We appreciate your positive evaluation and constructive feedback, particularly regarding the clarity of the paper. First, we would like to take this opportunity to address some misunderstandings, especially those related to Q1 and the interpretation of Figure 3(c) and the usage of recall@1 (Q2, Q3) with additional experiments in the Expr 4 setup confirm that both Separability and Recall@1 serve as indicators of clustering performance, consistent with our findings in Expr 1, 2, and 3. Then, we address the concern regarding the conclusion and respond to Question 4.

interpretation of Figure 3(c) & Question 1

We believe the phrase ”the two patterns” in your summary refers to the Cohesion and Separability patterns, and we’d like to clarify this potential confusion. As shown in Eq. (6) and Eq. (7), the train-unseen similarity governs Cohesion and Separability, and their relationships are illustrated in Figures 1, 2, 4, and 5. In contrast, Figure 3(c) stems from Section 4.2 and aims to illustrate that, depending on $\beta$, inputs can become distinguishable or indistinguishable in the learned feature $s_L$. This figure presents a pair of individual points (rather than class means or distributions), highlighting a key insight for the experimental section and the main conclusion: only semantically relevant classes contribute meaningfully to feature extraction.

Response to Question 1

From a distributional viewpoint, when class manifold distributions align with the spike direction $\beta$, they are separable; if they lie in subspaces orthogonal to $\beta$, they become indistinguishable, regardless of manifold structure (Corollary 4.7). Your consideration of manifolds is valid, and our key contribution is identifying the precise condition—alignment with $\beta$ that governs distinguishability.

We expect that, despite the nonlinear transformations in deep architectures (e.g., convolution or attention), inputs orthogonal to the learned discriminative directions tend to persist after training, owing to the structure of matrix operations. Such inputs could be indistinguishable to the model, which underscores the importance of identifying spike directions—a central goal of our analysis in section 4.2. Moreover, discovering such directions in deeper architectures would also be crucial for understanding model behavior.

Response to Question 2

Your question may stem from a few conceptual confusions. We first address your concern directly and then provide clarifications to ensure a precise understanding of the relationship between recall@1 and the alignment of unseen patterns to the spike direction.

First, recall@1 serves as a surrogate measure of the Cohesion and Separability of learned features (as detailed in our response to Q3). According to the interpretation given in Eq. (6), Eq. (7), and Section 4.1, this Cohesion and Separability is governed by the train-unseen similarity, which you referred to as the alignment of unseen patterns to the spike direction.

Therefore, our central claim is that recall@1 is governed by the alignment of unseen patterns to the spike direction, which is supported by our empirical results (Experiments 1–4).

We would also like to clarify the following technical points to avoid further misunderstanding:

1. Eqs. (6) and (7) involve more than mean representations.
   • As stated on Line 35, Cohesion and Separability represent standard clustering performance criteria [Clémençcon, 2011; Papa et al., 2015; Liu et al., 2017; Li and Liu, 2021]. (See also Line 705 for further background.)
   • Eqs. (6) and (7) provide approximations of the Cohesion and Separability of feature representations learned after a single gradient step (see Theorem 3.3, Definition 3.4, Line 173, and Appendix L).
   • Hence, these equations quantify the clustering performance of the trained features for a given unseen class.
2. The alignment of unseen patterns to the spike direction is distinct from recall@1.
   • This alignment is captured by $\beta^\top \mu$, as defined in Eq. (6) and Eq. (7), and reflects train-unseen similarity.
   • As in our analysis (Line 282), the alignment is captured through train-unseen similarity and semantic similarity, not recall@1.
   • Instead, this alignment determines the clustering performance, and thus indirectly affects recall@1, but the two are conceptually distinct.

Response for Question 3: Recall@1 serves as a practical surrogate for clustering performance, cohesion, and separability.

When both Cohesion and Separability are high, the nearest neighbor is more likely to belong to the same class, resulting in a higher recall@1 measured in inner product. This is a standard practice in metric learning (Wang et al., [2018], Zhai and Wu [2019], An et al., [2023]) for measuring clustering performance. While recall@1 focuses on the top-1 neighbor, we claim that it is a practical and reliable surrogate for evaluating the combined effect of Cohesion and Separability. Moreover, in Experiments 1, 2, and 3, we observed that recall@1 exhibits a consistent increasing trend that aligns with the behavior of Separability, further supporting its relevance in evaluating performance (See appendix Figures 25, 26, 27, and 28).

Additional Experiment of Separability

To address concerns about Recall@1 as a surrogate for clustering criteria, we additionally evaluated Separability in Definition 4.3 for Expr. 4, alongside Expr. 1–3. Following standard practice (Zhai & Wu, 2019), we trained with unit-normalized features and, thus, report Separability only.

We observed that models **best separate test data when trained on the corresponding domain (e.g., CAR to CAR), consistent with the Recall@**1, reinforcing our main argument of Expr 4. The results are provided based on the ResNet18 with the initialization setting. Further experiments under different setups also confirmed this pattern and will be included in the camera-ready version.

Lastly, as discussed in our response to Reviewer MzGZ, we provide supporting evidence for a meaningful connection between train-unseen similarity and semantic similarity. We also test the validity of our theory in both multi-layer and multi-step settings in our response for MzGZ.

Addressing the Weakness in the Conclusion

To address the concerns raised regarding the strength of our conclusion, we propose the following improvements to articulate our contributions better and clarify the broader implications of our work:

1. Extension of prior theory: Our analysis generalizes existing frameworks by incorporating non-centered sub-Gaussian input distributions (Refer Appendix I–J) and deterministic feature approximations for classification tasks (Refer line 650).
2. Versatile analytical framework: Our approach opens up several promising avenues for extension, including applications to multi-step training dynamics, and potential connections to Neural Collapse under arbitrary input distributions.
3. Future extension to multi-class softmax: A natural next step is to extend the notions of Cohesion and Separability to the multi-class softmax setting, incorporating techniques such as normalization and temperature scaling to more closely reflect practical neural network training.
4. First quantitative link between semantic alignment and clustering: To our knowledge, this is the first work to quantitatively connect train-unseen alignment with open-set clustering performance, rather than relying solely on hypothesis-level arguments (line 671).
5. A representation-driven view of coreset construction: Our work suggests a potential new direction for representation-aware coreset selection—for instance, reducing the number of training classes or training data not contributed to $\beta$ direction to lower computational complexity while still enabling successful clustering and feature transfer.

Taken together, these points clarify that our work offers not only strong theoretical insights but also practical relevance and extensibility across a range of future research directions.

Response to Question 4

We excluded the bias term in Eq. (1) for analytical simplicity. This follows the convention in several recent studies, such as Moniri et al. [2024] and Ba et al. [2022], where feature learning is analyzed without bias. As Ba et al. [2023] and Mousavi-Hosseini et al. [2023], incorporating the bias term is possible, but the simplified formulation in Eq. (1) is sufficient to address the research question.

We appreciate the detailed reviews aimed at improving our paper. In particular, we are grateful for your recognition that our theoretical results provide novel extensions of prior work to non-centered sub-Gaussian distributions and are well supported by empirical evidence. However, you expressed concerns regarding the gap between our theoretical assumptions and practical neural networks, as well as the claim that our theoretical findings do not offer significant novel insights. We would like to first address these concerns before responding to your questions.

On the Gap Between Theoretical Setup and Practice

We acknowledge the simplified nature of our model (1. two-layer, 2. 1-vs-1 classification, 3. perfectly class-balanced, 4. proportional regime $d \rightarrow \infty$ 5. one-step gradient descent, 6. learning rate 1). However, we support our claims with experiments on deep networks (e.g., ResNet) as in Section 5 and observe that the theoretical predictions remain consistent in practice. On top of that, we conducted additional experiments for the rebuttal by extending our assumption. Specifically, we performed experiments under the Expr 1 setting with (1) 2-layer networks trained for up to 5 steps, and (2) networks with 2, 3, 4, and 5 layers. In both cases, we observed that as the train-unseen similarity $\beta^\top \mu$ increases, Cohesion and Separability consistently improve. These results are included in the final part of our rebuttal to Reviewer MzGZ under Multi-Step and Multi-Layer Experiments.

Lastly, this setting parallels a large body of existing theoretical work (see Table 1) and is justified for the following reasons:

1. Two Layer networks

   We adopt this two layer network as it adequately enables analytical insight into feature learning and transfer, and has proven effective and sufficient in prior work for explaining key phenomena in deep models, such as feature learning and the superiority of neural networks (Goldt et al. [2020], Ba et al. [2022], Ba et al. [2023], Cui et al. [2024], Mousavi-Hosseini et al. [2023]).

2. 1-vs-1 classification: addressed in question 1

3. Perfectly class-balanced: addressed in question 2

4. Proportional regime: addressed in question 3

5. One step update

   As supported by a growing body of theoretical work (Appendix Table 1), one-step updates have been successfully used to explain feature learning effects in real neural networks. This setting is not an arbitrary simplification but a theoretically grounded framework (Moniri et al. [2024], Dandi et al. [2024], Demir and Dogan [2025]), especially given the well-documented strong influence of early-phase dynamics on final model behavior (Ba et al., 2022). Within this framework, we show that even a single gradient step reveals meaningful clustering through the emergence of the $s_L$ term which sufficiently addresses our core research question. (see Appendix L).

6. Learning rate 1

   We would like to point out a detail that may have been overlooked respectfully — please refer to line 119. While our theoretical results hold for learning rates $\eta = \Theta(1)$, we represent the learning rate as 1 purely to avoid notational complexity. This setting is equivalent to using a small learning rate (e.g., 0.01) in standard neural networks.

Novelty

We will explain below why our work goes far beyond merely rephrasing known intuitions.

1. A Novel Framework for Analyzing Classifier - Deterministic Spike-Based Closed-Form Decomposition:

   We establish a foundational framework for analyzing the feature behavior of classifiers on new inputs, rather than focusing on the commonly studied regression task (Ba et al. [2022], Dandi et al. [2023], Cui et al. [2024], Dandi et al. [2024]). Notably, our predominant feature component β admits a closed-form, deterministic expression, unlike the stochastic feature decompositions used in recent works such as Demir and Dogan [2025] (see line 650). As a result, we are able to directly analyze the output of the learned feature map in Section 4.

2. Counterintuitive Finding – Importance of Relevant Classes, not Many Classes:

   Beyond clustering, Corollary 4.7 in Section 4.2 and Experiments V–VI show that, contrary to common belief, which emphasizes the importance of more training samples, feature expressivity depends mainly on semantically related classes. This insight challenges the focus on quantity (Brown et al., 2020; An et al., 2023; line 337), suggesting that reducing training classes, not just samples, may improve feature transfer performance and efficiency. Moreover, our work suggests a potential new direction for coreset selection —for instance, reducing the number of training classes or training data not contributed to β direction to lower computational complexity while still enabling successful clustering and feature transfer.

3. Unifying empirical observations and prior theories - Theoretical Quantification of Feature Transfer Phenomena:

   Our work offers a theoretical contribution by unifying empirical observations and prior theories, clarifying when feature transfer succeeds or fails (Section 4.1). While feature transfer is known to work better when unseen inputs resemble training data, previous metric learning studies mainly focus on performance gains without consistent explanations (See 671). In contrast, our framework precisely identifies both when feature transfer works (Propositions 4.4 and 4.5) and when it breaks down, such as when multiple unseen classes map to the same spike β, causing Separability to fail despite high similarity. This quantitative characterization of transfer success and failure is a key strength of our work.

Response to Questions

1. To ensure compatibility with existing regression-based frameworks and to allow for a more tractable derivation, we adopted a 1-vs-1 classification setting. However, by avoiding the matrix formulation in Equation 2 and instead representing the labels y using one-hot vectors, our analysis could potentially be extended to general k-class classification problems as well.
2. Our proofs only require that $X$ is sampled from class-conditional distributions that are non-centered sub-Gaussian, so the proof technique for $\beta = \frac{1}{n} X^\top y$ remains unchanged even in imbalanced settings.
3. The assumption $d \to \infty$, commonly used in feature learning research (e.g., Ba et al. [2022]), facilitates theoretical analysis of neural networks. Ba et al. [2022] justify this regime as representative of modern neural networks with large datasets and architectures. We adopt it to isolate the key low-rank spike term (Theorems 3.1 and 3.3). From Section 4 onward, our results assume a fixed, finite, and sufficiently large sample size $N$, supported by experiments demonstrating the theory’s applicability to large, finite neural networks.
4. The notation $\mathcal{S}^{d-1}$ refers to the $(d-1)$-dimensional sphere embedded in $\mathbb{R}^d$. $W_0[i]$ is a $d$-dimensional vector indexed by $i \in [[N]]$. Sorry for the mistake.
5. To ensure consistency in the theoretical objects we analyze, we adopted the assumption used in Moniri et al. [2024]. While Ba et al. [2022] initialize parameters with Gaussian distributions, which would lead to differences in the proof techniques, we expect similar qualitative results under that alternative setup.
6. We apologize for the typographical error. The intended meaning is that the problem corresponds to K classification tasks, not P. Thank you for pointing this out.
7. This is a typographical error. The assumptions are the same as in Moniri et al. [2024], where σ′, σ′′, and σ′′′ are bounded. The boundedness of σ itself was not used in the proof. We apologize for the mistake and will make the necessary correction.
8. The Hermite polynomial $H_k$ is discussed in detail in Appendix E. For readability, we will include a reference to it in the main text.
9. Theorem 3.1 concerns an approximation of the gradient and is not a direct result about $F_0$. Additionally, although the term $\frac{1}{n}$ appears in $A_{ij} = \frac{c_1}{n} X_{ij}^\top y a_{ij}^\top$, as summarized in Remark K.4 and Equation 24, as $n \rightarrow \infty$, almost surely $||\beta||$ and $||a_{ij}||$ remain at constant scale, so $||A_{ij}||$ also stays at a constant scale and does not degenerate as $n$ grows.
10. $\beta$ refers to $\beta_{ij}$, but the distinction of i and j is not essential, so we omit the subscripts. For clarity, we will explain this in the main text. We apologize for any confusion caused.
11. Thank you for pointing out the typo.
12. It is a Gaussian distribution. We will make this explicit.
13. We simplified it as it does not play a significant role in our analysis. It is a term unrelated to $\beta$ and $\mu$ that appears in the proof, and it is defined in Appendix H. We will indicate where it is defined.
14. We defined cohesion and separability in Definition 4.3 as functionals (functions from functions to scalars) that take any feature extractor F (which may be $s_L$, $F_0$, or $F_L$). It can be computed using $s_L$ instead of F in the definition. Using F seems to confuse, so that we will replace it with a different symbol, such as G.
15. Sampling from the test distribution, which is standard practice in the metric learning field, will be clearly described in our work.

Response to Concerns on Related Work

Related work on transfer learning theory is discussed in Appendix D (line 661). These studies examine how changes in the teacher function structure affect sample complexity. To our knowledge, theoretical analyses of feature transfer—especially in metric learning or open-set clustering without further training—remain unexplored (see line 671). For clarity, we provide a more detailed discussion of these aspects in the main text.

We sincerely thank the reviewer for their thoughtful and accurate summary of our work. In this rebuttal, we address the concerns you raised regarding the weaknesses and questions.

Response to Question 1 and Major Weaknesses 1

We share intuition on multi-step training and deeper models. In addition, we provide supporting experiments showing that our theoretical predictions largely hold in both multi-step and multi-layer settings.

For multi-step training

We expect that multi-step training refines the spike component by amplifying directions that enhance Separability while suppressing others, causing class features to collapse more tightly and become better separated. This process may lead to effective information compression for classification and improved transferability. To this end, one may build on Dandi et al. (2023), who analyze the "staircase property" in multi-step training.

For the deeper model

While our current two-layer model already offers insight into how feature learning enhances clustering, building on this with deeper architectures could deepen our understanding. Existing techniques (e.g., Fan & Wang [2020]; Nichani et al. [2023]; Wang et al. [2023]) have already generalized conjugate kernel analysis to multi-layer settings, suggesting that our approach may extend similarly. Such extensions could help further explain how generally transferable features emerge. Notably, our multi-layer experiments showed that when $\beta^\top\mu$ becomes too large, especially in 3- and 4-layer models, Separability may not improve. Studying this critical threshold could be an interesting direction.

Response to Question 2 and Major Weaknesses 2

We appreciate your concern regarding the applicability of our theoretical framework to real-world networks. In real world validation of our feature transfer theory, we surrogate the theoretical quantity $\beta^\top \mu$ (train-unseen similarity) using a semantic similarity of the dataset domain. This analogy is justified, as datapoints in each dataset (CUB, CAR, SOP, ISC) exhibit consistent visual characteristics within the dataset. Nevertheless, to address your request and strengthen our claim, we conducted the following additional experiment.

Additional Experiment:

As you suggested, for each dataset (CUB, CAR, SOP, ISC) with three seeds, we trained a model and extracted the class-wise mean embeddings from the training set as a surrogate for $\beta$. Similarly, we computed class-wise mean embeddings from the test sets as a surrogate for $\mu$. Then, for each test class, we computed the maximum cosine similarity with any of the train class means, as follows:

# same procedure for mean_embeddings_train
for test_dataset in [cub, car, sop, isc]:
	embeddings, labels = extract_feature(model, test_dataset)
	mean_embeddings_test = torch.zeros(num_unseen_classes, dim)
	for uc in range(num_unseen_classes):
		mean_embeddings_test[uc] = embeddings[labels == uc].mean(dim=0).normalize()
		# get statistics
		sim = mean_embeddings_test @ mean_embeddings_train.T # (num_unseen, num_seen) 
		max_values = sim.max(dim=1) # (num_unseen) 
		median(max_values), mean(max_values), std(max_values)

Findings:

As shown in the table, the maximum cosine similarity between unseen and train class means is consistently highest when the train and test domains match (e.g., 0.84 for CUB → CUB, 0.78 for CAR → CAR). This supports our interpretation that semantic similarity across domains serves as a valid proxy for theoretical train-unseen alignment. We provide the results of ResNet18 trained initialization in the table below. We will include analyses of all setups and visualization in the camera-ready version.

Response to Question 3

We believe there may have been a misunderstanding regarding the "assignment" in Note 4.6. The assignment we refer to is a theoretical construct defined without training, computed via the sign $\beta^\top \mu = \frac{1}{n} (X^\top y)^\top \mu$ in data space. It reflects a linear decision boundary in the data space, not one learned through neural network training. We will clarify this in the revised manuscript to avoid potential misinterpretations.

However, your question raises a valid point about how stable this assignment is to perturbations in the training data itself, such as reduced class separability. In such cases, the magnitude of $\beta$ decreases, reducing Cohesion and Separability scores in Eqs. 6 and 7 due to the $||\beta||$ term. This aligns with the intuitive notion that noisier training data leads to less transferable features.

To illustrate, suppose the training data consists of two classes drawn from $c_1 \sim \mathcal{N}(\mu, I)$ and $c_2 \sim \mathcal{N}(R\mu, I)$, where $R$ is a rotation matrix. Then the spike direction converges (under large $n$) to $\beta \rightarrow \frac{1}{2} (\mu - R\mu)$ by law of large numbers.

$\beta = \frac{1}{n} X^\top y = \frac{1}{n} X_i y_i = \frac{1}{2} (\frac{n}{2} \sum_{c_1} X_i - \frac{n}{2} \sum_{c_2} X_j) \rightarrow \frac{1}{2} (\mu - R \mu) .$

In the extreme case where $R = I$, the classes are indistinguishable and $\beta = 0$, eliminating both Cohesion and Separability. When $R = -I$, $\beta = \mu$, which yields maximal separation.

Response to Minor Weaknesses

1. Specific key theoretical steps, especially the transition from Theorem 3.3 to Definition 3.4 (Dominant Feature), would benefit from a more intuitive explanation in the main text. We will include a summary and interpretation to aid the reader's understanding.
2. We acknowledge limitations of this approach in low-dimensional regimes. In the camera-ready version, we will discuss the need for alternative, more interpretable and efficient approximations better suited for these settings. This will clarify the scope of our approximation.

Multi step and Multi-layer experiments

To test the validity of our theory in multi-layer and multi-step settings, we applied Expr 1 and observed that increasing $\beta^\top\mu$ generally improves both Cohesion and Separability, consistent with the prediction in Section 4.1.

In deeper networks, however, excessively large $\beta^\top\mu=5 \cdot 10^3$ leads to reduced Separability in 3- and 4-layer cases, as unseen features become aligned yet too far in magnitude. This suggests an interesting direction for future work: identifying a potential critical threshold of alignment strength in deep models that balances directional consistency and proximity.

As we are unable to include figures, we provide tables instead. We report Cohesion and Separability in separate tables, where each column corresponds to a layer or step setting, and each row shows values for increasing $\beta^\top\mu$.

Multi-layer One-step, lr 0.1, and Expr 1 of Training Data 2 (Truncated Gaussian)

Cohesion

Separability

Multi-step Two-layer, lr 0.001, and Expr 1 of Training Data 2 (Truncated Gaussian)

Cohesion

Separability

We appreciate the recognition of the theoretical soundness and experimental rigor of our work. Your comments encouraged us to clarify key contributions more precisely.

Notably, we would like to respectfully highlight an aspect that may have been overlooked in the summary. Beyond clustering analysis, a key contribution of our work lies in extending the analysis of two-layer neural networks from regression to classification. To address this, we consider inputs drawn from a non-centered sub-Gaussian distribution, broadening applicability to classification tasks. To our knowledge, this is the first such tool to enable tractable, closed-form analysis of feature learning in classification. Moreover, this framework not only supports clustering results but more generally characterizes how feature expressivity for unseen classes depends on the number of semantically related training classes (see Section 4.2, Corollary 4.7).

On the Gap Between Theoretical Setup and Practice

We acknowledge the simplified nature of our model (two-layer, one-step gradient descent).

However, we support our claims with experiments in Section 5 on deep networks (e.g., ResNet) and observe that the theoretical predictions remain consistent in practice. We believe this combination of theory and empirics provides a compelling case for the relevance of our simplified setting.

On top of that, we conducted additional experiments for the rebuttal by extending our original 2-layer, 1-step setup. Specifically, we performed experiments under the Expr 1 setting with (1) 2-layer networks trained for up to 5 steps, and (2) Bigger networks with 3, 4, and 5 layers. In both cases, we observed that as the train-unseen similarity $\beta^\top \mu$ increases, Cohesion and Separability consistently improve. These results are included in the final part of our rebuttal to Reviewer MzGZ under Multi-Step and Multi-Layer Experiments.

Lastly, this setting is consistent with a large body of works (see Appendix D) and is justified for the following reasons:

1. Our assumptions enable analytical tractability and provide clear insight into the mechanisms of feature learning and transfer. In particular, our central research question—”Can we capture the presence of feature learning in classification and identify the conditions where features cluster effectively in new distributions?”—can be addressed using a single training step of a two-layer neural network.
2. One-step update captures early-phase dynamics that are known to be strongly influential in determining final model behavior (Early Phase of NN Optimization Section of Ba et al., 2022).
3. Shallow network has been successfully used in prior work to explain nontrivial phenomena observed in bigger networks, such as the superiority of neural networks over traditional kernel methods (Ba et al. [2022]), the mechanisms of feature learning (Dandi et al. [2023], Moniri et al. [2024]), the mechanisms of learning the internal structure of data (Ba et al. [2023], Mousavi-Hosseini et al. [2023]), the representational power of random features (Louart et al. [2017], Goldt et al. [2020], Cui et al. [2024]), lottery ticket hypothesis([1]), generalization, interpolation and double descent ([2, 3, 4]), overparameterization ([5]), Catastrophic Forgetting([6])

[1] Malach, Eran, et al. "Proving the lottery ticket hypothesis: Pruning is all you need." International Conference on Machine Learning. PMLR, 2020.

[2] Suzuki, Keita, and Taiji Suzuki. "Optimal criterion for feature learning of two-layer linear neural network in high dimensional interpolation regime." The Twelfth International Conference on Learning Representations. 2023.

[3] Abeykoon, Chathurika S. The Double Descent Behavior in a Two Layer Neural Network for Binary Classification. Diss. The University of Mississippi, 2023.

[4] Xu, Ruichen, and Kexin Chen. "Rethinking benign overfitting in two-layer neural networks." arXiv preprint arXiv:2502.11893(2025).

[5] Zhang, Yaoyu, et al. "Local Linear Recovery Guarantee of Deep Neural Networks at Overparameterization." Journal of Machine Learning Research 26.69 (2025): 1-30.

[6] Li, Boqi, Youjun Wang, and Weiwei Liu. "Towards Understanding Catastrophic Forgetting in Two-layer Convolutional Neural Networks." Forty-second International Conference on Machine Learning.

We hope that our findings, alignment in real-world networks, the empirical validity of our theoretical predictions in multi-step and multi-layer settings, and strong connections to prior literature linking simplified assumptions to deep models help convey the broader relevance of our work and encourage further interest in our research.

Response to Question 1

Deeper Networks

While our current two-layer model already offers insight into how feature learning enhances clustering, building on this with deeper architectures could deepen our understanding. Existing techniques (e.g., Fan & Wang [2020]; Nichani et al. [2023]; Wang et al. [2023]) have analysis for multi-layer settings, suggesting that our approach may extend similarly. Such extensions could provide insights into how neural networks learn to classify by selectively amplifying or suppressing directions in input space. Ultimately, this line of inquiry helps further explain how generally transferable features emerge across layers in deep models.

Multi-step training

Dandi et al. (2023) proposed an analytical framework for studying multi-step training dynamics in two-layer neural networks. They analyzed the so-called staircase property in such networks and demonstrated that features are learned progressively from the data structure, leading to a monotonic decrease in generalization error. Given the similarity between their setting and ours, their methods could potentially be adapted to study the evolution of classifier features across multiple training steps, beyond the one-step clustering analysis we focus on.

Novelty & Question 2

We understand your concern that the core result—features cluster better when unseen inputs resemble training data—may seem unsurprising. However, we believe our work goes significantly beyond rephrasing known intuitions:

1. Novel Theoretical Tools for Classifier Analysis – Extending to Non-Centered Sub-Gaussian Distributions:

   Our analysis extends prior work (see Table 1 in the Appendix) by generalizing feature learning theory from centered Gaussian settings to non-centered sub-Gaussian data, thus enabling application to classification tasks beyond regression. This requires new technical tools (see line 93, Appendices I–J) that allow analysis under arbitrary labels and finitely supported distributions.

   In Appendix I, we extend classical results on standard Gaussian expectations of Hermite polynomial products (O'Donnell [2021], Moniri et al. [2024]) to non-centered, non-unit-variance Gaussians. We then derive computable bounds of the above expectations for general sub-Gaussian inputs. In Appendix J, we generalize key results from Vershynin [2010, 2018]—originally for centered sub-Gaussians—to the non-centered case.

   These contributions are essential for analyzing models trained on non-Gaussian separable data and offer broadly applicable tools for future theoretical work.

2. A Novel Framework for Analyzing Classifier - Deterministic Spike-Based Closed-Form Decomposition:

   We establish a foundational framework for analyzing the feature behavior of classifiers on new inputs, rather than focusing on the commonly studied regression task (Ba et al. [2022] and Table 1) or training data (See line 685). Notably, our predominant feature component β admits a closed-form, deterministic expression, unlike the stochastic feature decompositions used in recent works such as Demir and Dogan [2025] (see line 650). As a result, we are able to directly analyze the output of the learned feature map in Section 4.

3. (Question 2) Counterintuitive Finding – Importance of Relevant Classes, not Many Classes:

   While it is widely believed that increasing the number of training samples improves feature quality (Brown et al., 2020; An et al., 2023; line 337), our results in Corollary 4.7 and Experiments V–VI suggest a more nuanced view—namely, that only semantically aligned classes contribute meaningfully to expressivity. This challenges the conventional emphasis on quantity and opens a promising direction aligned with recent work in coreset selection, where reducing the number of training classes rather than just the number of samples can enhance feature transfer.

4. Unifying empirical observations and prior theories - Theoretical Quantification of Feature Transfer Phenomena:

   Our work offers a theoretical contribution by unifying empirical observations and prior theories, clarifying when feature transfer succeeds or fails (Section 4.1). While feature transfer is known to work better when unseen inputs resemble training data, previous metric learning studies mainly focus on performance gains without consistent explanations (Chopra et al. 2005, Liu et al. 2018, El-Nouby et al., 2021, Caron et al., 2021, Ermolov et al. 2022). In contrast, our framework precisely identifies both when feature transfer works (Propositions 4.4 and 4.5) and when it breaks down, such as when multiple unseen classes map to the same spike β, causing Separability to fail despite high similarity. This quantitative characterization of transfer success and failure is a key strength of our work (See 671).

Response to Clarity

Finally, we appreciate your suggestion regarding the introduction and presentation. In the revised version, we will:

1. Simplify the introduction and text by breaking down longer sentences and providing intuitive context before diving into formalism.
2. Separate intuition and theory throughout the paper to improve accessibility.