Fading to Grow: Growing Preference Ratios via Preference Fading Discrete Diffusion for Recommendation

We sincerely thank you for your positive and constructive feedback. We greatly appreciate your recognition of our theoretical contributions, the novelty of the preference fading–growing framework, and the strong empirical results across multiple benchmarks. In response to your concerns, we have conducted a more in-depth and quantitative analysis of the model's scalability. We hope this additional analysis effectively addresses your concerns. Should you have any further questions or suggestions, we would be more than happy to engage in further discussion.

Comment 1: significant computational challenges when scaling to very large item sets due to the all-pair preference ratio modeling.

We conduct a quantitative analysis of PreferGrow's scalability to address your concerns raised in Comment 1. Additionally, we provide a more detailed explanation of the promising solution mentioned in the Limitations section.

I. PreferGrow adds only $\mathcal O(N)$ Complexity in network, loss and inference, comparable to contemporary diffusion-based recommenders. The table below compares model, loss, and inference complexities with other baselines. L: history length, N: #items, B: #neg samples, d: hidden dim, T: #diffusion steps

• Model Parameters: We never materialise the full $N^{2}$ preference‑ratio matrix; Equation (14) only requires $N$ scores, so the extra cost is $\mathcal O(Nd)$.
• Loss Computation: Leveraging the idempotent fading matrix $E$, Equation (4) simplifies (see App. C.1–C.4) to element‑wise mean, indexing, and a pre‑computable constant—also $\mathcal O(N)$. The seemingly complex score entropy loss consists of three parts: positive and negative terms (involving simple mean and indexing operations, $\mathcal{O}(N)$), and a constant term (ensuring non-negativity, dependent only on $t$, and pre-computable with $\mathcal{O}(T)$ complexity, $T=20$).
• Inference cost: Idempotency gives $E\cdot s_{\theta}(x_t,t,u)=\bigl(\mathbf p_T^{\top}s_{\theta}/\mathbf1^{\top}\mathbf p_T\bigr)\mathbf1$ reducing Equation (18) from $\mathcal O(N^{2})$ to $\mathcal O(N)$.
• Modeling Target: The $\mathcal{O}(N^2)$ complexity only arises from its desired modeling target. This more expressive target contributes to a higher performance ceiling for PreferGrow.
• Scalability to extremely large item set (billions of items): In this case, even $\mathcal{O}(N)$ becomes impractical. As we have mentioned in section 5, there might exists several alternatives that can mitigate this challenge including but not limited to sparse matrix operations and reducing the item ID space into $m$ smaller (size $c$) semantic ID spaces. And for the latter PreferGrow with SID alternative, we provide the analytical result in the last line in the table.

To better address this concern, we will add an additional appendix section for the analytical computational cost and extend the discussion of the scalability in section 5.

II. Incorporating Semantic IDs is a promising next step for PreferGrow. As noted in the Section 5 Limitations, incorporating Semantic IDs offers a promising direction for further scaling PreferGrow. Semantic IDs (SIDs) represent each item using $m$ codebooks of size $c$, thereby enabling the representation of up to $c^m$ unique items. Since $c^m \gg N \gg mc$, SIDs provide a compact yet expressive representation space. This allows PreferGrow on SIDs to reduce its computational complexity from $\mathcal{O}(N)$ to $\mathcal{O}(mc)$ .

However, existing works on SIDs — such as Tiger [1], ActionPiece [2], DDSR [3] and OneRec [4] — have not been open-sourced. To truly deploy PreferGrow in real-world industrial scenarios, one must also address the open challenge of large-scale multimodal SIDs pre-training. Extending PreferGrow to operate on SIDs is our ongoing future work.

We acknowledge that the discussion in the Limitations section was too general. We will re-analyze the limitations quantitatively and include the corresponding details in the Appendix. We hope that this quantitative analysis provides a clearer understanding of PreferGrow's scalability and helps alleviate your concerns regarding its potential computational challenges.

Thank you once again for your valuable feedback. We remain receptive and ready to engage with any further questions or suggestions you may have.

References

[1] Rajput, Shashank, et al. "Recommender systems with generative retrieval." NeurIPS 2023.

[2] Xie, Wenjia, et al. "Breaking determinism: Fuzzy modeling of sequential recommendation using discrete state space diffusion model." NeurIPS 2024.

[3] Hou, Yupeng, et al. "ActionPiece: Contextually Tokenizing Action Sequences for Generative Recommendation." ICML 2025.

[4] Deng, Jiaxin, et al. "Onerec: Unifying retrieve and rank with generative recommender and iterative preference alignment." ArXiv 2025.

We would like to express our gratitude once more for your positive and constructive feedback. This message serves as a friendly reminder as we approach the conclusion of the discussion phase. We sincerely hope that the improvements during rebuttal will be taken into consideration.

We also would like to share some recent advancements in our design of the fading matrix, which may help further address concerns regarding realism and model flexibility.

Improving Realism with Rank-$r$ Fading Matrix (Cluster-wise Replacement)

We have already presented the rank-1 idempotent fading matrix in the paper. $\mathbf{E}=\dfrac{\vec{p}_T \mathbf{1}^{\top}}{\mathbf{1}^{\top}\vec{p}_T}$.

Extending to rank-$r$ idempotent fading matrix

We now generalize this to a rank-$r$ idempotent fading matrix

Physical intuition

• Clustering: Partition the item set $\mathcal{I}$ ($|\mathcal{I}|=N$) into $r$ clusters $\{\mathcal{C}_1,\dots,\mathcal{C}_r\}$, where items in the same cluster are more similar.

To obtain the item clusters, one may apply K-means or other vector-based clustering methods to semantic embeddings of items; alternatively, $r$-way classification can be performed on the user–item bipartite graph or the weighted sequential interaction graph using GCN or similar approaches. Since these methods rely on additional side information beyond our current framework, we leave a full exploration of cluster construction to future work.

• Cluster-wise target distributions: Define

Because $\vec{p}_T^{~i},\forall i= 1,\cdots,r$ are disjoint, the resulting $\mathbf{E}$ is idempotent with rank $r$.

• Effect: During fading, each preferred item is replaced only by items within the same cluster, filtering out cross-cluster candidates and better respecting item heterogeneity. Hence the rank-$r$ fading matrix offers improved realism. Furthermore, rank-$r$ idempotency also reduces the cost of $\mathbf{E} \cdot \mathbf{x} = \left(\sum\limits_{i=1}^{r}\frac{\mathbf{x}^{\top}\vec{p}_T^{i}}{\mathbf{1}^{\top}\vec{p}_T^{i}}\right) \vec{1}$ from $\mathcal{O}(N^2)$ to $\mathcal{O}(rN)$.

We believe that PreferGrow equipped with a rank-$r$ fading matrix — cluster-wise replacement —combines theoretical elegance with practical realism. We will add this rank-$r$ extension as an extra appendix section and cite it in the main text. We hope this addresses your concerns about realism and inspires further research and applications.

We sincerely appreciate your thoughtful summary and encouraging evaluation of our work. Your valuable comments regarding the model’s scalability, training efficiency, and sensitivity to the personalization strength parameter $w$ have motivated us to conduct a more in-depth and quantitative analysis of these aspects. We hope that our analysis effectively addresses your concerns. If you have any further questions or suggestions, we would be glad to discuss them with you.

Comment 1: model’s scalability

We conduct a quantitative analysis of PreferGrow's scalability to address the concerns raised in Comment 1. Additionally, we provide a more detailed explanation of the promising solution mentioned in the Limitations section.

I. PreferGrow adds only $\mathcal O(N)$ Complexity in network, loss and inference, comparable to contemporary diffusion-based recommenders. The table below compares model, loss, and inference complexities with other baselines. L: history length, N: #items, B: #neg samples, d: hidden dim, T: #diffusion steps

• Model Parameters: We never materialise the full $N^{2}$ preference‑ratio matrix; Equation (14) only requires $N$ scores, so the extra cost is $\mathcal O(Nd)$.
• Loss Computation: Leveraging the idempotent fading matrix $E$, Equation (4) simplifies (see App. C.1–C.4) to element‑wise mean, indexing, and a pre‑computable constant—also $\mathcal O(N)$. The seemingly complex score entropy loss consists of three parts: positive and negative terms (involving simple mean and indexing operations, $\mathcal{O}(N)$), and a constant term (ensuring non-negativity, dependent only on $t$, and pre-computable with $\mathcal{O}(T)$ complexity, $T=20$).
• Inference cost: Idempotency gives $E\cdot s_{\theta}(x_t,t,u)=\bigl(\mathbf p_T^{\top}s_{\theta}/\mathbf1^{\top}\mathbf p_T\bigr)\mathbf1$ reducing Equation (18) from $\mathcal O(N^{2})$ to $\mathcal O(N)$.
• Modeling Target: The $\mathcal{O}(N^2)$ complexity only arises from its desired modeling target. This more expressive target contributes to a higher performance ceiling for PreferGrow.
• Scalability to extremely large item set (billions of items): In this case, even $\mathcal{O}(N)$ becomes impractical. As noted in Section 5, possible solutions include sparse matrix operations and reducing the item space to $m$ smaller semantic ID spaces of size $c$.

To better address this concern, we will add an additional appendix section for the analytical computational cost and extend the discussion of the scalability in section 5.

II. Incorporating Semantic IDs (SIDs) offers a promising next step, further reducing complexity from $\mathcal{O}(N)$ to $\mathcal{O}(mc)$. As noted in the Section 5, incorporating Semantic IDs offers a promising direction for further scaling PreferGrow. Semantic IDs (SIDs) represent each item using $m$ codebooks of size $c$, thereby enabling the representation of up to $c^m$ unique items. Since $c^m \gg N \gg mc$, SIDs provide a compact yet expressive representation space. This allows PreferGrow on SIDs to reduce its computational complexity from $\mathcal{O}(N)$ to $\mathcal{O}(mc)$ :

However, existing works on SIDs (Tiger [1], DDSR [2], ActionPiece [3], OneRec [4]) have not been open-sourced. To truly deploy PreferGrow in real-world industrial scenarios, one must also address the open challenge of large-scale multimodal SIDs pre-training. Extending PreferGrow to operate on SIDs is our ongoing future work.

We acknowledge that the discussion in the Limitations section was too general. We will re-analyze the limitations quantitatively and include the corresponding details in the Appendix. We hope that this quantitative analysis provides a clearer understanding of PreferGrow's scalability and helps alleviate your concerns.

Comment 2: training efficiency

I. After our quantitative analysis, we further attribute the longer training time of PreferGrow to a more fundamental mismatch between the modeling target complexity and the model capacity. As shown in the complexity table, PreferGrow models an all-pair preference ratio objective with a complexity of $\mathcal{O}(N^2)$, while the model itself has a network complexity of only $\mathcal{O}(d^2 + Nd)$. When $N=10{,}000$ and $d=256$, the modeling target becomes nearly 40× more complex than the model's representational capacity. We also analyzed training dynamics and observed that the last 50% of training time contributes only ~5% gain in NDCG@5, indicating redundancy in the training process. This is partially due to uniform timestep sampling. Designing a dynamically adaptive timestep schedule is an exciting direction for our future work.

II. Incorporating Semantic IDs (SIDs) offers a promising solution, effectively eliminating the complexity imbalance. PreferGrow operates on $m$ codebooks of size $c$, leading to a modeling target complexity of $\mathcal{O}(mc^2)$ and model complexity of $\mathcal{O}(md^2 + mcd)$. When $d = c = 256$, these complexities become well matched, resolving the current imbalance.

Comment 3: sensitivity to the personalization strength parameter $w$

The personalization strength $w$ is locally stable within a reasonable range and highly consistent across data splits, thus posing no practical challenge for tuning. As shown in Figure 4 of our paper, PreferGrow is sensitive to large-magnitude changes in the personalization strength parameter $w$ (e.g., from 0 to 20). On datasets like Steam, we observe that performance remains locally stable within a reasonable range (e.g., $w \in [5, 15]$), but drops notably when $w$ is set too small (e.g., $w=1$). This highlights the need to set $w$ appropriately for each dataset. To assess sensitivity more systematically, we tested PreferGrow under $w \in [0, 2, 5, 10]$ and evaluated the mean absolute error between the optimal $w$ values selected on the training, validation, and test sets. These optimal values are determined by uniformly sampling performance across 40 checkpoints throughout the training process and optimality is defined by maximizing the combined metric $HR@5 + HR@10 + NDCG@5 + NDCG@10$. Our findings show that the optimal $w$ is highly consistent across data splits, indicating that $w$ can be tuned on a validation (or even training) set like other standard hyperparameters. This ensures that tuning $w$ does not pose a practical challenge.

Comment 4: Do you have depolyed your method into real world industrial ?

We are actively working on building high-quality, large-scale multimodal Semantic IDs, which is a key challenge for real-world deployment. We plan to apply PreferGrow with Semantic IDs in industrial settings once this foundation is established.

Thank you once again for your valuable feedback. We hope that our efforts during the rebuttal period will be taken into account in your evaluation. We remain open and willing to discuss any new questions or suggestions you might have.

References

[1] Rajput, Shashank, et al. "Recommender systems with generative retrieval." NeurIPS 2023.

[2] Xie, Wenjia, et al. "Breaking determinism: Fuzzy modeling of sequential recommendation using discrete state space diffusion model." NeurIPS 2024.

[3] Hou, Yupeng, et al. "ActionPiece: Contextually Tokenizing Action Sequences for Generative Recommendation." ICML 2025.

[4] Deng, Jiaxin, et al. "Onerec: Unifying retrieve and rank with generative recommender and iterative preference alignment." ArXiv 2025.

We would like to share some recent advancements in our design of the fading matrix, which may help further address concerns regarding realism and model flexibility.

We have already presented the rank-1 idempotent fading matrix in the paper. $\mathbf{E}=\dfrac{\vec{p}_T \mathbf{1}^{\top}}{\mathbf{1}^{\top}\vec{p}_T}$.

Extending to rank-$r$ idempotent fading matrix

We now generalize this to a rank-$r$ idempotent fading matrix

Physical intuition

• Clustering: Partition the item set $\mathcal{I}$ ($|\mathcal{I}|=N$) into $r$ clusters $\{\mathcal{C}_1,\dots,\mathcal{C}_r\}$, where items in the same cluster are more similar.

To obtain the item clusters, one may apply K-means or other vector-based clustering methods to semantic embeddings of items; alternatively, $r$-way classification can be performed on the user–item bipartite graph or the weighted sequential interaction graph using GCN or similar approaches. Since these methods rely on additional side information beyond our current framework, we leave a full exploration of cluster construction to future work.

• Cluster-wise target distributions: Define

Because $\vec{p}_T^{~i},\forall i= 1,\cdots,r$ are disjoint, the resulting $\mathbf{E}$ is idempotent with rank $r$.

• Effect: During fading, each preferred item is replaced only by items within the same cluster, filtering out cross-cluster candidates and better respecting item heterogeneity. Hence the rank-$r$ fading matrix offers improved realism. Furthermore, rank-$r$ idempotency also reduces the cost of $\mathbf{E} \cdot \mathbf{x} = \left(\sum\limits_{i=1}^{r}\frac{\mathbf{x}^{\top}\vec{p}_T^{i}}{\mathbf{1}^{\top}\vec{p}_T^{i}}\right) \vec{1}$ from $\mathcal{O}(N^2)$ to $\mathcal{O}(rN)$.

We believe that PreferGrow equipped with a rank-$r$ fading matrix — cluster-wise replacement —combines theoretical elegance with practical realism. We will add this rank-$r$ extension as an extra appendix section and cite it in the main text. We hope this addresses your concerns about realism and inspires further research and applications.

We sincerely thank you for your time and valuable comments. We appreciate your recognition of the technical soundness of our approach, especially our effort to address the mismatch between continuous diffusion modeling and the inherently discrete nature of user interactions in recommendation systems.

Summary: ... to learn item representations ... the target item embedding corruption ... the corrupted item embedding ...

From your summary, it appears there is a key misunderstanding of our method. Specifically, in PreferGrow the variables $x_0$ and $x_t$ denote discrete item IDs — that is, $x_0, x_t \in [1, 2, \dots, N]$ — rather than continuous embeddings. In our notation, only $\mathbf{x}_0, \mathbf{x}_t$ represents the continuous embedding. We will clarify this distinction more explicitly in the revised manuscript. Following the pipeline illustrated in Figure 2, PreferGrow consists of the following key components:

1. Forward Perturbation Process: Given a preferred item $x_0$ (an integer item ID), we retain it as $x_t = x_0$ with probability $\alpha_t$, and replace it with another item $x_t \ne x_0$ with probability $1 - \alpha_t$. The replacement strategy is governed by a pre-defined fading matrix.

2. Modeling Target: The model learns preference ratios $s_{\theta}(x_t, t, u)$, which quantify how user $u$ prefers other items over the current item $x_t$. A positive value indicates user $u$ prefers other items more than $x_t$; a negative value suggests the opposite.

3. Backward Generation Process: From the estimated preference ratios $s_{\theta}(x_t, t, u)$, we compute the user’s preference distribution over all items: $p(i \mid u), i \in [1, 2, ..., N]$. The user embedding thus conditions the distribution $p(i \mid u)$, allowing PreferGrow to generate personalized item distributions tailored to each user—i.e., recommend different items to different users.

We hope that our detailed explanation helps you gain a clear understanding of PreferGrow.

Comment 1: the clarity and writing

We devoted considerable effort to carefully writing and organizing the technical parts of our paper, as well as visualizing the core pipeline in Figure 2. In fact, other reviewers (1sA3, 3oh1, XkWC, rfEc) all rated our Clarity as Good or Excellent. We believe that your concerns may stem primarily from a misunderstanding of our notation, specifically regarding the meaning of $x_0$ and $x_t$. In our response to your summary, we have clarified that $x_0$ and $x_t$ denote discrete item IDs, not continuous embeddings. We also re-explained the PreferGrow pipeline in detail. We hope that this clarification helps you better understand our method.

Comment 2: incorporation of user embedding

PreferGrow incorporates user embeddings through conditional modeling to support personalized recommendations. We clarify the role of the user embedding in PreferGrow, as introduced in the Preliminaries section. PreferGrow generates a discrete probability distribution over the item set, i.e., $p(i),\ i \in [1, 2, \dots, N]$. To support personalized recommendations, the model estimates a user-conditioned distribution $p(i \mid u)$, where $u$ denotes the user embedding. In our framework, $u$ is derived from the user's interaction history using the SASRec module; specifically, we use its final hidden representation as the conditional input to generate $p(i \mid u)$. Additionally, to compute the unconditional distribution $p(i)$ (i.e., without user information), we introduce a learnable non-preference embedding $\phi$, such that $p(i \mid \phi) = p(i)$. Actually, this conditional modeling approach is widely adopted in other diffusion-based recommendation models.

Comment 3: theoretical contributions

We respectfully disagree that the theoretical contributions are marginal, as they provide a necessary foundation for future research and model development. While it is true that our theoretical proofs focus on establishing the general correctness of the PreferGrow framework, we believe their value is far from marginal. First, a well-founded theoretical basis is crucial for enabling future developments—just as the success of DDPMs [1] and DPO [2] was grounded in solid theoretical foundations that paved the way for a broad range of follow-up work. Second, by providing clear and rigorous general proofs, we offer a reusable theoretical foundation that future studies can build upon—especially when deriving more specialized results tailored to recommendation tasks.

Comment 4: efficiency comparison.

Thank you for your helpful suggestion. To address your concern, we provide a qualitative comparison of computational complexity between PreferGrow and other baselines, covering model parameters, loss computation, modeling targets, and inference cost,as summarized in the table below. (L: history length, N: number of items, B: number of neg samples, d: hidden dim, T: steps of diffusion inference)

• Model Parameters: We never materialise the full $N^{2}$ preference‑ratio matrix; Equation (14) only requires $N$ scores, so the extra cost is $\mathcal O(Nd)$.
• Loss Computation: Leveraging the idempotent fading matrix $E$, Equation (4) simplifies (see App. C.1–C.4) to element‑wise mean, indexing, and a pre‑computable constant—also $\mathcal O(N)$. The seemingly complex score entropy loss consists of three parts: positive and negative terms (involving simple mean and indexing operations, $\mathcal{O}(N)$), and a constant term (ensuring non-negativity, dependent only on $t$, and pre-computable with $\mathcal{O}(T)$ complexity, $T=20$).
• Inference cost: Idempotency gives $E\cdot s_{\theta}(x_t,t,u)=\bigl(\mathbf p_T^{\top}s_{\theta}/\mathbf1^{\top}\mathbf p_T\bigr)\mathbf1$ reducing Equation (18) from $\mathcal O(N^{2})$ to $\mathcal O(N)$.
• Modeling Target: The $\mathcal{O}(N^2)$ complexity only arises from its desired modeling target. This more expressive target contributes to a higher performance ceiling for PreferGrow.
• Scalability to extremely large item set (billions of items): In this case, even $\mathcal{O}(N)$ becomes impractical. As noted in Section 5, possible solutions include sparse matrix operations and reducing the item space to $m$ smaller semantic ID spaces of size $c$.

To better address this concern, we will add an additional appendix section for the analytical computational cost and extend the discussion of the scalability in section 5.

We also supplement this with empirical training and inference statistics, demonstrating that PreferGrow achieves superior performance without introducing additional overhead. We provide the number of training batches of size 256 ( Train B) and inference GFLOPs as $|X|$ increases, compared with DDSR.

Overall, PreferGrow exhibits comparable algorithmic complexity to existing methods, while featuring a more expressive modeling objective, which contributes to its higher performance ceiling. Under similar complexity, PreferGrow achieves significant performance gains without adding inference overhead by learning more expressive preference ratios.

We sincerely hope that our response has effectively addressed your concerns. If so, we would be grateful for your consideration in raising the score. Should you have any remaining questions or suggestions, please feel free to share them—we remain committed to actively engaging with your feedback and continuously improving our work.

References:

[1] Ho, Jonathan, Ajay Jain, and Pieter Abbeel. "Denoising diffusion probabilistic models." NeurIPS 2020.

[2] Rafailov, Rafael, et al. "Direct preference optimization: Your language model is secretly a reward model." NeurIPS 2023.

We would like to express our gratitude once more for the valuable feedback provided by the reviewer. This message serves as a friendly reminder as we approach the conclusion of the discussion phase. We sincerely hope that the improvements during rebuttal will be taken into consideration.

We also would like to share some recent advancements in our design of the fading matrix, which may help further address concerns regarding realism and model flexibility.

Improving Realism with Rank-$r$ Fading Matrix (Cluster-wise Replacement)

We have already presented the rank-1 idempotent fading matrix in the paper. $\mathbf{E}=\dfrac{\vec{p}_T \mathbf{1}^{\top}}{\mathbf{1}^{\top}\vec{p}_T}$.

Extending to rank-$r$ idempotent fading matrix

We now generalize this to a rank-$r$ idempotent fading matrix

Physical intuition

• Clustering: Partition the item set $\mathcal{I}$ ($|\mathcal{I}|=N$) into $r$ clusters $\{\mathcal{C}_1,\dots,\mathcal{C}_r\}$, where items in the same cluster are more similar.

To obtain the item clusters, one may apply K-means or other vector-based clustering methods to semantic embeddings of items; alternatively, $r$-way classification can be performed on the user–item bipartite graph or the weighted sequential interaction graph using GCN or similar approaches. Since these methods rely on additional side information beyond our current framework, we leave a full exploration of cluster construction to future work.

• Cluster-wise target distributions: Define

Because $\vec{p}_T^{~i},\forall i= 1,\cdots,r$ are disjoint, the resulting $\mathbf{E}$ is idempotent with rank $r$.

• Effect: During fading, each preferred item is replaced only by items within the same cluster, filtering out cross-cluster candidates and better respecting item heterogeneity. Hence the rank-$r$ fading matrix offers improved realism. Furthermore, rank-$r$ idempotency also reduces the cost of $\mathbf{E} \cdot \mathbf{x} = \left(\sum\limits_{i=1}^{r}\frac{\mathbf{x}^{\top}\vec{p}_T^{i}}{\mathbf{1}^{\top}\vec{p}_T^{i}}\right) \vec{1}$ from $\mathcal{O}(N^2)$ to $\mathcal{O}(rN)$.

We believe that PreferGrow equipped with a rank-$r$ fading matrix — cluster-wise replacement —combines theoretical elegance with practical realism. We will add this rank-$r$ extension as an extra appendix section and cite it in the main text. We hope this addresses your concerns about realism and inspires further research and applications.

Thank you for your thoughtful and constructive comments. We appreciate your recognition of our framework’s originality, theory, clarity, and empirical impact. Your insights — especially on scalability and realism — guided valuable analyses and improvements outlined below.

Comment 1: Fundamental Scalability Flaw:

I. PreferGrow adds only $\mathcal O(N)$ Complexity in network, loss and inference. The table below compares model, loss, and inference complexities with other baselines. L: history length, N: #items, B: #neg samples, d: hidden dim, T: #diffusion steps

• Model Parameters: We never materialise the full $N^{2}$ preference‑ratio matrix; Equation (14) only requires $N$ scores, so the extra cost is only $\mathcal O(Nd)$.
• Loss Computation and Inference: Thanks to idempotency, $\mathbf{E} \cdot \mathbf{x} = \left(\vec{p}_T^{\top} \mathbf{x} / \vec{1}^{\top} \vec{p}_T\right) \vec{1}$ reduces the cost from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$. The seemingly complex score entropy loss consists of three parts: positive and negative terms (involving simple mean and indexing operations, $\mathcal{O}(N)$), and a constant term (ensuring non-negativity, dependent only on $t$, and pre-computable with $\mathcal{O}(T)$ complexity).
• Modeling Target: The $\mathcal{O}(N^2)$ complexity only arises from its desired modeling target. This more expressive target contributes to a higher performance ceiling for PreferGrow.
• Scalability to extremely large item set (billions of items): In this case, even $\mathcal{O}(N)$ becomes impractical. As noted in Section 5, possible solutions include sparse matrix operations and reducing the item space to $m$ smaller semantic ID spaces of size $c$.

To better address this concern, we will add an additional appendix section for the analytical computational cost and extend the discussion of the scalability in Section 5.

II. Incorporating SIDs is a promising next step to further reduce complexity from $\mathcal{O}(N)$ to $\mathcal{O}(mc)$. SIDs encode each item via $m$ codebooks of size $c$, allowing up to $c^m$ items with only $\mathcal{O}(mc)$ cost. Extending PreferGrow to operate on SIDs is our ongoing future work.

Comment 2: Theoretical "Elegance" at the Cost of Realism:

I. The assumption of a shared target distribution $\vec{p}_T$ is widely adopted in mainstream generative models, such as VAE, DDPM, and RecFlow, which typically assume a common Gaussian target. Although this is a strong simplifying assumption, it has nonetheless demonstrated impressive generative power across diverse domains like text, images, and videos. In our case, the shared $\vec{p}_T$ also ensures robust cold-start generation, whereas DDSR relies on the user’s historical interaction distribution, making it difficult to generalize to cold-start users.

II. PreferGrow can also personalize the target distribution to reflect individual user preferences by setting $\vec{p}_T = \text{MLP}(\text{SASRec}_u)$. More importantly, PreferGrow explicitly models the non-preference user state vector $\vec{\phi}$, enabling cold-start generation via $\vec{p}_T = \text{MLP}(\vec{\phi})$, even in the absence of historical interactions.

III. Incorporating semantic IDs into PreferGrow opens the door to modeling item heterogeneity more effectively. For instance, if the semantic ID of a shirt is [32, 32, 63] and that of a refrigerator is [13, 24, 17], then all clothing items may share the prefix [32, 32, *]. This allows fine-grained control in the fading process — for example, fading a shirt into other clothing can be achieved by modifying only the last ID. Similarly, fading earlier IDs enables broader transitions in item space, supporting a more flexible and semantically aware perturbation process.

Comment 3: Questionable Reliability of Experimental Evaluation:

To ensure the reliability of our findings, we have conducted a comprehensive re-evaluation of all experiments under multiple random seeds $[100, 200, 300, 400, 500]$ and statistical significance testing (p<0.001). The new results consistently confirm the significant performance gains of PreferGrow over baselines under varying seeds. All updates will be included in the revised manuscript. We sincerely thank you for highlighting this important issue. Some of the results (NDCG@5) are shown below.

Comment 4: Marginal Contribution vs. Disproportionate Complexity:

I. PreferGrow’s overall complexity is $\mathcal{O}(N)$ and its performance gains have been statistically verified. Please refer to the earlier discussion.

II. PreferGrow’s another contribution lies in its unified modeling objective that serves both generative modeling and ranking.

Specifically, the preference ratio $\log\frac{p_t(y)}{p_t(x_t)}$ is both ranking-oriented and a discrete approximation of the score function $\nabla \log p_t(x_t)$ in generative diffusion models:

Optimizing preference ratios thus promotes correct rankings and improves log-likelihood estimation. The use of preference ratios and idempotent fading is not added complexity, but a principled design that offers both theoretical grounding and practical gains.

Comment 5: Further Discussion on Scalability:

Under similar complexity, PreferGrow achieves significant performance gains without adding inference overhead by learning more expressive preference ratios. We provide the number of training batches of size 256 ( Train B) and inference GFLOPs as $|X|$ increases, compared with DDSR.

Only a sparse non-preference vector $\vec{p}_T$ is needed to construct a sparse, idempotent fading matrix. However, based on our analysis, incorporating semantic IDs is a more promising direction. PreferGrow with SIDs reduces model complexity to $\mathcal{O}(mc)$ and target modeling to $\mathcal{O}(mc^2)$, further enhancing scalability and efficiency.

Comment 6: The Deeper Connection Between "Preference Fading" and Negative Sampling:

Even a simple uniform replacement yields strong performance, further demonstrating the effectiveness and robustness of the PreferGrow framework. We appreciate your insightful question. As discussed at the end of Section 3.1.2, prior negative sampling techniques can indeed inspire the design of the fading matrix. We agree that incorporating more mature or complex negative sampling strategies has strong potential to enhance performance. We have already explored this by introducing a learnable vector as the non-preferred state, which also shows promising gains (Table 1).

Comment 7: Sensitivity Issue of Personalization Strength w

The personalization weight $w$ is locally stable and consistent across data splits, making it tunable like a standard hyperparameter. As shown in Figure 4, PreferGrow is sensitive to extreme values (e.g., $w = 0$ or $20$) but stable within a practical range ( $w \in [5, 15]$). We further evaluated $w \in [ 0, 2, 5, 10 ]$ and found minimal deviation in the optimal $w$ across training, validation, and test sets, confirming its robustness.

Comment 8: Explanation of $x\_{-1}$ in the Point-Wise Setting:

Yes, $x_{-1}$ is a learnable embedding. Using a real item as a global negative may conflict with some users' preferences, while a learnable embedding offers a consistent and conflict-free negative reference.

Thanks again for insightful feedbacks.

We would like to express our gratitude once more for the valuable feedback provided by the reviewer. This message serves as a friendly reminder as we approach the conclusion of the discussion phase. We sincerely hope that the improvements during rebuttal will be taken into consideration.

We also would like to share some recent advancements in our design of the fading matrix, which may help further address concerns regarding realism and model flexibility.

Improving Realism with Rank-$r$ Fading Matrix (Cluster-wise Replacement)

We have already presented the rank-1 idempotent fading matrix in the paper. $\mathbf{E}=\dfrac{\vec{p}_T \mathbf{1}^{\top}}{\mathbf{1}^{\top}\vec{p}_T}$.

Extending to rank-$r$ idempotent fading matrix

We now generalize this to a rank-$r$ idempotent fading matrix

Physical intuition

• Clustering: Partition the item set $\mathcal{I}$ ($|\mathcal{I}|=N$) into $r$ clusters $\{\mathcal{C}_1,\dots,\mathcal{C}_r\}$, where items in the same cluster are more similar.

To obtain the item clusters, one may apply K-means or other vector-based clustering methods to semantic embeddings of items; alternatively, $r$-way classification can be performed on the user–item bipartite graph or the weighted sequential interaction graph using GCN or similar approaches. Since these methods rely on additional side information beyond our current framework, we leave a full exploration of cluster construction to future work.

• Cluster-wise target distributions: Define

Because $\vec{p}_T^{~i},\forall i= 1,\cdots,r$ are disjoint, the resulting $\mathbf{E}$ is idempotent with rank $r$.

• Effect: During fading, each preferred item is replaced only by items within the same cluster, filtering out cross-cluster candidates and better respecting item heterogeneity. Hence the rank-$r$ fading matrix offers improved realism. Furthermore, rank-$r$ idempotency also reduces the cost of $\mathbf{E} \cdot \mathbf{x} = \left(\sum\limits_{i=1}^{r}\frac{\mathbf{x}^{\top}\vec{p}_T^{i}}{\mathbf{1}^{\top}\vec{p}_T^{i}}\right) \vec{1}$ from $\mathcal{O}(N^2)$ to $\mathcal{O}(rN)$.

We believe that PreferGrow equipped with a rank-$r$ fading matrix — cluster-wise replacement —combines theoretical elegance with practical realism. We will add this rank-$r$ extension as an extra appendix section and cite it in the main text. We hope this addresses your concerns about realism and inspires further research and applications.

Since the concerns have been resolved, we hope that our improvements will be taken into consideration, and we would greatly appreciate it if you could increase the score.

We sincerely thank you for your time and thoughtful comments. We appreciate your recognition of our discrete diffusion framework, the principled design of the idempotent fading matrix, and the alignment of preference ratio modeling with ranking objectives.

Comment 1: the reference items at each step appear to be mostly independent of one another

The reference items at each step are in fact not independent in PreferGrow. As explicitly shown in Equation (7) of our paper, we define a probabilistic transition matrix from step $s$ to step $t$, which models how reference items evolve over time and the progress involves item correlations. The perception of independence may stem from a misunderstanding. For example, in item-level diffusion models, the item embedding and the Gaussian noise are independent. However, the inter-step correlation is introduced by the gradual increase of noise during the forward process, which perturbs the item embeddings progressively [1] [2]. Similarly, in PreferGrow, inter-step correlation arises from the changing probability of replacing the current item. As the diffusion progresses, the probability of remaining at the current item decreases, while the probability of transitioning to other items increases. This smooth shift reflects a temporal dependency, analogous to how noise accumulation introduces correlation in continuous diffusion models. We will add a brief note in the revision to clarify this confusing point.

Comment 2: PreferGrow degenerates into weighted negative sampling under idempotent fading, losing the key strengths of diffusion.

PreferGrow introduces correlated reference items, enabling indirect optimization of $\log p\_{\text{data}}(x\_0)$ without sacrificing the core advantages of diffusion models. It is true that during preference fading, when the current item is replaced by others, PreferGrow may appear similar in mechanism to weighted negative sampling. However, this does not mean it loses the core advantages of diffusion models. The core advantages of diffusion models lies in their ability to introduce a sequence of correlated latent variables $\{x\_t\}\_{t=1}^{T}$ across timesteps through noise injection. These latent variables form a correlated trajectory that enables the indirect optimization of the log-likelihood of the original data distribution $\log p_{data}(x\_0)$, which is intractable to optimize directly but becomes easier to optimize in each diffusion-recover with intermediate latent variables. The inter-step correlation of latent variables has already been addressed in our response to Comment 1. Here, we emphasize that PreferGrow retains the log-likelihood optimization property. Specifically, the modeling target in PreferGrow is the preference ratios, proved to be a discretized version of the score function $\nabla \log p\_{t}(x\_t)$ [3].

As a result, optimizing the preference ratios under score entropy loss is equivalent to optimizing the log-likelihood of the latent variables, which effectively serves as a surrogate for the original data log-likelihood ( proved in SEDD [3]). In summary, PreferGrow preserves the core advantages of diffusion modeling —— correlated latent variables across timesteps and their role in indirectly optimizing the log‑likelihood of the original data distribution. Moreover, its resemblance to a weighted negative‑sampling mechanism in the forward process further enhances its suitability for recommendation tasks, given the well‑established effectiveness of negative sampling in recommendation.

Comment 3: Instead of using the unified fading‑matrix settings described in Appendices C.1–C.3, the fading matrix should retain the underlying semantic or structural correlations.

In addition to the unified fading‑matrix settings, we introduce an adaptive setting with a learnable fading matrix $\vec{p}_T=\text{Softmax}(\vec{\theta})$ that can adaptively adjust the replacement probability for other items. The empirical results for this adaptive variant are reported in Table 1, and the theoretical details are provided in Appendix C.4. Furthermore, PreferGrow can also personalize the target distribution to reflect individual user preferences by setting $\vec{p}_T = \text{MLP}(\text{SASRec}_u)$. More importantly, PreferGrow explicitly models the non-preference user state vector $\vec{\phi}$, enabling cold-start generation via $\vec{p}_T = \text{MLP}(\vec{\phi})$, even in the absence of historical interactions.

Thank you once again for your valuable feedback. We hope that our efforts during the rebuttal period will be taken into account in your evaluation. We remain open and willing to discuss any new questions or suggestions you might have.

Comment 4: The idempotent matrix E should not be designed in such a uniform way. At the very least, some degree of semantic or structural correlation should be preserved in the matrix

I. Our use of a uniform fading matrix was intentional—to validate PreferGrow’s effectiveness under the simplest setting. Even a simple uniform fading matrix $E$ yields strong performance, further demonstrating the robustness and effectiveness of the PreferGrow framework. We also appreciate your insightful suggestions.

II. To capture structural correlations, the fading distribution $\vec{p}_T$ can be designed using advanced negative sampling techniques. As discussed in Section 3.1.2, prior negative sampling strategies can inspire the design of $\vec{p}_T$. We agree that incorporating more sophisticated methods has strong potential to enhance performance. Furthermore, adaptive designs such as $\vec{p}_T = \text{Softmax}(\vec{\theta})$ and $\vec{p}_T = \text{MLP}(\text{SASRec}_u)$ (mentioned in our earlier response) can also capture meaningful structural patterns.

III. To model semantic correlations, integrating Semantic IDs offers a promising direction we are actively exploring. For example, if the semantic ID of a shirt is [32, 32, 63] and that of a refrigerator is [13, 24, 17], then all clothing items may share the prefix [32, 32, *]. This enables fine-grained fading—e.g., fading a shirt into other clothing by modifying only the last ID. Similarly, modifying earlier IDs allows broader transitions in item space, enabling a more flexible and semantically-aware perturbation process.

References:

[1] Ho, Jonathan, Ajay Jain, and Pieter Abbeel. "Denoising diffusion probabilistic models." NeurIPS 2020.

[2] Song, Yang, et al. "Score-based generative modeling through stochastic differential equations." ICLR 2021.

[3] Lou, Aaron, Chenlin Meng, and Stefano Ermon. "Discrete diffusion modeling by estimating the ratios of the data distribution." ICML 2024.