features can be decomposed into a component derived from random initialization and a \"spike component\" that depends on the training data. Based on the properties of this spike component, the paper concludes that clustering Cohesion and Separability are determined by the similarity between the training data and unseen data distributions. Specifically, higher similarity improves Cohesion, while Separability changes depending on the assignment of unseen classes. It was also shown that feature expressiveness diminishes for inputs that are irrelevant to the training data. These theories are validated through experiments on both synthetic data and real-world benchmark datasets.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The framework of decomposing learned features via a \"spike component\" and explaining clustering performance (Cohesion, Separability) with \"train-unseen similarity\" is novel and provides deep insights into the mechanics of feature transfer. It moves beyond the empirical rule-of-thumb that simply \"training on large datasets is better.\"\n- The paper carefully validates its theoretical claims in two stages: (1) rigorous validation on synthetic data that meets the theoretical assumptions, and (2) practical validation in real-world scenarios with deep models (ResNet). This dual approach demonstrates both the theory's validity and its practical implications, making the paper highly persuasive.\n\n**Major Weaknesses**\n- The analysis is confined to \"a two-layer network trained with a single gradient descent step.\" Modern deep learning models are much deeper and are trained through complex dynamics over multiple epochs. How the insights from this simplified setting generalize to deeper, more complex models requires further discussion. While the experiments with ResNet on real data show consistent trends, a gap remains between the theory and practice.\n- The core theoretical metric, \"train-unseen similarity\" (β⊤μ), is clearly defined. However, in experiments on real data (e.g., Expr. IV), it cannot be directly computed and is replaced by the intuitive concept of \"semantic similarity.\" The relationship between these two metrics, while plausible, is not quantitatively analyzed, leaving the applicability of the theory somewhat ambiguous.\n\n**Minor Weaknesses**\n- While it is standard for major proofs to be in the appendix, some key theoretical steps (e.g., the transition from Theorem 3.3 to the definition of the Dominant Feature) feel a bit rushed in the main text. Including more intuition or a brief sketch of these steps in the main body would improve clarity for the reader.\n- As noted in Appendix A, the technique of approximating the activation function with polynomials has its limits. Specifically, it can require high-dimensional spaces for computation, and in finite dimensions, a discrepancy can arise between the approximation and the actual network's behavior. A more in-depth discussion in the main text on the validity of this approximation and its potential impact on the results would further strengthen the paper's credibility.\n\nQUESTIONS:\n- Your work focuses on a single gradient descent step. How do you anticipate the behavior of the spike component sL would change if multiple training steps or epochs were considered? Specifically, how might your conclusions (especially regarding Cohesion and Separability) be extended or modified as more complex features are learned?\n- In Expr. IV, you interpret the theoretical \"train-unseen similarity\" as practical \"semantic similarity.\" Would it be possible to perform a more quantitative analysis to bridge this gap? For instance, one could try to approximate the spike direction β from a model trained on one domain (e.g., CAR) and compute its inner product with the class means μ from another domain (e.g., CUB).\n- Your analysis of Separability shows that the \"assignment\" of unseen classes to training classes is critical. This assignment can be interpreted as being based on the learned decision boundary. Do you have any insights into the stability of this assignment itself (e.g., its robustness to initial conditions or minor perturbations in the data)? If the assignment is unstable, it seems that predicting Separability would also become difficult.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}