Nonparametric Modern Hopfield Models

We thank the reviewer for the valuable comments. We have revised our paper to address each point.

The revised draft is in this anonymous Dropbox folder. All modifications are marked in BLUE color. Thanks!

1. Critical Discussion on Limitations (Newly Added Appendix B.2 https://imgur.com/a/R4PLZMs)

• Fixed Feature Mapping (Reviewer Concern: Realistic Embedding Scenarios)

  • We base our theoretical analysis on a fixed feature mapping.
  • This choice streamlines convergence proofs but restricts the model if features evolve.
  • Yet, this condition is easy to meet. Moreover, in practice, static memory model do not have evolving feature map.

• Robustness to Random Masking (Reviewer Concern: Potential Retrieval Failures)

  • Our retrieval guarantee assumes the correct memory item remains in the support set.
  • If the random mask excludes it, retrieval may fail.
  • While this assumption is restrictive, it is necessary for tractable analysis.

• Well-Separated Patterns for Exponential Capacity (Reviewer Concern: Data Correlations)

  • Our exponential capacity analysis requires patterns to be sufficiently distinct.
  • This simplifies theoretical arguments, yet real datasets often have correlated patterns.
  • Yet, this setup is standard for characterizing capacity in literature (Krotov & Hopfield, 2016; Demircigil et al., 2017; Ramsauer et al., 2020; Iatropoulos et al., 2022; Hu et al., 2023; Wu et al., 2024b; Santos et al., 2024a;b). Moreover, it can be relaxed by optimizing the Hopfield-energy landscape for larger memory capacity.

• Unproven Extensions in Section 5 (Reviewer Concern: Additional Convergence and Capacity Proofs)

  • We outline several model extensions but do not provide full proofs (no convergence and capacity guarantees).

We see these points as opportunities for growth. Addressing them will extend our framework to broader settings and make retrieval more robust. We also invite the reviewer to check the newly added Appendix B.2 for detail in our latest revision.

2. Experiments and Sparsity (Methods And Evaluation Criteria)

• Our current experiments already simulate the conditions the theory addresses, especially regarding sparsity of patterns.
• Namely, our experiments already simulate both sparse (MNIST) and denser (CIFAR-10) data. This aligns with our theoretical assumptions.
• Our current numerical results already justify our theory. Figure 2 in Appendix G.1 shows sparse models perform better on MNIST. This matches our claim that higher sparsity improves retrieval capacity and reduces error (Proposition 4.1, Theorem 4.1).
• In MIL supervised experiments, sparse models converge faster (Corollary 4.1.1). These results match the modern Hopfield literature (e.g., Hu et al. 2023, Wu et al. 2024, Ramsauer et al. 2020).

We have revise the draft to emphasize these:

• Revised G.1 "Results": https://imgur.com/a/OS2q77m and
• Revised Fig 2 "Caption": https://imgur.com/a/JjxZwxv
• Revised G.2 "Results", "Captions" of Fig 3, Fig 4: https://imgur.com/a/50mFsX5

3. Technical Clarifications

• Monotonic Decrease of $E(x)$ (Condition T1)
  Our convergence analysis assumes that each update decreases (or does not increase) the Hopfield energy $E(x)$. This is a standard property of energy-based Hopfield models and guarantees deterministic convergence to a fixed point. It does not apply to stochastic or non-monotonic updates. We emphasize that the Hopfield memory model serves a different purpose than deep learning models: here, minimizing energy corresponds to pattern retrieval, not parameter learning. Comparing this condition to stochastic optimization in deep learning (e.g., SGD) is not appropriate—the objectives and dynamics differ fundamentally.

• Inequalities Below Eq. (3.2): We confirm these are component-wise inequalities. We clarify this in the revised text (https://imgur.com/a/wu2eW9A).

We appreciate the reviewer’s time and attention to details. The revisions should address the concern about missing limitations and clarify our theoretical and experimental contributions. We hope the updated manuscript meets your expectations.

Thanks for the constructive feedback. We have revised our paper to address each concern in detail.

The revised draft is in this anonymous Dropbox folder. All modifications are marked in BLUE color.

1. Empirical Clarity and Figure Explanations

Comment: The reviewer found our experimental setup unclear regarding retrieval steps, masking strategy, and theoretical connections. They also requested stronger links between figures and the underlying theory.

Response:

1. One-Step Retrieval: We now explicitly state that all experiments use a single feed-forward (one-step) retrieval without iterative updates: https://imgur.com/a/lAJLKR6
2. Masking Strategy: The masking is done on similarity score level following random masked attention. It sometimes removes the target memory if done randomly (e.g., Random Masking). This is what the statement in Fig2 caption is about. In contrast, top-$k$ masking retains the most relevant entries and preserves the correct memory in the set.
3. Linking Theory to Experiments: We have revised Appendix G.1 and G.2 to include direct references to the theorems and remarks in the main text:
   • Figures 2 & 3 (Thm 4.1, Prop 4.1, Coro 4.1.1): These results highlight memory capacity limits and robustness under sparsity or noise.
   • Figure 2 (Remark 4.2): Explains how different noise levels affect retrieval.
   • Figure 4: Demonstrates quick convergence and stable performance under $\epsilon$-sparse retrieval. We note that random masking can remove the target memory, violating the $\mu \in \mathcal{M}$ assumption (Theorem 3.2).

In the revised draft, we have updated the figure captions and added clarifications such that readers can immediately see the theoretical motivations for each experiment.

Examples of these changes appear in the updated Appendix G.1 ("Results") and G.2 ("Captions" for Figures 2–4). E.g., https://imgur.com/a/OS2q77m and https://imgur.com/a/JjxZwxv and https://imgur.com/a/50mFsX5.

2. Focus on Hopfield Memory Models & Novelty

Comment: The reviewer questions the distinction between Hopfield networks and Hopfield memory models, and the novelty of our approach.

Response: We emphasize that Hopfield memory models and Hopfield networks/layers are conceptually distinct:

• Hopfield memory model: A fixed content-addressable memory model with no training; retrieval is based on similarity to stored patterns. This is our focus.
• Hopfield network (e.g., Hopfield Layer): A neural layer integrated into deep learning, trained with backprop. This builds on the “Dense Associative Memory–Transformer Attention” correspondence (Ramsauer et al. 2021), later generalized by Hu et al. (2023) and Wu et al. (2024).

Our work extends the prior modern Hopfield memory model by framing it as a nonparametric regression problem. This allows efficient variants (Sec 3–4) and connects it to attention mechanisms (e.g., Performer) under a rigorous, unified theory. To our knowledge, this unification has not been done before and constitutes a novel contribution.

We have added a remark regarding above points https://imgur.com/a/E93HSM5. In the final version, we promise to move this remark to Sec 2 (background) if space allows.

3. Terminology & Background

Comment: The reviewer recommended refining terminology and providing enough background to ensure clarity.

Response:

• Cite Hoover et al. 2024: https://imgur.com/a/Ix7cp7n
• Question: Masked Target? Yes, if a pattern is masked, it will not be memorized by the SVR. This allows us use different masking strategies to accompany different sparsity patterns in sec 3.2 & E.
• Nonparametric: Yes, it does describe NPH. Despite having a matrix $W$ in the primal formulation of SVR, the model parameter is not fixed. This is more obvious in the dual form of SVR. In sec 3, we cast the retrieval dynamics $\mathcal{T}$ as SVR. The number of memories is proportional to to the number of support vectors, and hence proportional to model size. Thus the model size is not fixed. This is nonparametric (Remark 3.4). We had revised the Appendix C to strengthen this: https://imgur.com/a/E3Fnic5.
• Background in Main Text: We agree that the related work section can provide more background to main text . When finalizing the paper, we will try our best to integrate this background if space allows.
• “Memory-Enhanced Functionalities” (L415, right): We have added a remark to clarify this in Appendix F (https://imgur.com/a/E93HSM5). In the final version, we promise to move this remark to Sec 2 (background) if space allows. This remark ties them to the benefits of content-addressable memory systems in real-world applications.

Thank you for the review. Happy to discuss more :)

Reviewer’s Comment (Claims and Evidence & Experimental Designs or Analyses)

Concerns

• Claim2: The theoretical results (noise robustness, tighter retrieval error bounds) do not appear numerically verified in detail.
• Claim3: The experimental section doesn’t explicitly connect to the theory.

The paper lacks explicit discussion tying the figures/tables in Appendix G.1–G.4 to the theoretical claims.
Empirical validation of the theoretical retrieval error bounds is missing.

Authors’ Response

In response to Concerns of Claim 2, numerical verifications of the retrieval error bounds appear in Sec. G.1 and are summarized in Fig. 2 (pages 29–30).

In response to Concerns of Claim 3, we believe there might be an oversight. In the “Results” paragraphs, as well as the captions of Figs. 2 and 4, we explicitly linked our numerical findings to their underlying theoretical claims.

We feel that these “Results” paragraphs and captions are sufficiently informative (also check our concluding remarks, line 411-420, right).

We apologize for any confusion caused. We have revised the draft with better clarity (e.g., https://imgur.com/a/OS2q77m and https://imgur.com/a/JjxZwxv and https://imgur.com/a/50mFsX5.)

Reviewer’s Comment (Relation to Broader Scientific Literature)

Certain properties, such as tighter error bounds and stronger noise robustness, might be seen as natural extensions of classical Hopfield networks. What’s truly novel here?

Authors’ Response

1. Nonparametric Framework: Prior works do not formulate Hopfield retrieval as a soft-margin SVR, which is our key novelty. By bridging Hopfield networks with regression theory, we gain new insights and design flexibility (e.g., kernelized or masked retrieval).
2. Sub-Quadratic Sparse Hopfield: We extend the dense model to a computationally efficient variant while retaining classical benefits (exponential capacity, fixed-point convergence).
3. Sparsity Analysis: Our results detail when sparse retrieval can outperform dense retrieval, complementing prior sparse Hopfield analyses (Hu et al., 2023).

Reviewer’s Comment (Essential References Not Discussed)

... Amari (1972) for earlier foundational work...

Authors’ Response

Thanks for the suggestion. We had updated the draft accordingly (https://imgur.com/a/IYMKyu3).

Reviewer’s Comment (Questions for Authors)

1. Are the theoretical retrieval error bounds actually verified numerically?
2. How does this approach compare to Hu et al. (2023), which reportedly achieves better capacity and robustness in some cases?
3. Could you clarify the main takeaways of the figures/tables in Appendix G.1–G.4?
4. Are these “tighter” bounds and “stronger noise robustness” truly novel beyond standard Hopfield extensions?

Authors’ Response

1. Numerical Verification (Q1)

   • Yes. Sec. G.1 (Fig. 2) (pp. 29–30) shows empirical vs. predicted retrieval error for varied sparsity levels, confirming Theorem 4.1. Please also see https://imgur.com/a/JjxZwxv. We can add further numeric comparisons if requested.

2. Comparison with Hu et al. (2023) (Q2)

   • Fundamentally Different Mechanisms: Hu et al. rely on a data-dependent entropic regularizer (still $O(n^2)$). Our approach imposes data-independent masks for sub-quadratic retrieval.
   • Focus on Efficiency: We do not claim consistent accuracy superiority; rather, we stress efficiency (Thm 3.2).
   • Extended Theory: Our SVR-based approach and explicit sparsity scaling (Theorem 4.1, Remark 3.6) broaden the theory beyond Hu et al.’s entropic analysis.

3. Main Takeaways in Appendix G (Q3)

   • Sec. G.1–G.4: We evaluate different sparse strategies (top-$k$, random masks, etc.) under noise and large-scale tasks. Each figure corresponds to Thm 4.1 or Coro 4.11 or Prop 4.1, showing that careful sparsity can match or surpass dense retrieval in sparse regimes.

4. Novel Theoretical Insights (Q4)

   • We recast retrieval as a soft-margin SVR problem, yielding margin-based error bounds and stronger noise-robustness than purely energy-based or entropic methods (e.g., Hu et al., 2023).
   • This framework supports a diverse range of sparsity schemes—top-$k$, random masking, or learned patterns—yet still guarantees fixed-point convergence and exponential capacity.
   • Explicit sparsity-dependent bounds show how selectively ignoring irrelevant patterns can improve retrieval, providing concrete criteria for balancing accuracy and efficiency.
   • Consequently, these “tighter” bounds and “stronger” robustness are not minor tweaks to classical Hopfield models; rather, they stem from the SVR-based margin analysis and support sub-quadratic implementations for large-scale memory retrieval.

Thank you for the review. Happy to discuss more :)

The revised draft is in this anonymous Dropbox folder. All modifications are marked in BLUE color. Thanks!

Reviewer’s Comment (Claims and Evidence 1)

In the abstract, they claim the sparse model inherits the appealing theoretical properties of its dense analogue...Results like Corollary 4.11 showing the error of sparse recovery being lower than dense recovery only applies to patterns kept in the sparse model. Target patterns which are masked out may not get properly recovered. This seems to be borne out by the empirical results.

Authors’ Response

When we say the sparse model “inherits” the theoretical properties of the dense version, we mean it retains the key Hopfield guarantees (e.g., alignment with transformer attention, rapid convergence to fixed-point memories, exponential storage capacity). Our results show that introducing sparsity does not break these fundamental properties.

Yes, performance can degrade when target patterns are masked out. This reflects a general accuracy–efficiency tradeoff (see lines 421–431). However, our numerical results (Fig. 4, Sec. G.5) show that, in practice, the degradation is mild: sparse masking achieves decent accuracy while significantly improving efficiency (similar to how linear/random feature approximations to softmax perform in other contexts).

Reviewer’s Comment (Theoretical Claims)

The theoretical claims seem correct but they are of limited use if the sparse model is only good for a subset of patterns. Am I missing something?

Authors’ Response

We do NOT claim the sparse model always outperforms the dense model in all situations. Instead, we specify conditions where sparse retrieval can yield better accuracy (e.g., when many irrelevant memories introduce noise). Under those conditions, our empirical results (e.g., Fig. 2, p. 30) support the theoretical claim https://imgur.com/a/JjxZwxv.

Reviewer’s Comment (Claims And Evidence 2)

From what I can understand, the sparse Hopfield model in Hu et al. (2023) is a very different thing where the regularizer is chosen so that attention is sparse. Sparsity by random masking is not the same thing. Thus, their claims of sparsity-induced advantages ring a bit hollow to me.

Authors’ Response: Sparse Hopfield Model – Sparsity Masking vs. Regularization

We appreciate the comparison. To clarify, our model uses sparse masking (Definition 3.3), with random masking as one special case (Example 1 in line 275). In Remark 4.1 (line 368), we compared our approach to [Hu et al. 2023]. Here is a brief overview:

• Different Sparsity Mechanisms

  • Ours: Data-independent, predefined masking for efficiency.
  • Hu et al. 2023: Data-dependent, regularizer-based sparsity (but still with $O(n^2)$ complexity like [Ramsauer et al. 2020]).

• Focus on Analysis
  We study how controlling the sparsity dimension $k$ affects retrieval accuracy (Theorem 4.1). Hu et al. (2023) do not provide that level of interpretability.

• No Claim of Superiority
  We do not assert that random/sparse masking categorically outperforms regularizer-based methods; we emphasize their analytical tractability and efficiency.

Reviewer’s Comment (Questions For Authors)

Why is $\Phi$ defined in Lemma 3.1 to weigh $\xi_{\mu}$ by the attention between $x$ and $\xi_{\mu} + \delta \xi_{\mu}$, rather than $\xi_{\mu}$ itself (Eq. 3.5)?

Authors’ Response

1. SVR-Based Nonparametric Interpretation for MHM (lines 172–202; 159–169; Appendix C.1)

   • We cast Hopfield retrieval as a soft-margin SVR.
   • Each stored pattern $\xi_\mu$ is paired with $\xi_\mu + \delta\xi_\mu$.
   • This enforces the SVR model to map any noisy query near $\xi_\mu$ back to $\xi_\mu$.
   • Equation (3.2) enforces $\epsilon$-retrieval (Definition 3.2).

2. Aligning with Basin of Attraction (lines 258–274) and $\epsilon$-Retrieval

   • Hopfield models attract queries near $\xi_{\mu}$ back to $\xi_{\mu}$.
   • $\xi_{\mu} + \delta\xi_{\mu}$ defines a local basin around each pattern.
   • The SVR constraints ensure $\lvert T_{\mathrm{SVR},\Phi}(x) - \xi_{\mu}\rvert \le \epsilon$ within that basin, enabling noise-robust retrieval (from partial or corrupted queries i.e., Definition 3.2.)

3. Avoiding Trivial SVR Solutions

   • Using $(\xi_{\mu}, \xi_{\mu})$ only would let the system learn the identity (no actual denoising).
   • Including $\delta\xi_{\mu}$ forces true associative memory behavior.

Overall, using $\xi_{\mu} + \delta\xi_{\mu}$ ensures the nonparametric retrieval aligns with Hopfield’s attractor dynamics, supports noise-robust memory retrieval, and prevents trivial SVR solutions.

Thank you for the review. Happy to discuss more :)