Structured Spectral Reasoning for Frequency-Adaptive Multimodal Recommendation

We sincerely thank you for your thorough and insightful review, and appreciate the valuable comments and discussions you provided. We are especially grateful for your deep understanding of our work, it is clear that you have fully grasped the core ideas of our approach, and the questions you raised are highly meaningful. We have prepared the following responses in the hope of addressing some doubts.

Q1| About Application Scenarios

Our framework is designed to be general and extensible, accommodating various types of auxiliary signals by modeling them as graph-defined fields and analyzing their frequency characteristics. In that sense, choosing images‑and‑texts for the first study is an instantiation rather than a restriction. The same framework remains valid so long as two conditions hold:
1) the side information can be represented on the same (or an aligned) node set;
2) frequency carries meaning that the downstream task cares about.

(1) Knowledge‑graph Recommendation as a Natural Extension

Knowledge graphs (KGs) offer rich relational structures that naturally align with our spectral reasoning paradigm. Each relation induces its own Laplacian, whose spectral components reflect both hierarchical and fine-grained semantics, analogous to the frequency bands in multimodal signals.

In this setting:

• Our spectral decomposition generalizes to relation-wise spectra, enabling structured reasoning without relying on handcrafted metapaths. This allows the model to respect the inherent relational geometry of the KG while avoiding manual path enumeration or rigid schema constraints.
• Causal masking offers a principled alternative to heuristic pruning. In scenarios where certain relations introduce shortcut correlations (e.g., nationality as a proxy for brand), this mechanism promotes stable generalization through invariance learning.
• The cross-band fusion operator can be viewed as soft multi-hop reasoning across relation bands, compositing semantic signals in a flexible, data-driven manner. This enables the model to learn adaptive reasoning chains over KG structures.
• Finally, the alignment module ensures coherence between learned ID embeddings and structured KG entities, reinforcing consistency across behavioral and factual spaces. This helps bridge the gap between collaborative signals and external knowledge.

(2) Beyond KGs: Extending to Knowledge and Structure

The frequency-centric paradigm naturally extends beyond multimodal recommendation. It offers a unified lens to model diverse auxiliary signals such as structured knowledge, user attributes, or interaction contexts by interpreting their graph spectral signatures.

• Contextual signals (time, location, device) possess multi‑scale periodicities that manifest as distinct spectral signatures; causal masking filters volatile bursts while retaining structural seasonality.
• Social graphs contribute an overlay network whose spectrum encodes community structure; cross‑band operators let social influence modulate item signals without hand‑tuned fusion weights.
• Spatio‑temporal forecasting problems already lean on spectral thinking; our causal perturbation adds a principled route to stability against distribution shifts.

Together, these principles position our framework as a broadly applicable, theoretically grounded solution for modeling structured signals across a wide range of recommendation and other scenarios.

(3) Why Multimodal Recommendation

We chose to first validate our framework in the domain of multimodal recommendation because this scenario presents some of the most urgent and practically consequential challenges in real-world systems. In particular, multimodal-rec suffers from two critical and widely acknowledged pain points: (i) modality‑specific noise—irrelevant backgrounds, inconsistent photos, sparse or generic text that inject high‑frequency artefacts; (ii) semantic misalignment between image/text content and true user preferences. These issues are well documented in academia and industry, yet existing solutions still rely on heuristic fusion or ad‑hoc filtering, which break down in cold‑start or noisy settings. Our goal was to propose a general and theoretically grounded paradigm that can systematically handle these issues.

Moreover, multimodal Rec offers a unique combination of stress-testing and interpretability:

• It serves as a stress test because it represents a high-noise, high-stakes setting where naive fusion fails frequently and where robust generalization is difficult.
• It also provides a microscope: the availability of visual and textual features allows us to interpret frequency behaviors (e.g., distinguishing between coarse object shapes and fine-grained textures), offering qualitative validation of the causal reasoning process.

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Q2 | Relationship to the Spatial Domain

Both DGCF and GRCN begin with the same diagnosis we do: a single interaction graph entangles multiple, sometimes spurious, factors of user preference.

• DGCF tackles the problem by factorising each node embedding into several intent sub-vectors.
• GRCN refines the graph by pruning low-confidence edges during training.

These are spatial operations: they alter the neighbourhood or the embedding space directly.

Our approach keeps the diagnosis but changes the coordinate system. We ask:

Where does the entanglement live in the graph spectrum?

We find that global semantics, useful local patterns, and high-frequency noise separate naturally along the eigen-axis.
This shift leads to three practical advantages:

1. Orthogonal partitions without manual design
   Laplacian eigenvectors are mutually orthogonal, so frequency bands are independent by construction.
   DGCF must enforce disentanglement with extra regularisers and hyperparameters;
   GRCN must learn heuristics to decide which edges to drop.

2. Causal filtering rather than edge-by-edge pruning
   Spectral Causal Masking down-weights entire bands that violate an invariance objective,
   removing whole classes of spurious signals (e.g., background textures)
   instead of individual edges. This is data-driven yet more stable than threshold-based pruning.

3. A single mechanism that generalises
   The same band decomposition works whether the signal is an image embedding,
   a sentence vector, or a KG relation embedding. This allows us to plug in new side-information
   without redesigning intent counts or pruning rules.

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Q3 |  Reason for Using CP

To enable efficient and flexible reasoning across spectral bands, our model uses a band-to-band kernel represented as a third-order tensor. However, directly learning a full 3D tensor incurs high memory and computation cost, especially as the number of bands grows.

To keep the model lean, we trial‑tested four standard tensor factorisations. For a band‑to‑band kernel tensor of size BBd:

A single experiment on Amazon‑Baby (R@20 = 0.1103) shows that CP with rank 8 achieves similar accuracy (0.1095) using only one-fifth of the parameters, while Tucker drops to 0.1068 and doubles the parameter count; other decompositions perform worse in both accuracy and size.

Beyond the numbers, CP offered two practical advantages: (i) its separable factors train stably without rank scheduling, unlike high‑rank Tucker; (ii) the three factor matrices are easy to inspect and prune, each column corresponding to a latent “band‑interaction motif.” Given this accuracy‑to‑size balance and optimisation stability, CP became the default throughout the paper and other decompositions remain compatible with the framework.

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Q4  |  Reason for Gated Attention

Spectral bands differ not only in content but also in energy scale and reliability. A plain soft‑max attention would force the model to assign all probability mass somewhere, even when a band is mostly noise. By inserting a learnable gate, we decouple two roles:

1. Relevance estimation (attention weights): Which other bands matter for the current band?
2. Reliability gating (sigmoid gate): How much of that information should actually be let through?

This separation lets the model suppress unreliable bands without renormalising the whole distribution, giving finer control in noisy multimodal settings.

Here is a supporting ablation on the Amazon-Baby dataset to justify our use of the gated attention mechanism for cross-band fusion:

Gated attention therefore offered the best trade‑off between performance, stability, and complexity, and became our default choice.

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Once again, we deeply appreciate your careful and detailed review. Your suggestions have been very inspiring, and we will incorporate additional analysis and clarifications based on your feedback in the revised version.

We sincerely thank you for your thoughtful review of this paper. Your comments are highly valuable to us. For the concerns you raised, we hope the following responses can provide effective clarification.

[Reply to Questions]

1. More Evidence of Scalability

We appreciate your suggestion about scalability of our approach. In response, we provide new empirical results demonstrating that SSR scales effectively across diverse datasets and maintains competitive efficiency in both training and inference.

(1) Generalization to a new distributionally distinct dataset

To test SSR’s robustness under different content modalities and interaction patterns, we conduct additional experiments on a TikTok short video dataset, which features rich and noisy multimodal inputs (visual, acoustic, and textual) and exhibits distributions significantly different from Amazon.

SSR achieves the highest accuracy, showing a +5.2% gain in Recall@20 over the strong CL-based method AlignRec, and outperforming the spectral baseline SMORE as well. These results confirm SSR’s ability to generalize and scale effectively in real-world short video scenarios.

(2) Practical runtime efficiency across baselines

Beyond the theoretical complexity discussed in Appendix D, we have benchmarked actual runtime performance. SSR maintains competitive training efficiency and comparable inference speed:

SSR converges faster than SMORE while achieving higher accuracy, and operates within a similar memory and inference cost envelope. Unlike many CL-based methods that entangle all modal features into a global latent space, our approach isolates causal and informative bands, allowing stable learning even under distribution shifts.

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2. Robustness of Frequency Decomposition and Causal Masking

Yes, we conducted three controlled perturbation experiments on the Amazon-Baby dataset to verify that frequency decomposition and Spectral Causal Masking (SCM) improve robustness against different forms of noise and spurious signals. Starting from the clean scores of SMOER (R@20=0.1035) and SSR (R@20=0.1096), we compare three variants:

(1) SMORE (spectral baseline without decomposition/masking),
(2) SSR w/o SCM (our model with decomposition but no causal masking),
(3) Full SSR (with both decomposition and SCM).

Findings:

• Full SSR consistently outperforms both SMORE and the ablated variant across all perturbation types.
• The performance drop under perturbation is smallest for full SSR, demonstrating that SCM effectively filters unstable high-frequency features, and that frequency decomposition provides a structure for isolating and correcting noisy modal signals.

These results provide evidence that the proposed frequency-aware design is robust to noise and spurious modality-specific variations.

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3. Concrete Example for Interpretability

To provide evidence of interpretability beyond aggregate metrics, we present a band‑wise qualitative case study on the Amazon‑Baby dataset, showcasing how different frequency components influence the final ranking decision.

User context: The user has recently browsed breast‑pumping accessories and milk‑storage solutions.
Candidate pair: Item A = “Medela CSF Milk‑Storage Bags (50 pack)” , Item B = “Fisher‑Price Nesting Action Vehicles (toy set)”

How each band affects the decision

• Low‑band (global semantics). Encodes product category: “milk‑storage bags” vs. “toy vehicles”. Removing it hurts B more, because the toy’s global semantics are less relevant to the user’s recent interest.
• Mid‑band (functional cues). Captures keywords like “pre‑sterilised”, “BPA‑free”, and the zipper seal icon on Item A’s package, exactly the features matching breast‑feeding needs. Masking this band flips the ranking, proving it is the decisive component.
• High‑band (noisy details). Contains packaging glare or marketing adjectives; its removal scarcely changes the ranking.

Recommendation explanation:

We prioritised Item A because its mid‑frequency features, such as ‘pre‑sterilised milk‑storage’ and the zipper‑seal graphic, match user's recent searches for milk‑pumping accessories. When we hide these features, the ranking reverses in favour of the toy set, confirming that the mid‑band information drives this recommendation.

This counterfactual analysis illustrates that each spectral band carries distinct semantics, and that mid‑band alignment, highlighted and preserved by our frequency decomposition with SCM, directly explains why Item A is recommended over Item B.

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[Reply to Cons]

1. Complexity and Data Support:

While our method introduces spectral decomposition and causal masking, both are applied in a modular and efficient manner. As shown in our runtime benchmark results (see Q1), SSR achieves comparable training time and inference speed to prior methods such as SMORE and MMGCN. Specifically, SSR reaches the best validation performance faster, with no major inference overhead. This demonstrates that the proposed method, though conceptually novel, is practically efficient and easy to deploy.

2. The proposed Improvements are not Experimentally Verified:

We have conducted targeted perturbation experiments (see Q2) designed to test the robustness of our frequency decomposition and SCM modules under modality-specific noise and spurious correlations. These controlled trials confirm that both components provide complementary robustness benefits. Further, to demonstrate the interpretability promised by our design, we present a band-wise qualitative case study (see Q3), analyzing how different frequency components affect final item ranking in a real-world setting.

3. Missing Baseline Analysis

We sincerely appreciate your suggestion. For all key analysis experiments (e.g., robustness tests, interpretability study), we have included baseline comparisons such as SMORE and ablated variants (e.g., SSR w/o SCM) to ensure fair and intuitive evaluation. These comparisons are reported in Q2 and Q3. We will further clarify this in the revised version to improve readability.

4. Limitations:

We sincerely appreciate your suggestion. Below, we outline some limitations of our current method that could inspire future improvements and extensions：

Although Structured Spectral Reasoning (SSR) performs well and is robust on multiple datasets, it has several limitations. First, the model relies on a one‑off computation of Laplacian eigenvectors as its spectral basis; on extremely large or rapidly evolving graphs, frequent node or edge updates would require incremental eigensolver techniques or polynomial approximations to avoid expensive recalculation. Second, the current implementation uses frozen CLIP/BERT multimodal embeddings for efficiency, meaning any bias or noise in the upstream encoders is passed unchanged to SSR; end‑to‑end fine‑tuning or lightweight distillation could further improve representation quality. Finally, SSR’s interpretability is confined to the band level—it can quantify how low, mid, and high frequencies contribute to the score and how SCM re‑weights them, but it cannot pinpoint specific image regions or text tokens as pixel‑ or token‑level attention maps do, limiting its usefulness in scenarios that demand extremely fine‑grained auditing; future work could combine differentiable visualisation or local adversarial analysis to reveal semantic sources within each band.

We hope the above responses have addressed your concerns. If you have any further questions, please feel free to leave additional comments. We would deeply appreciate it if you could consider revisiting your evaluation of our work. Thank you again for your kind support.

We sincerely thank you for your careful and thoughtful review of our work. Your comments are highly valuable to us. For the concerns you raised, we hope the following responses can provide effective clarification.

[Reply to Questions]

1. Inconsistency between Figures 3 & 4:

Thank you for your careful reading regarding Figure 4. Figure 4 measures distance between modalities (see the y-axis and title), rather than similarity. The high frequency band exhibits larger cross‑modal distances, which is consistent with Figure 3’s visualization of noisy, less‑aligned distributions in the high‑frequency space.

2. About Visual Clarity:

Thank you for the suggestion. We will refine all figures in the revised version by improving visual clarity and adding more informative annotations and legends to aid understanding. Specifically, we plan to highlight key modules in the pipeline and use clearer labels for axes and metrics.

3. Cross-modal Recommendation Baselines:

Several recent recommender systems treat cross‑modal alignment as their main contribution, e.g. MENTOR (AAAI2025) aligns image and text features at ID‑, item‑, and graph level; MMSSL (WWW 2023) introduces cross‑modal contrast with adversarial perturbations; AlignRec (CIKM2024) shrinks the modal gap via global auto‑encoding; and DiffMM (ACM’MM 2024) adds diffusion‑based augmentation before contrastive alignment. All four methods operate in the time domain and align the entire feature vector, so high‑frequency noise or spurious correlations are blended into the objective. By contrast, our SSR first decomposes each modality into equal‑energy spectral bands, uses Spectral Causal Masking to drop non‑causal (often high‑freq) bands, and then applies band‑wise spectral contrast; this frequency‑aware pipeline preserves useful interaction signals while avoiding noise propagation. Empirically, the same implementation setup shows that SSR improves Recall@20 by ≈ 5 pp over MENTOR and AlignRec, and by ≈ 10 pp over MMSSL and DiffMM across three Amazon domains and a TikTok‑style dataset, while using fewer parameters than AlignRec and similar training time to SMORE. These results confirm that fine‑grained, spectrum‑aware alignment is both conceptually cleaner and practically more effective than existing full‑band cross‑modal algorithms.

4. Actual Training and Inference Runtimes:

Beyond the asymptotic complexity in §4.4, we have benchmarked actual runtime performance. The table below compares several multimodal baselines in terms of average training time per epoch, time to best validation performance, and inference throughput:

As shown, our SSR maintains competitive training efficiency and inference speed. It reaches the best validation result faster than SMORE while achieving higher accuracy and better robustness, as discussed in §4.2 and §4.4.

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[Reply to Weaknesses:]

1. Clarity of the Decomposition module:

Thank you for noticing the ambiguity. The three frequency‑specific representations (low, mid, high) are indeed derived from the same modality via a per‑modality graph Fourier transform. Due to space constraints, we did not elaborate on this detail in the original version. Thank you for pointing it out—we will include a clarifying description in the revised version at key locations (e.g., §3.1 and Figure 1’s caption) to avoid potential misunderstandings.

2. Missing Citations on Cross‑modal Alignment:

Thank you for the suggestion regarding cross-modal recommendation references. Our spectral alignment module was indeed inspired by prior work on cross-modal alignment, including AlignRec [16], MMSSL [56], MENTOR[57], and CLIPER [58]. We designed our SCR module to build on these ideas while incorporating frequency‑band semantics. We will explicitly cite these works in the Related Work section and discuss their differences from our method in §3.

3. Dataset Diversity:

Most of the baselines we compare against (e.g., DualGNN [27], SLMRec [30], LATTICE [33], BM3 [31], FREEDOM [29], MMIL [35], AlignRec [16], SMORE [5]) were originally designed and benchmarked on Amazon datasets with standardized preprocessing protocols. To ensure fair comparison, we used the same three Amazon subdomains (Baby, Sports, Clothing), which also provide natural distribution shifts between domains. Beyond Amazon, we have conducted preliminary experiments on a TikTok‑style recommendation dataset. The results (Recall@20) are: MMGCN = 0.0730, SLMRec = 0.0845, AlignRec = 0.1174, SMORE = 0.1185, Our SSR = 0.1237. SSR shows a consistent improvement of +4.2 pp over SMORE. We are currently running the full suite of baselines on this dataset and will include the complete comparison in the final version. Additionally, we fully agree with your suggestion and are planning to expand our benchmark suite with new standardized datasets such as KuaiRand‑Rec and MIND‑News.

4. Inconsistency between Figures 3 & 4:

Thank you for your careful reading regarding Figure 4. Figure 4 measures distance between modalities (see the y-axis and title). The high frequency band exhibits larger cross‑modal distances, which is consistent with Figure 3’s visualization of noisy, less‑aligned distributions in the high‑frequency space.

5. Significance Analysis; Code and Dataset not Released:

We have conducted statistical significance testing using paired t-tests (Recall@20 and NDCG@20) comparing SSR to SMORE. On all three datasets, the p-values are < 0.005, well below the 0.05 threshold. We will include the complete test results in the revised version. Regarding the dataset and code: we used the same publicly available MMRec‑processed datasets as the baselines. We are also ready to release our code, and we can release at any time if it’s needed.

6. Figure 4 not Referenced / Duplicated with Appendix

Thank you for pointing out the reference mismatch between Figure 4 and the Appendix. Our original intention was to refer to Figure 4 in Section 4.6, and present Figure 5 in the Appendix as a detailed counterpart. We mistakenly cited the appendix figure in the main text and will fix this in the revision. Thank you again for the helpful observation.

We hope the above responses have addressed your concerns. If you have any further questions, please feel free to leave additional comments. We would deeply appreciate it if you could consider revisiting your evaluation of our work. Thank you again for your kind support.

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[56] Wei, W., Huang, C., Xia, L. and Zhang, C., 2023, April. Multi-modal self-supervised learning for recommendation. In Proceedings of the ACM web conference 2023 (pp. 790-800).

[57] Xu, J., Chen, Z., Yang, S., Li, J., Wang, H. and Ngai, E.C., 2025, April. Mentor: multi-level self-supervised learning for multimodal recommendation. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 39, No. 12, pp. 12908-12917).

[58] Wu, X., Huang, A., Yang, H., He, H., Tai, Y. and Zhang, W., 2024. Towards Bridging the Cross-modal Semantic Gap for Multi-modal Recommendation. arXiv preprint arXiv:2407.05420.

Dear Area Chair,

Thank you for your continued engagement and for providing these insightful questions. We appreciate the opportunity to clarify our experimental choices and future plans. As the discussion period concludes, we have made every effort to provide a thorough and comprehensive response to all the points you have raised.

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1. On the Choice of Datasets (Amazon vs. MovieLens/Netflix)

We are grateful for your question regarding the inclusion of popular recommendation benchmarks like MovieLens and Netflix. We agree that these are seminal datasets that have driven significant progress in the field. Our decision to initially focus on the Amazon suite was guided by our specific research context, multimodal recommendation, and we would like to elaborate on our rationale.

Initial Rationale:

The primary reason for prioritizing the Amazon datasets stems from our focus on evaluating models in an inherently multimodal setting. The Amazon datasets are particularly well-suited for this, as they naturally bundle user-item interactions with rich, corresponding modal features (product images and textual reviews). This allows for a clean and direct evaluation of a model's ability to leverage multimodal information, adhering to what has become a de facto standard preprocessing protocol in numerous recent multimodal recommendation papers.

In contrast, datasets like MovieLens and the public Netflix Prize dataset are fundamentally "unimodal" in their raw form, primarily containing user-item interaction data (e.g., ratings). To use them in a multimodal context, researchers must augment them with external data, a non-trivial process:

• MovieLens: This is a classic user-item rating dataset. To incorporate multimodality, one must source and align external information, typically by linking movie IDs to databases like IMDb, TMDb, or OMDb to fetch posters (visual), plot summaries (textual), genres, and other metadata. While valuable, this process introduces variability depending on the external source, the API version, and the data cleaning pipeline. Pioneering works like MMGCN [1] have indeed constructed their own multimodal versions of MovieLens, which we highly respect.
• Netflix Prize Dataset: The official public dataset is similarly focused on user ratings. The rich multimodal content associated with Netflix titles (e.g., posters, video trailers, subtitles, audio) is proprietary and protected by copyright, making it unavailable for direct, large-scale public release by Netflix. As with MovieLens, researchers (such as in MMSSL [2]) have creatively constructed multimodal versions by linking to external, legally accessible sources.

Frankly speaking, because a large portion of the reference baselines in the multimodal recommendation literature are standardized and many of them use only the three Amazon multimodal datasets, we were concerned that if we ran experiments on just one of them, we might be criticized for not covering all three. Coupled with strict page limits and the non-trivial computational cost of running and reporting on multiple datasets, we therefore prioritized the Amazon datasets as our highest priority in the initial submission.

That said, we genuinely believe it is important to include more widely recognized datasets in order to make our evaluation broader and more convincing. Therefore, we commit that in the camera-ready version we will include experiments on at least one additional popular dataset drawn from either MovieLens or Netflix, in addition to our existing benchmarks.

Notably, responding to earlier reviewer comments on dataset diversity, we have already conducted a full set of experiments on the TikTok dataset, as curated by the authors of MMSSL [2]. This dataset represents a completely different domain (short-form video) from e-commerce and provides a strong test of our model's adaptability. Our preliminary results are still promising:

In conclusion, we commit to expanding our experiments to the benchmarks you mentioned. This means the final paper will present results on at least five datasets: the three Amazon benchmarks, TikTok, and either MovieLens or Netflix. We believe this significantly strengthens the empirical evidence and demonstrates the robustness and versatility of our proposed method.

[1] Wei, Yinwei, et al. "MMGCN: Multi-modal graph convolution network for personalized recommendation of micro-video." Proceedings of the 27th ACM international conference on multimedia. 2019.

[2] Wei, Wei, et al. "Multi-modal self-supervised learning for recommendation." Proceedings of the ACM web conference 2023. 2023.

2. On the RecVAE Baseline

We highly appreciate the significance of RecVAE [Shenbin et al., WSDM’20], which is a well-recognized and influential work in the development of variational autoencoder-based collaborative filtering. The model introduces several elegant and well-motivated enhancements over the classical Mult-VAE framework, including a composite prior, user-specific KL divergence scaling, and alternating encoder–decoder training, and has inspired a broad range of subsequent research in recommender systems.

We acknowledge that RecVAE represents one of the most advanced non-multimodal baselines in the recommender systems literature. However, our current work has been primarily situated in the multimodal recommendation domain, where the majority of widely adopted baselines focus on modality alignment, fusion, and graph-based modeling. Because RecVAE does not directly operate in the multimodal setting and requires substantial additional adaptation, it was not included in our initial set of experiments. Our baseline pool was drawn almost entirely from prior multimodal recommendation studies to ensure modality-consistent comparisons.

That said, we fully agree with the AC that including RecVAE would further strengthen the completeness of our evaluation. We have therefore initiated an emergency experimental run to incorporate RecVAE into our benchmark suite. We commit to adding RecVAE results into the camera-ready version, and we will keep pushing experiments to completion under the tight AOE deadline of August 8. If any results are available before the final submission cut-off, we will immediately update them — even at the very last minute.

Thank you once again for your invaluable guidance throughout this process. Your feedback has been instrumental in helping us improve the quality and completeness of our work.

Best regards,

The Authors

We sincerely thank you for your thoughtful review of our paper. Your feedback has been helpful, and the questions you raised are highly valuable. Below we provide our responses to your concerns, and we genuinely hope they help clarify any confusion.

Reply to Weakness 1 : “GFT may be computationally prohibitive.

Thank you for raising this very valuable question, it is indeed something we have carefully considered. As a result, we introduced several runtime optimizations during implementation. Specifically, our implementation keeps the Graph Fourier Transform lightweight in three ways:

• Sparse, low‑rank eigenbasis. We compute only the top‑$k$ Laplacian eigenvectors with a sparse Lanczos solver, not the full spectrum. For the graphs in our experiments this preprocessing finishes quickly on a standard CPU and requires only modest memory. We also note there that performance is stable when k is varied in {32,64,128}, and chose 64 as a trade-off.

• One‑time, reusable preprocessing. The eigenbasis is generated once and reused throughout training and inference, so its cost is amortised and does not affect per‑epoch throughput.

• Scalable fallback. For very large graphs we provide a Chebyshev‑polynomial approximation that eliminates eigendecomposition altogether while preserving almost the same accuracy.

After this step, each forward pass reduces to a simple dot product with a small, quantised eigenbasis, making the runtime overhead negligible. Empirically, our end‑to‑end training time matches strong spectral baselines, confirming that GFT preprocessing is not a bottleneck. In addition, please refer to the “4. Actual Training and Inference Runtimes” section in our Reply to Reviewer YRJP for detailed runtime efficiency results.

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Reply to Weakness 2 : – “Why choose equal‑energy spectral bands?”

The motivation for using equal-energy bands stems from both theoretical considerations and empirical performance:

• Balanced information allocation. In spectral analysis of graph signals, low-frequency components typically dominate total signal energy. Using equal-width frequency intervals would result in high- and mid-frequency bands containing very little information, leading to unstable training or underutilized components. By contrast, equal-energy partitioning ensures that each frequency band contains a comparable portion of the total energy, allowing all bands to contribute meaningfully to learning.

• Robustness and performance. We conducted controlled experiments comparing different band-splitting strategies on the Amazon-Baby dataset. The results are summarized below:

Equal-energy bands outperform both static and learnable alternatives in terms of both overall accuracy and robustness under perturbation.

• Adaptability. This method adapts naturally to different datasets without requiring manual tuning, as the cumulative energy profile reflects the inherent spectral characteristics of the graph.

• Interpretability. Because each band carries roughly one-third of the total energy, we can more intuitively analyze where semantic alignment or noise concentration occurs, as shown in Figures 3 and 4.

These results validate our decision to use equal-energy partitioning as a principled and effective design choice. We will add more ablation details to Appendix F in the revised version.

Once again, we greatly appreciate your careful review. If you have any further questions, please feel free to leave additional comments. If possible, we would also be grateful if you could consider giving a more positive evaluation of our work. We truly appreciate your support and feedback.

Dear Reviewers (especially cgiv and KUNi),

Please feel free to respond to the above comment as well: have you read the theoretical results in the appendix, and if so, what did you think of them?

Best regards,

AC

Dear Area Chair and All Other Reviewers,

We are sincerely grateful for your detailed feedback and for providing clear, constructive options for us to move forward. We completely agree that the theoretical sections in our initial version lacked the necessary rigor. Your critique was not only correct but also tremendously helpful.

First and foremost, to ensure the paper meets the highest standards, we promise to remove all theoretical claims in the camera-ready version if the paper is accepted.

However, your insightful comments provided such a clear and actionable roadmap that we felt compelled to attempt the first option you proposed. Guided by your specific advice, we have performed a complete overhaul of our theoretical contributions. We believe this new version, which was directly inspired and shaped by your critique, now possesses the rigor and clarity we were previously missing.

We present these comprehensive revisions below, not as a rebuttal, but as a testament to the value of your guidance. We would be honored if you would consider them, and we eagerly await your final judgment on whether they have earned their place in the paper.

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Regarding the Circular Definition and Vague Setting of Theorem 2 (Spectral Causal Masking):
We acknowledge that the original formulation suffered from a circular definition of "non-causal" and that the learning setting was not precisely defined. We have addressed this in three major ways:

1. Formal Definition of Spuriousness: We have introduced Assumption 1, which formally defines "spurious" frequency components based on a standard conditional independence criterion from the causality literature. This breaks the logical circularity by providing an external definition of spuriousness that is independent of our model's behavior.

2. Clarified Learning Setting: We have introduced Assumption 2, which explicitly states that the graph topology is considered fixed and given in our analysis. This resolves the ambiguity you correctly pointed out regarding the randomness of the graph and its spectral components. The randomness in our setting now arises solely from the sampling of user-item pairs.

3. Reframed Theorem Statement and Proof: We have reframed previous Theorem 2 (now Theorem 1, as the original Theorem 1 has been removed) to be a rigorous statement about how our Spectral Causal Masking (SCM) objective encourages the learned predictor to become invariant to the spurious components defined in Assumption 1. The new proof demonstrates that minimizing the SCM loss pushes the model towards a state where its predictions are insensitive to these spurious features. We also explicitly mention that we use a standard BCE loss as a surrogate for the ranking metrics, a common practice we now state clearly.

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Regarding the Trivial Nature of Theorem 1 (Linearity of Decomposition):
We concur with your assessment that labeling this basic property as a "Theorem" was an overstatement, , and we have accordingly removed the original Theorem 1.

• We have downgraded this result to Proposition 1 and rewritten it with a clear, formal statement of its purpose: to provide the mathematical justification for the decomposability of linear operators used in our framework. We have also added an Implication section to clarify its precise role in our methodology, ensuring its contribution is not overstated.

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Regarding the Circularity and Lack of Specificity of Theorem 3 (G-HSNO Approximation):
We agree that the original statement was circular and, more importantly, that it failed to justify why a low-rank approximation is particularly suitable for our problem setting.

• Non-Circular Statement: We have rewritten the original Theorem 3 as Theorem 2 (following the removal of the previous Theorem 1), a standard universal approximation statement, removing the circular logic. It now clearly states that the G-HSNO architecture can approximate any bilinear frequency interaction operator.

• Justification for the Low-Rank Inductive Bias: This was your most profound point. We have added a new dedicated discussion subsection: "Justification for the Low-Rank Assumption in Recommendation". In this section, we argue why a low-rank interaction model is a suitable inductive bias for this specific problem. We hypothesize that cross-frequency interactions in recommendation are not arbitrary but are governed by structured semantic rules (e.g., global context modulating fine-grained details), which can be efficiently captured by a low-rank model. This reframes the low-rank choice from a mere efficiency heuristic to a principled modeling decision.

[3] Wu, Felix, Amauri Souza, Tianyi Zhang, Christopher Fifty, Tao Yu, and Kilian Weinberger. "Simplifying graph convolutional networks." In International conference on machine learning. Pmlr, 2019.

[4] Cossock, David, and Tong Zhang. "Subset ranking using regression." International conference on computational learning theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.

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Dear Authors,

Many thanks for your reply and for your detailed comments and renewed attempts at the proofs. I appreciate your efforts to engage not only with the reviewers but also with me. I have a few comments/questions.

Please note there is an error in the expression of the mutual information: the conditioning in $I(y; x^{(m)}\| x\_{\neq m})$ doesn’t show, though I believe this is due to Markdown issues.

Secondly, you mention in your rebuttal that “In finite datasets we cannot observe $I(y; x^{(m)}\| x\_{\neq m})$ exactly”. There are several problems with this. First of all, I don’t think that is obviously true in the strict sense. It is certainly possible for random variables taking values over a discrete set to be independent (whether there exist graphs which can induce such representations in the frequency space is different question).

Secondly, I understand that it is true in practice: you probably never have a single example where it makes any sense to define the independence, because there are no two examples $x_1,x_2$ such that $x_2^{(m)}\neq x_1^{(m)}$ but $[x_1]\_{\neq m}=[x_2]\_{\neq m}$. This further underscores the the problem with this definition: it can really only serve, at most, as some idealised intuitive concept. This limitation should be clearly explained in the camera-ready version or resubmission, together with the finite sample identifiability remark you added.

Regarding the assumption of fixed graph topology. I strongly recommend you include this assumption, together with ample explanations of how very, very strongly limiting it is in any camera-ready version or resubmission. If someone were to attempt to claim, for instance, generalization bounds (or implications of the result for generalization purposes in a setting where it is natural to think of the graph as random) for node-based graph learning whilst hiding the fact that the graph is assumed to be fixed, that would be unforgivably misleading to an extend that previous acceptance of the practice in the literature would not be a valid excuse. In particular, I am having trouble understanding the statement “This assumption is commonly adopted in spectral GNNs and graph-based recommendation theory [1,2], where the graph Laplacian serves as a deterministic frequency basis.” Could you clarify where in the references in question the assumption is made? I object to the term “graph-based recommendation theory” here: learning theory usually refers to generalization analyses, which to the best of my understanding are absent from both [1] and [2].

I will continue to answer the other parts of the proofs in the next couple of hours.

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Regarding Theorem 1 (SCM Quantitatively Enforces Invariance)

We greatly appreciate your careful reading and precise critique. In our effort to formalize the proof, our revision made the statement technically correct. Yet, as you rightly point out, this formalization exposed the weakness of its substantive meaning.

As you explained, $S(f)$ measures robustness to removing specific "non-causal" bands, while $\mathcal{L}\_{\text{SCM}}$ measures robustness to broad, random spectral perturbations. The theorem merely established an upper-bound connection between the two, essentially noting that one particular masking event (removing all non-causal bands) is included in the expectation that defines $\mathcal{L}\_{\text{SCM}}$, and thus its probability-weighted effect bounds $S(f)$.

This inequality does not, and cannot, prove that minimizing $\mathcal{L}\_{\text{SCM}}$ discriminates between causal and non-causal bands. In hindsight, we recognize this is far from the causal identification mechanism we had hoped to convey, and retaining it could risk misleading readers.

Therefore, we will remove this theorem and its proof entirely from the final version. Instead, we will present SCM purely as a stability-promoting heuristic: it encourages invariance to spectral perturbations, without making any formal claims about causal discrimination.

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Regarding Theorem 2 (Quantitative CP-Rank Bound)

We appreciate your detailed comments, which helped us identify substantive flaws in our original formulation. Upon closer examination, we realized that several parts of our argument were not sufficiently rigorous or well-grounded for our specific setting. In particular, our bound relied on results for generic tensors that do not transfer directly to the structured cross-band interaction tensors in our model, and parts of our reasoning implicitly conflated matrix rank properties after matricization with CP rank properties, an unjustified step in this context. Moreover, our citation of Chiantini–Ottaviani’s Theorem 1.1 was misplaced, as that result addresses uniqueness conditions rather than rank bounds.

Given these issues, we have decided that the most responsible course of action is to remove Theorem 2 (Quantitative CP-Rank Bound for G-HSNO) and all associated proofs and discussion from the final version. Instead of making formal CP-rank claims, we will retain the low-rank aspect purely as a modeling heuristic, motivated by the structured nature of cross-frequency interactions in our task. In the camera-ready version, we will clearly state this without attaching unsupported formal guarantees, ensuring the presentation remains accurate and transparent.