Fading to Grow: Growing Preference Ratios via Preference Fading Discrete Diffusion for Recommendation
Authors: Guoqing Hu
An Zhang
Shuchang Liu
Wenyu Mao
Jiancan Wu
Xun Yang
Xiang Li
Lantao Hu
Han Li
Kun Gai
Xiang Wang
Venue: NeurIPS 2025 poster ()
Keywords: Recommendation
Discrete Diffusion
Preference Ratios
Primary area: deep_learning
TL;DR: preference fading discrete diffusion tailored for recommendation via modeling preference ratios

Abstract: Recommenders aim to rank items from a discrete item corpus in line with user interests, yet suffer from extremely sparse user preference data. Recent advances in diffusion models have inspired diffusion-based recommenders, which alleviate sparsity by injecting noise during a forward process to prevent collapse of perturbed preference distributions. However, current diffusion‑based recommenders predominantly rely on continuous Gaussian noise, which is intrinsically mismatched with the discrete nature of user preference data in recommendation. In this paper, building upon recent advances in discrete diffusion, we propose \textbf{PreferGrow}, a discrete diffusion-based recommender modeling preference ratios by fading and growing user preferences over the discrete item corpus. PreferGrow differs from existing diffusion-based recommenders in three core aspects: (1) Discrete modeling of preference ratios:
PreferGrow models relative preference ratios between two items, where a positive value indicates a more preferred one over another less preferred. This formulation aligns naturally with the discrete and ranking-oriented nature of recommendation tasks.
(2) Perturbing via preference fading: Instead of injecting continuous noise, PreferGrow fades user preferences by replacing the preferred item with alternatives---physically akin to negative sampling---thereby eliminating the need for any prior noise assumption.
(3) Preference reconstruction via growing: PreferGrow reconstructs user preferences by iteratively growing the preference signal from the estimated ratios. We further provide theoretical analysis showing that PreferGrow preserves key properties of discrete diffusion processes.
PreferGrow provides a well-defined matrix‑based formulation for discrete diffusion-based recommendation and empirically outperforms existing diffusion‑based recommenders across five benchmark datasets, underscoring its superior effectiveness.
Our codes are available at \url{https://anonymous.4open.science/r/PreferGrow_Commit-2259/}.

Fading to Grow: Growing Preference Ratios via Preference Fading Discrete Diffusion for Recommendation
Guoqing Hu
An Zhang
Shuchang Liu
Wenyu Mao
Jiancan Wu
Xun Yang!
Xiang Li
Lantao Hu
Han Li
Kun Gai
Xiang Wang
University of Science and Technology of China 'Independent Researcher
(HugoChinn))maiil wst.. edu. cn {an_zhane zhang, xyang21]@ust...dduu cn
(wenyumao2, wujcan, xiangwangg122 com com
(xiangli.thle th11, hulantao, lee. hae.hh)l))gmmi..... com, gai kun@qq.
Abstract
Recommenders aim rank items from discrete item corpus in line with user interests, yet suffer from extremely sparse user preference data. Recent advances in diffusion models have inspired diffusion based recommenders, which alleviate spar- sity by injecting noise during a forward process prevent the collapse of perturbed preference distributions. However, current diffusion-basee recommenders pre- dominantly rely on continuous Gaussian noise, which is intrinsically mismatched with the discrete nature user preference data recommendation. this paper, building upon recent advances in discrete diffusion, recommender system that models preference ratios by preferences over the discrete item corpus. PreferGrow
propose PreferGrow. a
discrete diffusion- fading and growing differs from existing diffusion bastud recommenders in three core aspects: (1) Dis- crete modeling of preference ratios: PreferGrow models relative preference ratios between item pairs, rather than operating in the item representation or raw score simplex. This formulation aligns naturally with the discrete and ranking-oriented nature of recommendation tasks. (2) Perturbing via preference fading: Instead of injecting continuous noise, PreferGrow fades user preferences by replacing the preferred item with peermatiess--hysiallll akin to negative sampling-therede eliminating the need for any prior noise assumption. (3) Preference reconstruction via growing: PreferGrow reconstruets user preferences by iteratively growing the preference signals from the estimated ratios. PreferGrow offers well-defined matrix based formulation with theoretical guarantees on Markovianity and re- versibility, and it demonstrates consistent performance gains over state-of-the-art diffusion-basee recommenders across five benchmark datasets, highlighting both its theoretical soundness and empirical effectiveness. Our codes are available btttps://githnn cow Hoom/Hugo-ao Chimm/Preferroou
1 Introduction
Recommender systems aim to rank items from a discrete item set that align with user interests, where ones in the user-item interaction matrix denote observed interactions, and zeros indicate unobserved
"Corresooning authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

Embedding Layer
Item-level Diffusion
User
preferred item
- dense embedding
Add Gaussian Noise into Embeddings
Model Gradient of likeliood of Perturbed distribution (x,
Xt
without negative sampling
Vlog
(1o
Vlog
Preference score-level Diffusion
user preferred item
One-hot vectorization
preference scores
17
28
Preference Scores
03
O
24
Pr(t,) E (0.
24
26
O
25
Simplex Constraint
02
89
48
23
p+(r) 2
48
X
X₁
X
Σ
p.6) P' =1
Constrained within probability simplex
Recover clean preference scores from perturbed
PreferGrow: Discrete Diffusion Modeling Preference Ratios
preferred item Xg
Preference Fading Discrete Diffusion
Preference Ratios
0
Xo
Replace with
PrGrlu)
(0, +0 os)
0.4
0.2
FMA equally
0.2
0.4
0.2
0.2
p(2, xo) 0.6
P:(Y())
(-00,
(2(
0
Retain
0.2
0.2
0.2
0.4
p( xo) 0.4
Remove Constraint
Figure 1: Comparison of diffusion-based recommenders in terms of modeling targets and perturbation strategies. Item-level diffusion recommenders (top) add Gaussian noise into dense item embeddings but overlook negative signals. Preference score-level diffusion recommenders (middle) perturb one-hot preference vectors under the constraints of the probability simplex. PreferGrow (bottom) directly models discrete preference ratios via preference fading.
or missing entries. In real-world scenarios, this interaction matrix often extremely sparse 1 2) which poses a significant challenge for recommenders in accurately modeling user preferences 345) Recent advances diffusion models (DMs) offer promising solution: by injecting noise during the forward process, DMs transform sparse data into smoother, denser distributions while preventing collapse into isotropic zero values (68 Motivated by these properties, diffusion-bassd recommenders [9/37] have proliferated in recent studies, showing strong potential in addressing data sparsity and improving preference modeling.
Current diffusion-based recommenders predominantly adopt continuous noise as the perturbation mechanism by adding noise into either dense item embeddings or one-hot preference score vectors. As shown in Figure [] item-level diffusion recommenders [ apply Gaussian noise to user- interacted item embeddings during the forward process, and lear score function the gradient of the perturbed distribution bog-likelihhode However, these methods overlook the negative signals in recommendations, lacking mechanisms such as negative sampling to differentiate user preferences [9] [7). In contrast, preference score-level diffusion recommenders (28)37) model users' preference scores among the full item corpus, i.e., user interacted one-hot vectors, by perturbing them within the probability simplex in the forward process. While the reverse process attempts to reconstruct the preference scores, the simplex constraints- non-negativity and normalization-initee
recover item embeddings during the reverse process.
additional optimization difficulties (38). Worse still, both diffusion-baar based recommenders assume a prior noise in the sampling process, e.g., Gaussian [ Bernoulli [29]. which may poorly reflect the inherently discrete and sparse nature of the user preference data in recommendation scenarios.
propose PreferGrow, discrete diffusion-bacee recommender that models the preference ratios by fading and growing user preferences through forward and backward processes. Unlike existing continuous diffusion-basse
Building recent advances in discrete diffusion models 39 38
40

---

recommenders that model preference scores (e.g,, probabilities that users interacted with items), PreferGrow directly models relative preference ratios over discrete item set. This formulation aligns more naturally with the discrete and ranking-oriented nature of recommendation tasks and avoids the strong constraints imposed by the probability simplex. the forward perturbation, PreferGrow fades user preference by replacing the preferred item with alternatives, enabling explicit negative sampling and alleviating data sparsity without relying on predefined noise distribution. In the backward generation, PreferGrow reconstructs user preferences by iteratively growing the preference signal from the estimated ratios. We further provide a theoretical analysis showing that these forward and backward processes preserve key properties of discrete diffusion models.
PreferGrow offers unified and theoretically grounded framework for discrete diffusion based recommendation, characterized by well-defined matrix-based formulation, flexible and interpretable preference ratio modeling aligned with diverse negative sampling strategies, and consistently superior empirical performance.
Theoretical foundation: At the core of the formulation for PreferGrow is an idempotent fading matrix used to replace preferred items during the forward perturbation process. We provide closed-form solution of this preference fading matrix and theoretically prove that its idempotent property is critical for ensuring both the Markov property and reversibility of the diffusion process.
Flexible modeling: By parameterizing the preference fading matrix, PreferGrow flexibly supports point-wisc, pair-wise, and hybrid preference ratios modeling. These variants correspond to distinct and physically interpretable negative sampling strategies, enabling the framework to adapt to diverse modeling targets in recommendation.
Empirical validation: Extensive experiments on five benchmark datasets demonstrate that Prefer- Grow consistently outperforms existing diffusion- bassed recommenders, validating the practical effectiveness of its theoretical foundations.
2
Preliminaries
User preference data is represented as a pair (u, i). where u denotes the user and is the preferred item. The preference score p(ilu) indicates the probability that the user interacts with item i. In sequential recommendation, denotes the item sequence that the user has interacted with.
Diffusion-based Recommenders generally consist of three major components: a forward noise- adding process for perturbation, a generative modeling target, and backward denoising process for reconstruction. Specifically, item-level ceffusion-based recommenders encode a preferred item i into its dense embedding Xg: and instantiate three components follows 41
forward perturbation: X; = VaiXo + VI act4, where a₁ E [0, 11] 4 ~ 1).
modeling target: network (Xe, t, Po Vx, log lo //te
1-a;
Vi-ar"
backward generation: x, Va. x -al (x
v1
- u), <
Vau
Preference score-level diffusion-based recommenders represent the preferred item as one-hot preference score Xg, modeling user preference over the entire item space. Early works [28] 30 34 adopt Gaussian noise priors ()::)): however, Gaussian perturbations are incompatible with the probability simplex constraints inherent to preference scores. To address this mismatch, subsequent studies 22 35 introduce discrete priors replace the Gaussian assumption that preserves the
simplex structure throughout the diffusion process. For instance, RecFusion 29 adopts a Bernoulli prior ((::)), resulting in a binomial diffusion formulation:
forward perturbation: x, B 0x (ar)) 2
ar E [0,
modeling target: B (xx-1:80 (x; f, u)) ~ p p(x,-1 1 x().
backward generation: x,-1 ~ B (x;-1; Se Xx, 1, u)).
The other approach employs Categorical prior Cat(:;;
forward perturbation: (x xx-1) Cat(x;; Q,x0) where Q -I'= Q
3

---

Non preference User Modeling
Preference Fading Forms Reference Ratios
Modeling Preference Ratios with Reference
-Traning Pipeline
User Condition
log Pro #)
Growing Preference Ratios
Aa M,,, PV
044
0.2
0.2
egg
drop probability
History
User
0.4
0.2
aporent-opochh
Tt
Feet
None Condition
0.2
0.2
0.4
Learnable Embedding
Nompreference Condition
Pos add
Reference Ratios
Personalized Contrast
Preference Growing from Preference Ratios -
Recovering Preference Scores from Preference Ratios
-against Non preference User
Inference Pipeline
Growing Preference from Non- Preference
Psp Pat e @
contrast
1,a)
5y(K),
Soliy T.0)
D3
07
25
User
Usero
27
25
W- (f ) SulK, 0))
21
26
Personalization Enhancement
.66
39
26
Figure 2: The overall training and inference pipeline of PreferGrow under the pair-wise setting.
modeling target: se t, u) == X0-
Q XxxOQ, 188 (x.,,..)) backward generation: Pt. (X:-1 Xo = sx,, (,x,,,,,)) Cat X1-1; x Q,5e (x,,,
Preference Ratios log p(al)) characterize the relative preference between items, and positive value indicates more preferred item i, and a less preferred item i.s Notably, preference modeling in RLHF 44]444 and DPO (45 grounded the Bradley-Terry model [49], which explicitly models preference ratios in Equation ( This underscores the effectiveness of preference ratios in more faithfully capturing user preferences.
plip plip > id [)) p(ip u) + p(ia u) 1 + exp (-log
1
a (log p(ty (1), p(ia u)
(1)
p(uplu) p()
Discrete Diffusion Models previously are formulated based on the following Kolmogorov forward and backward equations [50] 51 38):
OP c/= Ōt
aP elc Ot
Q.P f/st
RRP
where P denotes the transition probability matrix from time to time f. Q2 is the forward transition rate matrix time f. and R, the reverse-timg transition rate matrix at time s. Discrete diffusion models further model the ratios of data distributions through the following score entropy loss [3
Lgg Exorpaane BIEUTO, Ee (oo
Σ Qe(reny) Isg 2r, xr,y) v) VEN
(3)
IsE(xo, Iz, v) = (0(())))) Peo(yl
(y( + poo( [log Pro(y)oo))
1],
(4)
Pt (2, mo)
Peto(xe (1)o(1)
where X' denotes the discrete state space, and se (y,, n.) ) the output of neural network that estimates the preference ratio between and H, at timestep I.
3 Method
As illustrated in Figure 2 the pipeline PreferGrow consists of three stages: the preference fading discrete diffusion process (Section B.I), the modeling of preference ratios with reference via score entropy loss (Section 3.2p, and the preference growing reverse generation process (Section 3.3). Additionally, PreferGrow models non-preference user and achieves personalized enhancement (Section 3.4p. Theoretical proofs are provided in Appendix(C C while the computation score entropy loss under different fading matrix settings is detailed in Appendix D

---

3.1 Forward Perturbation: Preference Fading Discrete Diffusion Process
PreferGrow operates directly on the discrete item set X, wherein 2g represents the preferred item of user u. PreferGrow first fades user preference based on whether to retain or not to retain the preferred item Zo. The whole fading process further achieved by progressively decreasing the probability of retention 0z from timestep 0 to T, fading user preferences towards non-preference.
3.1.1 Preference Fading Forms Reference Ratios
Building upon recent advances in discrete DM [39 40 we introduce a preference fading discrete diffusion model tailored for recommendation. For preference data (u, zo). we fade user preference by retaining the target item 2, = zo with probability at and replacing 2g according to discrete distribution E(X, xo) as the following equation:
(()o = azdza (zt) + (1 - au E(r,, 2t X.
(5)
where Dirac delta function 8mo (zx) equals I when 21 2o. and 0 otherwise. Also, in matrix form:
P:jo = = + (1 a,
(6)
where I is the identity matrix, representing retention, and is a matrix whose column sums equal defining the replacing mode for perturbation. For simplicity, we refer E fading matrix. We further demonstrate that the idempotence of the fading matrix E ensures the Markov property and reversibility of preference fading discrete diffusion process, which is crucial for the reverse generation.
Theorem 1. Suppose at [0, T] - [0, 1] is a strictly decreasing function with og = 1 and or E NxN is idempotent, i.c., E² = E, the following properties hold:
Markov property: (Poo) is Markov process and satisfies the Chapman- equation:
01 I (1 (1=0)
P Prls Os
E
ax
Pt/r P P//.Ps/rs S//r for all 0 S sStts
(8)
Invertibility: Each Pals is invertible, and its inverse is given by:
Og I
P 1/3
E.
Ot
Upon obtaining x, through preference fading, as illustrated in Figure we can compute the reference ratios for all items X' timestep with fading awareness ar as follows:
Pyo(y zo, u)
azd2o (v) (1 04 E(y,
Te (20 Vz, v) = log
Prjo (Ft Fo, u. = log On (xt) + (1 Ore E(xe,
Vy E X.
(10)
As shown in Figure when the preferred item Fo is retained, i.e., Se To. the reference ratio re (T = To. y + zop) is negative, reflecting tendency to stay with the current item. Conversely, when Zn replaced, i.e., H, o zo, the reference ratio rz It & Zp. y = 2n) tends be positive, highlighting the preference for the positive item Jo. In this case, the faded item can be interpreted as negative item, establishing a conceptual connection to mechanism of negative sampling.
3.1.2 Design Paradigms of the Idempotent Fading Matrix
Given that designing such an idempotent fading matrix is crucial, derive that, if the preference fading process converges to unified state, the fading matrix can be expressed in closed form.
Proposition 1. Suppose the preference fading Markov process (Pajo converges unified non-preference state pr. Then, the fading matrix E can be expressed in closed form as:
IT
E
(11)
I FT
5

---

Equation 1 gives rank-1 solution for the fading matrix E. Most existing discrete diffusion models 38
40 5 implicitly adopt such rank-1 instances (i.e., the absorbing and uniform cases). In Appendix D examine several representative rank-1 configurations of E that, in physical sense, correspond to distinet negative sampling strategies and thus induce different forms of preference ratios. Specifically, by parameterizing the rank-1 preference- fading matrix, PreferGrow flexibly supports point-wise, pair-wise, and hybrid preference-ratio modeling. These variants align with interpretable regative-sampling schemes, enabling the framework to adapt to diverse recommendation objectives. For each setting, we provide analytical solutions for the fading matrix, closed- expressions for the reference ratios, and simplified training losses. We believe prior works negative sampling [36 57 offer valuable guidance for designing more effective rank-1 fading matrices; a systematic exploration is left for future work. Beyond the rank-1 case, Appendix derives a general rank-r solution for E. We present closed-form characterization together with a discussion of its physical interpretation. Notably, rank-r fading matrix induces quantization of the item space i.e., partition into components. comprehensive study of this quantization mechanism and its algorithmic implications deferred to future work.
3.2 Modeling Target: Preference Ratios with Reference via Score Entropy
We begin by computing the reference ratios Tz Tt. y) induced by preference fading, defined in Equation (10). These reference ratios are then employed to guide the modeling of the user- conditioned preference ratio log p<(]//)) where denotes the user context. Concretely, given training instance (u, 2o) in sequential recommendation. we encode the interaction history with sequential recommender to obtain a user embedding u SeqRee(u) (instantiated as SASRec (58) in our experiments). We then parameterize learnable function se estimate the preference ratio for each y E X' at time t:
Pe(alu)
se (a t, (,) y MLP concat(x,, t, u)), E.X.
(12)
where and X, are the embeddings of item y and the faded item 2t, respectively, and t. is the embedding of timestep 1. For the training objective, we adopt the well-known score entropy loss for discrete diffusion modeling (38), as defined in Equations (3) and 4 To legitimately employ it, first verify that PreferGrow satisfies the applicability conditions of score entropy loss.
The preference- fadiing discrete diffusion (PelooseseeST with an idempotent fading matrix E satisfies the Kolmogorov forward equation:
Proposition 2.
Q: Paje-
(13)
Öt
Here the rate matrix Qt and ai are defined succinctly by
Q = lim at 3(1) (E-), I), ae := exp (- ((F)
OP::
(14)
with B(r) 0 and I the identity matrix.
To better explain the effectiveness of the Score Entropy (SE) loss in recommendation, we theoretically analyze its connection to the Binary Cross-Entropy (BCE) loss (56].
Proposition 3. Define the soft label Tyra,l= = ply > Zt xa) = or Fy, x)) as in Equation ① where a denotes the sigmoid function, and use it as the label for the BCE loss. Treating (se L, u)y) as the prediction for the BCE loss yields the soft-label BCE objective:
LICE = yyy= log a (ses(Ze, 1, u))) (1 10--mom Ty)zz(1o)) log(1 - (ss((1i, u,)))))
From a gradient perspective, the SE loss and the soft-label BCE loss are related by:
V V.e (1 ( (e(( V se CyBCE-
(15)
Consequently, the SE loss and the soft-label BCE loss share the same descent direction (up to positive scalar) and attain the same optima. Hence, although originating from discrete diffusion modeling. SE loss is equally well suited to ranking-orientel recommendation tasks as the BCE loss.

---

3.3 Backward Generation: Preference Growing from Preference Ratios
After estimating the preference ratios via score entropy, we reverse the preference fading process to grow user preferences from the non non-preference state. We first demonstrate that this reverse process of preference fading, referred to the preference growing process, Markovian and satisfies the Kolmogorov backward equation. Together with Proposition 2 this implies that PreferGrow with an idempotent fading matrix inherits the key properties of previous discrete diffusion models [51 38)
Theorem 2. If the preference fading process (P (Pooo YE-o with an idempotent fading matrix E converges to unifted non non-preference state pr, the reverse preference growing process (ParT) > holds:
Markov property:
(P ( is also Markov process:
P P
Ph
+ PT f/s
VOSSSIST.
(16)
Kolmogorov backward equation: Preference growing {P P satisfies:
ÖP s|t
R.Pa/er s/tr
VOSsSIŞT.
(17)
at
OP
)
1 Pe
Rg
lim
Q
ps
Q
Ps
OI
(18)
Ös
P5
The matrix Pt
denotes exponential preference ratios with entries given by
polylu) (.
Ye,
X.
At this point, we equipped with the approximated preference ratios se to grow user preferences. Given a user condition u, we first sample an item 27 from the non-preference state pr, and then compute the reverse transition matrix fr to progressively grow user preference over time
(2 = y) = (1r =) v) Σ
Prjs (x) = ylæ = 2) soletttn)) VOSSSIST. (19) ZEX'
we iteratively grow user preferences backward until reaching zo. Finally, based on the preference scores at timestep 0, we recommend the top- K items with the highest scores.
As illustrated in Figure 2
3.4 Modeling Non-preference User for Personalization
3.4.1 Non-preference User Modeling
To enhance the personalization of preference ratios, we introduce the modeling of non prefereeee users-i.e.. cold-start users with no interaction history- during training. This allows the model to perform personalized contrast against the non-preference user during inference. thereby reinforcing user-specific signals for personalized recommendations. As illustrated in Figure [] we randomly drop the user condition u with fixed probability during training, and replace it with a learnable embedding that represents the non-pree oon-preference user
3.4.2 Personalized Contrast against Non-preferencee User
The non-prefere user defined such that Prjo(ra/zo Φ) = Prjo (re (o)). According Bayes'
theorem (z ro, u) ((o((a())))) p(u[zo( v.), there holds the following ratio condition:
p(u(me)
Pap(a, = y| u) Pei( = y|ao) p(u x; =y)
(20)
Peo (z, 2 @0.
Peo x (2)) p(u Iz x)
We then estimate the ratio of the likelihood with a personalization strength parameter w as
59
p(u y) p(u x)
PHo(2, = y|=0. Φ)
Pejo(r = y|zo, u)
(21)
pao(== x r/ の) Prjo x
Therefore, the personalization-enhanee preference ratios with strength computed as follows: S 2, u)g = w so t,,,,,,,,,, u), + (1 w) s0 (,0e,,,00 4)
(22)

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Table Performance comparison across different datasets and baselines.
Dataset
Moviesl gms
Steam
Beauty
Toys
Sports
Method
HR
NDCG
HR
NDCG
HR
NDCG
HR
NDCG
HR
NDCG
69-5
0.0905
0.0502
0.0321
0.0192
0.0326
0.0221
0.0340
0.0258
0.0154
0.0101
SASRec
0,0709
0.0760
0.0572
0.0272
0.0438
0.0257
0.0438
0.0289
0.0230
0.0126
0,2899
0,1059
0.0368
0.0595
0.0297
0.0499
0,0305
0.0143
695
0.0925
0.0576
0.0258
0.0163
0.0170
0.0106
0.0093
0.0072
0.0081
0.0030
Caser
0.1605
0.0794
0,0446
0.0223
0.0219
0.0122
0.0129
0.00833
0.0093
0,0043
0,2592
0,1042
0,0736
0,0295
0.0295
0.0141
0.0185
0.0098
0.0149
0.0057
63.5
0.0892
0.0551
0,0255
0.0156
0.0130
0.0082
0.0124
0.0078
0.0070
D.0048
GRURec
0.34
0.0757
0,0441
0.0216
0.0188
0.0101
0.0180
0.0092
0.0107
0.0059
20
0.2501
0.1000
0.0769
0.0298
0.0313
0.0132
0.0304
0.0117
0.0171
0.0076
65
0,0676
0,0437
0,0109
0.0073
0,0300
0.0247
0,0381
0,0304
0.0095
0.0084
DreamRec
10
0.1083
0.0568
0.1557
0,0088
0.0353
0.0265
0,412
0,0114
0.0112
D.0089
0,1610
0.0701
8.0218
0.1044
0.0402
0.0277
09400
0,0327
0.0149
0,0098
65
0.0538
0.0349
0,0167
0.0105
0.0335
0.0272
0.0866
0.0088
0.0168
0.0130
PreferDiff
0.0852
0.0450
0.0434
0.0304
0.0494
0,0343
0.0144
20
0,1270
0.0555
0.0526
0.0203
0.0577
0.0340
0.0644
0.0380
0.0256
0.0155
0.0266
0.0121
0,0268
0.0143
0.0095
0.0055
0.0053
0.0028
0.0063
0.0035
DiffRec
0.0889
0.0320
0.0657
0.0268
0.0218
0.0094
0.0133
0.0059
0.0117
0.0052
0.1768
0,0541
0.0355
0.0128
00226
0,0078
0.0072
68.5
0.0871
0.0533
0,0252
0.1157
0.0291
0.0217
0,0386
0,0302
0.0146
0.0109
DDSR
e0
0.1523
0.0742
0,0437
0,0216
0.0434
0.0262
0.0479
0,0532
0,0213
0.0130
@2
0,2450
0,0975
0,0736
0.0291
0.0608
0.0306
0.0618
0.0367
0.0298
0.0151
PreferGrow
Impr.
65
0.1409
0.9911
0.0615
0.0406
0.0420
0.0322
0.0441
0.0331
0.0207
0,0142
Hybrid
10
0.2229
0.1174
0.0935
0,0508
0.0532
0.0358
0.0519
0.0356
0.0267
0.0162
20
0.3355
0.1458
0.1413
0.0628
0.0708
0.0402
0.0625
0.0383
0.0343
0.0181
0,1413
0.0912
0.0583
0.0375
0.0396
0.0310
0,0413
0.0304
0.0168
0.0127
Adaptive
@ 10
0.2240
0.1177
0.0914
0.0481
0.0508
0.0347
0.0513
0.0337
0.0207
0.0140
0.3362
0.1460
0.0610
0.0373
0.0642
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PrelerGrom-Adaptie PolatWice
w/o coapreference
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(a) Abalation Adaptive on Steam.
(b) Ablation of Hybrid on Steam.
Figure 3: Ablation Study Steam, showing the percentage of the relative effectiveness.
4
Experiment
We compare PreferGrow with a variety of baselines under the all-ranking evaluation protocol, including classical recommenders (SASRec [58] Caser (60), GRURec (61]), item-level diffusion-
based recommenders (DreamRec [9, PifferDiff (27)), and preference score-level diffusion-
recommenders (DiffRee [28] DDSR 335 across five benchmark datasets. Details of the datasets
are provided in Appendix G.I discussions of the basclines are presented in Appendix G.2, and the
training and evaluation settings are described in Appendix G.3
4.1 Overall Comparsion
From the results Table we observe that PreferGrow consistently outperforms all baselines. We attribute the effectiveness of PreferGrow to three factors: 1) its discrete diffusion process aligns with the discrete nature the recommendation scenario; 2) the preference fading noise perturbation mechanism, akin to negative sampling; and 3) preference ratios modeling without the simplex
8

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0.18
Effect of nea-pedferene ratio
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Effeet of mon-preferenee uar ratoo
Effect ofbybrid oveffiiicat
Hight
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(a) p of Adaptive Steam.
(b) p of Hybrid on Steam.
(c) of Hybrid Steam.
personalizatio strength
persseslizccon strength w
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0.34
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(d) w of Adaptive on Steam.
(e) Hybrid on Steam.
(f) Steps Hybrid on Steam.
Figure 4: Hyperparameter analysis of PreferGrow on Steam. Red stars denote the best results.
constraints, which better captures user preferences. Additionally, the hybrid and adaptive fading matrix settings, inspired by the negative sampling strategy, also demonstrate the flexibility.
4.2 Ablation Study
As presented in Figure 3 we perform a thorough analysis and evaluation of each key component within PreferGrow assess their individual significance. The ablation study is conducted using the following three variations: (1) w/o-PoinsWise, PreferGrow without the general hard negative item x-1: (2) w/o-PairWise, Point-Wise PreferGrow with pT ēー1: (3) w/o-Nonpreference, PreferGrow without modeling the non-preference user Φ, resulting in no personalized enhancement during the backward generation, as well as the combination of the aforementioned ablations. Overall, PairWise preference ratios modeling and non-pree non-preference user modeling are crucial for the effectiveness of PreferGrow, while PairWise preference ratios modeling adds an additional refinement.
4.3
Analysis
We further investigate the effect of key hyperparameters on the performance of PreferGrow (Figure 4. Overall, PreferGrow is fairly robust to the non-preference user proportion p, the hybrid coefficient and the number of sampling steps, while exhibiting relatively higher sensitivity to the personalization strength w. Importantly,
chosen at inference time, so no retraining is required. On the one hand, w is locally stable within reasonable range (see Figure 4): on the other hand, we show that the optimal highly consistent across data splits (Appendix G.4). Taken together, these properties ensure that tuning is practical and does not introduce significant overhead.
5 Limitations
While effective, PreferGrow still has several aspects that warrant further optimization.
Higher modeling complexity. As shown in Table & targeting preference ratios incurs substantially higher complexity than prior diffusion-based recommenders. On the one hand, preference ratios are more expressive and thus raise the potential ceiling of the model; on the other hand, they also increase the learning difficulty under finite model capacity- for example, leading to longer training time than previous diffusion-basee methods (Appendix G.3 A principled remedy is extend PreferGrow to rank-r fading matrix. By inducing a quantization structure, this formulation can markedly reduce the complexity of preference ratios while preserving their expressive power, thus achieving better expressiveness pppresiveness--rdduddngy trade-off. We identify this as a primary avenue for future work. In addition, we observe that the final 50% of training yields only about 5% gain in NDCG@5, suggesting substantial redundancy in the training process. A likely cause is uniform timestep sampling: different timesteps reflect different degrees of user preference fading and thus vary in the difficulty of modeling preference ratios. Incorporating this difficulty by

---

beginning with easier timesteps (mild fading) and gradually advancing to harder ones may further accelerate convergence.
High computational complexity. As shown in Table 8 PreferGrow has (NN) complexity for both loss computation and inference, where N represents the size of the item corpus. While this recommenders, it becomes impractical when scaling to
is comparable to prior diffusion- extremely large item sets (c.g., billions of items), where even ((N) is prohibitive. A promising next step incorporate a quantization structure such as Semantic IDs (SIDs) to reduce the cost from (NN) to ((m)). SIDs represent each item with codebooks of size c, enabling up to c items while maintaining only (mec) computation. Extending PREFERGROW with rank-r fading matrix to operate directly on SIDs is our future work.
6 Conclusion
Building upon the previous diffusion based recommenders and discrete diffusion models discussed in Appendix A this paper introduce a new discrete diffusion-based recommender tailored for discrete and sparse recommendation scenarios, named PreferGrow. In summary, PreferGrow distinguishes itself from existing diffusion-based recommenders three key aspects: (1) Discrete Diffusion: operates directly on the discrete item set, fully aligning with the discrete nature of recommendation. (2) Preference Fading: fades user preferences by replacing the preferred item with others, akin to negative sampling, thus removing the need for prior noise assumptions.
(3) Preference Ratios: It estimates preference ratios by modeling the logarithmic ratios of user-item interaction probabilities, cireumventing the constraints of the probability simplex. PreferGrow features well- defined theoretical formulation and demonstrates superior performance in experiments. We further discuss the broader impacts of PreferGrow in Appendix B
Acknowledgments
This research is supported by the National Natural Science Foundation of China (62572449).
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A
Related Work
Table 2: Comparison of Diffusion- Recommenders
Modeling
Forward
Negative
Representative
Target
Perturbation
Sampling
Work
Score Function
Gaussian on Embedding
X
DreamRec 9
Continuous
Score Function
Gaussian on Embedding
PreferDiff
Preference Scores
Gaussian on One-hot
DiffRec (28
Preference Scores
Bernoulli on One-hot
RecFusion 62
Discrete
Preference Scores
Categorical on One-hot
V
DDSR [35
Preference Ratios
Fading on Items
PreferGrow (Ours)
Diffusion-based recommenders 9437 utilize forward perturbation in diffusion models to address data sparsity [6). thereby better adapting to sparse recommendation scenarios. They typically consist of three core components: the modeling objective, the forward noise addition, and the corresponding backward generation process, which are summarized in Table [ Current research on diffusion-basee recommenders can be classified into two primary approaches: one involves adding noise to dense item embeddings at the item level, and the other focuses on perturbing the one-hot interaction probability vector for all items, which is referred to as preference scores. The first research line, pioneered by DreamRec [9] and DiffuRee [T0,, encodes user-pre sser-preferred items as dense embeddings and adds Gaussian noise to these item embeddings. DreamRec (9] models the score functions (the gradient of the log-likelihood of the perturbed distribution) without negative sampling, whereas DiffuRee [TO] incorporates a recommendation loss that includes negative sampling. Building on them, PreferDiff 27 introduced an optimization objective derived from BPR loss (63, which integrates multiple negative samples into the generative modeling framework. However, the application of continuous Gaussian noise to positive preferred items, in contrast to the discrete nature of negative samples, creates an inherent mismatch, making it difficult to optimize both simultancously during training, leading to a trade-off [10
64 27
[25] Additionally, other works have introduced more sophisticated or applied them to different recommendation incorporates multi-interest models, DiQDiff (23
module designs [1]
12
16
20
23
25
26
tasks 13 22 24). For instance, DimeRec 664
introduces semantic vector quantization, and DiffuASR [33 applies item-level diffusion sequential recommendation data augmentation. The second research line (28)37 involves converting user preference data into one-hot vectors, which are then mapped to preference scores within the probability simplex. DiffRec (28) add continuous Gaussian noise to the preference scores, and then learn to recover the clean preference scores from the perturbed ones. Consecutively, LD4MRec (30) refines the design for efficient multimedia recommendation. and D3Rec (34) introduces targeted category preferences to control diversity during inference. However, the constraints of the probability simplex- non-negativity and normalization- significant challenges in accurately estimating the preference scores 38 To address these challenges while considering the probability simplex constraints,
RecFusion 62 assumes a Bernoulli noise prior, completing the binomial diffusion process and subsequently modeling the parameters of the reverse binomial distribution to facilitate the reverse generation of preference scores. On the other hand, DDSR 355 adopts categorical noise prior, as proposed in (65), and directly recovers clean preference scores from the perturbed ones. Nonetheless, the constraints of the probability simplex may limit the effectiveness of preference score modeling. Moreover, both diffusion- based recommenders rely on prior noise assumptions, such as Gaussian [9] or Bernoulli noise 229), which may not be optimal for recommendation scenarios where user preference data is inherently discrete.
Discrete Diffusion Models 65
have made substantial advances recently. Initially, D3PM (65] proposed a discrete diffusion framework based on an arbitrary probability transition matrix, trained with the evidence lower bound of the log-likelihood. Subsequently, LDR 54 extended this framework to a continuous-time setting using the Kolmogorov forward and backward equations. However, modeling score functions in such models presents challenges, as the gradient of the data distribution is undefined. To address this, CSM (39] introduced Concrete Score discreiiscttizatioon of score functions and the ratios of data distributions modeling objectives
54
39
55
38
40
66
22

---

for discrete diffusion models. Building upon these advances, SEDD [38 further bridges discrete diffusion models and ConereteScore by introducing the score entropy loss. Expanding on these developments, we present PreferGrow, matrix-based discrete diffusion framework which perturbing data by retaining or replacing items within discrete corpus. The idempotent property of the replacement matrix (or fading matrix) is central to PreferGrow, and we demonstrate that it satisfies the Kolmogorov forward and backward equations as LDR [54], aligning it with prior works. Additionally, we introduce design paradigm for the idempotent replacement matrix, which unifies previous approaches, including absorbing and uniform settings.
B Broader Impacts
Theoretically, PreferGrow introduces a well-defined discrete diffusion model building upon prior work. While designed for recommendation, PreferGrow is also applicable to other discrete domains, such as molecular design in chemistry and protein structure prediction. There are many potential societal consequences of our work, none which we believe warrant specific attention at this time.
C Proofs of Main Results
Proof of Theorem and the preference fading discrete diffusion process denoted as Pejo a,I+ 0; ,, Vt E [0, Then we can rewrite preference fading discrete diffusion process as for oxo = 1:
[0, T] + [0. 1] is strictly decreasing function with 0g = 1 and or 0
P Pajo
I (1 00
E, Vt E [0,
(23)
Ogg
For any well-defined P $/a.0 with 0 < < the following limiting distribution constraint must hold:
Pe
Σ
Vx,, E X.
(24)
x,EX
In matrix form, this constraint reduces to P r|O particular solution of this equation is:
P
P//s,0
P
310 for all 0 < < t. Note that one possible
0t I (1
Pi|s,0 == Pejs
00 S ST
(25)
0s
We will show that this indeed satisfies the constraint under the condition that E is idempotent.
Qt
Ot
0x
Paj
P
Pajo
+ (1
E
E
(26)
0s
00
O0
Qt
2
00
1+
01
04
E
000
0ts
40
00
0ta
0t I+ E²
Qt
E
+
Qt
a2
EE
E2)
00
00
the fading matrix is idempotent + E²
dt I+ (1
0x
E
a0
Prjo.
Similarly, for all 0 < $ s t < T. there holds the Shuppann-Komogoroo equation:
O't
04
Oxs
0ss
Prop P e/rr
+ (1
E
+ (1
E
(27)
ass
ax
Or
ar
04 I E²
O2
E
+
02
E
--E22
ar
a a2
ar
Qr
the fading matrix is idempotent
E2
at "+ (1
at E ar
ar
= Pppr. t/r
23

---

We further note that Pt v4 s T is invertible, with inverse given by:
P-1
0,
+
0ts ai
E, 44
SaStET.
(28)
f/s
P
P
I I + (1
0l E
I
a, E
(29)
f/s
04
at
02
E
E²)
02
Ot
the fading matrix is idempotent = E ===b E
vO Irr SErissi 8 s T, combined with limiting distribution constraint f/r = P Pl/s. Palr- there are:
Pr/sr
Pej Prpp P- d/r
a1 I ++ (1
E
I
(1
E
(30)
0r
Or
a22
a2
at 1+ E
E
+
(
E²)
02s
0a
02
022
0x
the fading matrix is idempotent
E E
ag I+
E
O²2
Ots
Pr|s-
T. In summary, the Pr|s.r Piis indicates ( = (=))(z)(2.,, 44 S S t S.
preference fading discrete diffusion process Markovian but not time-homogeneous, satisfies the Chapman-KKlmogone equation, and is reversible.
I
Proof of Proposition 7 at [0, [0, 1] is strictly decreasing function with og and aT = 0. ar is further defined as SJ B(r)dr with (T) > The preference fading discrete Markov II+ 2 J vO < < t < T. We first show that the
diffusion process is denoted as Pa/= a, preference fading discrete Markov diffusion process converges to unified non-preference state pr is well-defined. The transition rate matrix Q is computed as follows:
OP
1/s
Q = lim
(31)
0t
i
0t I I+ (1
O't } a₃
lim
x Öt
0₂
lim
a1
I
(1
E
dos
x- da,
02
a₂
0t
1
lim
-
I
E
(-B(1))
a
0.
0,
= (() (E (E-I).
The rate matrix Q₂ characterizes the velocity of probability transitions at time t, encompassing both the direction and rate of transition. As shown Equation (32). Q at any time E [0, T] shares a consistent transition direction I- - E, while the transition rate ((t) varies over time. This dime-dependent rate results transition direction ensures that all diffusion paths converge to the same non-preference state, making the process well-defined. Specifically, the stationary distribution H, at time satisfies the equilibrium condition Q, Ó. Since different Q matrices differ only by a scalar factor (t)), they yield the same stationary solution म. Consequently, the Markov process converges to common steady-state distribution म, iee,, the non-preference state pr:
non- homogeneous preference fading process. However, the shared
# ー -
(T Ō
(32)
24

---

Given that (I E O, each column of E satisfies the non -on-frefeeence state equation (I - E) pr 0. To ensure unique solution pr, therefore assume that all columns of E are identical, i.c. E x pr IT. Considering the idempotence constraint E E, we then have:
Pr I" E
(33)
PT
E² Pr I PT I (IT))
(34)
pT. (1 pT) IT
PT IT
1'Pr
E.
Proof of Proposition 2 compute &... 8t
We have Pale
21 + (1
E and Q. = -((1) (E I). Then we
as follows:
OP, :
a
0t I + (1
at a2
E
(35)
Ot
ōt
at
0t E
aa
(1
da
as
a2
Ot
(-B(t))
04
0
at
((t)
(E I).
0ts
There holds the Kolmogorov forward equation:
0 I + (1
at )
Q.Prr= = B(() (E I)
(36)
as
a,
= -((1 | at + (2 Clt
1)E + (1 at
a2
the fading matrix is idempotent + E E
3(1) at
(E-1)
dP c|s
Proof of Proposition given as:
We begin by considering the expression for the loss function Lsz, which is
Lse F4, 3) (o(s,tt)) se ( t, u)y ("e(momeee))) ("e(pommee)) (rx (Zo. 3z, v) 1). (37)
Then, we compute the gradient of this loss with respect se, which will help establish link between this and the binary cross-entropy loss.
V ne LSE (o (((0001)))))
(38)
= (1 + ((lloonen))) - (1 + (oo(tsd (1 + ca(enddu)))(1 + (i(eo.=1)))) jol (2,, u)y) a(re(2o. v))]
25

---

Next, we consider the soft binary cross-entropy loss LICCE, which is defined as:
LSACE(TO U2, v) -y->yy> (mo log (zr, - - (mo (1sello (,))), (39)
where the soft label Ty>>m, (nne = ply > Iz zo) = (re Fe, y)) the probability of the preference of y over dig, and o(.) is the sigmoid function.
Now, we compute the gradient of LICCE with respect to se:
V ses LIBCE 2 (1 - (s,
(100)))) u)y
(40)
a(se(Er, t, u)2) Tyrzilzo
= as t, u)₂) ((r T2, v))
We can now relate the gradients both loss functions:
V so Est (1 + (0)(), + ((((
se fARCE.
(41)
Proof of Theurem The reverse preference growing process denoted as Ω = (PoT)o- collection F of subsets of Ω is called a o-algebra on Ω if it satisfies: 1) E F. 2) F
its complement A Ω \ also belongs to F. 3) If { [A,] S F, then the countable union U An E F. We first show that the preference growing process satisfies the Markov property. Fr is a o-ageebaa of the preference growing process Q2 = (P P VA E F, and 0 < sttstt
(es, P(A)
(a/20 A)
(42)
(2((A)
p(A)
(Ms, (1s, A)
A)
Parte, (Alat, M>)q((A(r))
(nsllalsssp (((r))
Parele, (Alzz,, 2o)
the preference fading process Markovian
Psoee (Alzrz)
(=(xx)
(x,) Peja (are
the Bayes' theorem + Pa Po(w)a)
p(())) Peje (dre(tr))) = (x)e(
pa(xs) (ejo(()ll..) in matrix form 9O SS We rewrite the equation (xx)211/))) pi(ae)
(x()
(43)
Pe (xt)
Po(#, = Pc(a; =x) Va, YE.EER E
denotes the matrix with entries
P1
P Plle
Ps
PT
D1
Pt
P
Ps 7
Ps
P a|t
Pr
1
-P-1
Pr
3
P
1/s.
26

---

The reverse- reverse-time preference growing process progresses from = T to s = 0, and thus 0g is an increasing function, transitioning from ar 0 to ag I. We then compute the reverse transition rate matrix R, as follows:
0Pan a/6
R lim
1-75 Üs
OP s/c
das
lim
(44)
[-+ Oa.
Os
0
da.
-lim
t-s5 das
P-1
1/s
Pt
Pt
P
Os
[-+8
lim
OP
aas
f/s
Pt
Pt
P
f/s
+
P
1/8
Pt
Pt
1
OPT
Oas
t/s
da
Os
0P
r/3
0
088
0g
1
+
E
(I
E)
0a.
das
at
at
01
lim
P
lim
I + (1
03
E
I
[-88
d/s
(+
04
OP
4/s
Qt I
+
(1
E
E
0a
aak
lim P
lim
+
(1
E
I
f/s
1-48
8-3
023
02g
da
lim
(I E)
Pt
I
Pt
E
at
Pr
Pt
0
Os
Q = -(1) (E 1), Q = (()
(E
8ag agB(s) >
Note that from time to time < t, 0g is increasing.
0
8s
1
((8) (ET
Ps
B(s)
(E 1)
16
15
1
1
Q
Ps
Q
pt
I
ps/
pe
Moreover, we compute
dl and
Plc 0s
as follows:
a I + (1
aa.
E
(45)
Øs
aa,
a
a2
Ōs
da.
Note that from time / to time < l, as is increasing. 0s
a.B(s)) > 0
I
0t
E
(s)
a
((s)
Qr (E I)
04
(E
1)² = E² - 2E + I = = (E - (E-(EE (E- 0
=B(s)
Q1 E 1)²
+ Bs) - -IE
[at (1 E
18(s) (E- 1)]
as
a
= -PiiqQs.
27

---

OP
0
P
P-1
1
P
(46)
Os
0s
b/2
Pr
6/2
0
Ös
Ps
Pt
P
f/s
0p.
0P
1/3
P 1/4
+
P.
Ös
P1
Os
OP
40
The reverse time begins with = T. +
Q2
Paju
08
0p.
ÖP 10
Po
..PPap P0
-
Q0.
08
0s
OPr|+
OP
+/s
Pis
Q2
(Ptls Qo) =Q_ P..s
0s
Ös
1
P.
(Q. P P(p))
Qs
Ps
P
pt
p1
Finally, we prove that the preference growing process satisfies the Kolmogorov backward equation:
Pe 7
R.P a/t
P.
Q Pt
p.
P
(47)
The idea behind this step is prove that the elements each position of the matrices are identical, thereby establishing their equality.
The details are provided in Equation
48
49
and
50
{
1
P&
QQ PT.))
Q P.
+
P
Pr
p
OP x/t
Is
Q 2
Vs
15
ph.
PT
f/s
(x,
(48)
Pa(x))
P2(2)
Σ 9e x))
Ptis
(b1))
p2(2)
Pz(
Pc(x))
Σ
(0,2, x (2)1((())
Pr(y)
zeX
-{1
c (Q_ P.T.)
) (1,
Pi
Q
Ps
I (I,y)
(49)
P
P=(z)
Σ
ao 2)
6m(2)
Ps(y)
ZE
go (ax (((
= on()
Σ
(x(
1ex
28

---

1
Q.
Ps
1
p.
P.jp (x,1)
(50)
Ps
Pt
P((
Ps(z)
Σ 82(2)
Σ
(2
4s
Papa (y12)
(x(x)
Pc(y)
=E
lEX
(1) (x(x)
Σ lo (o 1)
(x) pi(y) (1))(()
Σ ge( 1) Po(1)
= Ptplylz))
pc(y)
1E.t
- -1100 Qs Ps
3 P (2, Vz, E. X.
Pi
D Details of Different Fading Matrix Setting
D.1 Point-Wiss Setting
model point-wise preference ratios. Specifically, we introduce an auxiliary general hard negative item æ-1, which represented as learnable embedding. The unified non-preference state corresponds to the general hard negative item x-1, that is, f7 = č_ E N++,, where ě denotes the one-hot vector associated with ø-1- In this case, the reference ratios Fr (Zo, 24 { x-1), y {æo, x-1 11) capture only the relative preference between the positive item Ta and the general hard negative -- Thus, by using x-1 as common reference, we derive the point-wise preference ratios.
In the setting of masked discrete diffusion models 40 52 53
Let pr ē- N+1 Then, the rank-1 fading matrix E is defined as follows:
E
In this case, the reference ratios re(#o, xe {oo, x-1). E {oo, x-1)) model only the ratios between the general hard negative x-1 and real items X.
re Fe, xz) = ri To, 20, log
if Ir
(1,,,, v)
--cz
if 1 and 30
a
20,, = log
if y so and Dt x-
1-or
B(t)
if y x-1 if x and Te x-1 otherwise
Qr(,,)) = (() (E 1) B(t)
We eastimates the preference ratios log (vlu)) (e( with se e0 4, 4)y:
IsE(Io, Fi, y/u) ( - - (e
se (xt, t, u) e (r D1, 1).
For one user preference data (u, æo), with faded item T,. we compute C (sE 21.
LSE = Σ
Qe(2e.)) v) (so Ft, v/w)
29

---

x1) T1, + Σ
Qc(2t, v) IsE (2o. 21, (20, v/u)
sol 1,, = (r, 04) = 0 = (sE((0, Bee malu) = 0
Σ
Qe(Te, v) ( 2e, y/u)
Qi(Ei # I-1.9 + 24) 0 (2o 2o ylu) = = 21 2-1
Q0(x-1,00) ISE (2 x (.
= 3(t) [ 1-ac
at
sa t, u)y + 1 at
at
Log
Ott
1 at
D.2 Pair-Wise Setting
In the field of natural language processing, uniform discrete diffusion models are generally considered inferior to masked discrete diffusion models [38 40). However, in the context of recommendation. uniform discrete diffusion models correspond to pair-wise preference ratios, which resemble ran- dom negative sampling, and are in fact superior to point-wise masked discrete diffusion models. Specifically, we define the unified non con-prefeeenc state as having equal probability over all items, i.e. pr I RN, where I denotes the all-ones vector. In this setting, the reference ratios re Ft E X. y E X') capture the relative preferences among all item pairs.
Let pT I E NR,, we have fading matrix E as follows:
E
In this case, the reference ratios r, (z Ft X, y X)) model the ratios of all item pairs.
ri Dt, 22) =0
if if
(z zo. y # ro)
log(1 + N
oz 1-a: Cr
y * Zg and H; = Z0 if = xo and _ + xo otherwise
(T Z2,8) y) =
Ti (10 Tt + zo, zo) = log(1 + N
I-o.
re (1 24 F so, y + mo) =
( [(tt({ N -1 if = y ()()
(x, 3(1) (E 1)
otherwise
We eastimates the preference ratios log (vlu) with se (I,, t, u)y; pe uj
Isæ(zo, T4, v/u) ( - - se (me, l, u), + e (x [ra x1,8) 1].
For one user preference data (u, æo), with faded item Ie, we compute Cse (2o, 2t, y/u):
LSE (z. Is Ft, ulu)
yEX
= (1, + Σ
Qe v) 1gr (r, ylu)
so( t2 u)2, = re(2o 21, = + (0, = 0
= Σ Qr(2e,)) v) (5g( 21, 4/u)
22) 3(1) 1
N Σ 1SE(Fo,
= 3() N
30

---

1
Σ
N
N
BEA
(0(x,,t,
Σ
Σ
se (1z, u22
N
N
(E
VEX
a1
14 = To, denote
0t
1
1
=B(t).
(1)
(log(1 + +) + +1)
A
1+
if f 20, denote N
at
8())
(alandi Be (e u) se( u)o0 + (1++ (log) (1 + A) -1]
D.3 Hybrid-Wist Wise Setting
Note that the point-wise and pair-wise settings are independent of each other. Therefore, we can define a hybrid non-preference state as pT = A(I R 0) (1 - A)ē- E R=+1, simultancously modeling point-wise and pair-wise preference ratios. Since X is typically large, the hybrid coefficient should be designed as 10-">, where vI Z+
Let pi O(I, + (1 --)ē-1 1=+1, we have fading matrix follows:
N
X
In this case, the reference ratios Ti (z0, 2t. X.y E X) model the ratios of all item pairs.
0
if
log ai
else if y 2o and _₁ .-
[1-a,][1-X)
log
else y + vn and 2, x- else y = --1 and 2t Zo
[1-X]
(1-0:)(-X)
log (1-(1-a))
Tr (r y)
(1 (1-a)) +
log
else y _ and _, x
a++(1-a))
log
(1-a:)(1-
else y = - and st # fo- 1 No
(1-a0)
a1+(1-ai) R
log
else y = zo and xt * {x- Ho
(1-0r)
0
else y * 1-- xo) and 32 F { No)
((1)-1))
if - x-1 if + x_1
3(1) -1)
Qx(,,)) = ((t) (00))(EE11))) (E-I)-
8()(1- A)
if + y and x = Ø-1
3(0(2)
if y and x #
We estimate the preference ratios log p=(viu) (0: with se se t, u)y:
Ise(Io, Fz. y/u) co(oedu)) n (( se (Nx, t, u) e (20. (r Fz, v) 1).
For one user preference data (u, 2o), with faded item Fe. we compute L (2o, 2t, ylu):
LSE Qu( v) (sE(E0, Ne. vlu)
VEX
31

---

x1( 2)) E
Qe(Er: y) -I, vlu)
vEXi(
sa(ai, 1, W),, = (1 = = (t,
Σ
Qu v) (20, Iz, F2. y/u)
Q = -I-1, + vz) ((1) - ar 3(1) (
- Qe(In. y # 21)
Σ
y
(o(
Σ
N
N
VEX
1
3 (2,,t,
(0
se (Zt, u)
N
N
(ar)
VEX
al
A
A
if of = F-1, denote = N
and
I- at
X-A A
+
N
= 3())
N(1 - A) e
A Se (te(xe, l, u)
A
se (Ie,, L u)zo
N
1)
1
log
+8(t)
A
1 + N
log
A
N 1-
A
at
A A
if = zp, denote
N I 1-
and
az
A-A +N
=(())
A. - X. ( - (1
A se (20 (2), u)
N
{1-
1 X
+B(1)
A .A. Alo.-oo----- 1) .A. (log(
N
N
in * (2o, I.) denote A = N. 1- az
at
and A
A A
A A +
N
= -())
X A.aaleedii)) X (a
N.A n A s0 u) 1 +11 -1----
A
-B())
se ( t, 2,
N
1
1) - N) .2)
+8(1)
(log + log 1) (1 A) (1 log(
N ))
D.4 Adative Setting
Furthermore, we can adaptively update the non-preference state pr initialized with above settings. Under adaptive setting, Pr jī = softmax(0), where Ø are learnable parameters.
Let pT R- or N-1, with Σ Me we have fading matrix E follows:
xeX
:
P|
By arbitrarily specifying distribution f7 satisfying
Σ Mr = 1, we instantiate any desired
2ex
non-preference state, which is physically associated with different negative sampling strategies. In
32

---

this case, the reference ratios (z E N, y E X) are computed as follows:
re(ao, It, xt) = 0
if I₁ =
re(Jo, Jo. y # m0) -log( 01 01
if y # so and 2t 20 if y = zo and 1y f x0 otherwise
re(oo, Ft, v)
11-a---p (1 (e
It # Io, vo) = loog( (1 (1-a)):0
re te # zo, y # zo) log
Nx
(,(t)( N -1) if x = y B(t) N
Qx(,,)) = 3(1) (E -)
otherwise
We eastimates the preference ratios log pe(vlu) with se (1 L u)y; pe( u)
ls Fe, vlu) (d((ndt)) - - (e( se (x L, u), e a00 (r. xr,8) 11.
For one user preference data (u, zo), with faded item Iv. we compute LSe (zo, 24, ylu):
CSE
Σ Qu( v) Iss( ze. ylu)
VEN
= Qi(11, xz) (sE F1, sz/u) +
E
Qe v) Iss( 2t, 24, vlu)
solzn,, t, u) = 1e(00, yEX)(z()
(10, se/u)
Σ
Qe(Te, Iss(( v/u))
vEA
Qe(2,, y + xx) = B(t)) Mrre
B() N
Σ 1s
vEA)(a))
1
Σ (os) -
N
(xX
vEX
S6 (0( =
Σ
My "e (1 u)g =
Σ
My se (2ee, t, u)y
vEXM
yEX
i Σ My log By
VEN
0t
if 2, = To. denote
1-a4
1
B(())
so (z),t, u)
+ ua.
+3(1)-
Mxr
Ilit (Mz, 1)) (log(us, + 1) Mrx log [ay]
Mn, +
if ot * 2o, denote 2 Ot
1-at Ctt
=3(1)- |mm (0((m)) - B 1, u) 800 4, 4)
+.((t) - [A Maz - E1+ logg((())) + (E + Mzo) log ( + (rx) (Mzz) log(4zr))
E Rank-r Solution of Fading Matrix E
Theorem 3 (Nonnegative Idempotent Decomposition). Let E E N be entrywise nonnegative and idempotent (E² E) with rank = r. Then the following statements are equivalent:
(A) E is nonnegative and idempotent with rank (E)
33

---

(B) There exist r nonzero, pairwise disjoin nonnegative vectors P ,'."""" E R and the corresponding support-indicator vectors 8 supp(F") {0,1) such that
(²(5t) E,
E
E
(51)
(8')
where (T for each and supp(p") == t , N} p 0). Equivalently, for any column index j {I....,N
p ifj supp(7*) for some i (1,...,r}.
Ej =
0,
ifj & U supp(p").
Moreover, writing the support indicator as I restricted to supp(p"). the decomposition 51 the compact form
admits
IT supp(F")
E =
2' (F') =0 (i
supp(rsy
Proof (Sufficiency (B) (A)). Entrywise nonnegativity is immediate from p 2 0 and 8
Idempotence follows from disjoint supports: for we have (5") pd 0, hence
(5 (33)
(5') 1 (5
E,E, (3t) p' (55) PJ (5 pc (30) p
O.
For = j we obtain
E (T( (
(5') (5*)
E,,
(T
so E = EE, E, is (pan (p") and supp( are disjoint, hence the one-dimensional subspaces are independent; therefore rank(E) (E) = >, rank E,)
Σ, E, E. To compute the rank, observe that the column space of
Proof (Necessity (A) (B)). Since is idempotent (E² E), its eigenvalues are or so the image Im (E) has dimension Because E 0, every column of E is either the zero vector or a nonnegative fixed point E (indeed, for the j-th standard basis vector εj, we have E(E Eτ, 2 0). Group the nonzero columns of E by proportionality: put indices k in the same class if the j-th and k-th nonzero columns are positive multiples of each other. This is an equivalence relation, and it produces exactly classes, say Si...., Sr, because each class contributes one linearly independent direction in Im()). Pick one representative nonzero column from each class and denote it by p 2 (i 1, ..... r). By construction, for every j S, the j-th column of E equals p", and for j & U.S. S; the column zero. The classes are disjoint in support by definition. Let g = 1g, be the indicator of S, The matrix that places column 5 on S; and zeros elsewhere is
( (5
E,
with (5 Tp' 0.
Therefore
E
E
which is exactly the Equation 51 Disjoint supports immediately give (5") pJ = for + j.
Physical intuition:
Clustering. Partition the item set X' ((2 N) into clusters Cı,...,Cr. Cr, where items in the same cluster are more similar. In practice, one may obtain {C,) via rector-based methods (e.g., k-means semantic embeddings) via r-way classification the user-item bipartite/weighted sequential graph. Since these constructions require side information beyond our core formulation, full exploration is deferred to future work.
Disjointness means the supports do not overlap: supp(ō") supp((")) for i
34

---

Cluster-wise target distributions. Define for each cluster a nonnegative target vector
(+) > 0 i E Ci, (T =
otherwise,
1....,r. r.
(3+) Because (pT))= have disjoint supports, the resulting fading matrix E = DE=1 (5*) p1
is
idempotent with rank(E) (Theorem where g" 1
Effect. During fading, each prefemed item replaced only by items within the same cluster, which filters out cross-cluster candidates and better respects item heterogeneity. This rank-r structure also improves efficiency. Using the decomposition. for any vector x R.,,
(5
Ex
Σ
so computing (8*)T& T costs ([[[: and the combination step costs OrrN), reducing the overall complexity from the naive O(N²) OrrN).
Takeaway. PreferGrow equipped with rank-r fading matrix - i.c., cluster-wise replacement combines theoretical elegance with practical realism: confines replacements within semantically coherent groups while offering scalable (rrN) implementation.
F
Algorithms for Training and Inference
Algorithm 1 Training Algorithm of PreferGrow
Input: preference data D (u,, non-preference user ratio p, retention probability or 3(T)dr = log(1 - (1 -
a2 = e Já s(r)dr with f 3(r)dr =
and the non-rreference state Py for preference fading.
Output: estimated Preference Ratios se (x,, t, u) and the non-preference user Φ.
repeat
(u, mn) ~
D preference user-item pair. sampling timestep i uniformly. D non pre-rrreerence user modeling. retain or replace for preference fading.
Uniform T})
u ø with probability p
It Pr p=(l0=)))) Er + (1- (1-ar)p7,, Jr X
Compute the score entropy loss LSE E Evex Qz y) (g Iz, ylu) as Appendix D
Take gradient descent step VaLSE and update parameters.
until converged
Algorithm 2 Inference Algorithm of PreferGrow
user condition u, sampling timesteps S, { with ts and To = O, personalization strength w, estimated Preference Ratios (a u) and the non-preference user Φ. Output: grown preference scores ((rolu), xo E of user u.
Input:
JT Erg PT E X
the non-preference state.
for S do
š (o, (1 + (1) Tos u) se To, Φ) personalization enhancement. Pr.-117. (Ir.- [er,, u) Pr, (2r, [IT..) ExEx Pr. 1.-1 (Zr (2) (e
an Sar, (Fre-)) + ar.-1 prr (I as Equation D Prito ((ra)) a1d1
P Pralton (IT))12)==
10 11 (2) (1 (zo PP (x+--)) as Equation
as ExExP Polo, (2rma))) eetottoorr (Bl 0 B .
ar,
a., EeEx ctol tw.)),
+(1 ax--1) pr (TT,
az,
Fyo-1 ~ Pra-str, (I |r,, u), Fr--1 X.
reverse preference growing.
end for return p(zolu) = Prolti (2 (rg (2ro
u), xo E X
grown preference scores of user
35

---

Table 3: Statistics of datasets after preprocessing.
Dataset
# users
# items
# Interactions
sparsity
Movies
6040
3883
1001456
04.27%
Steam
39795
9265
2949605
00.80%
Beauty
22,363
12.101
198,502
00.07%
Toys
19,412
11,924
138,444
00.06%
Sports
35,598
18,357
256,598
00.04%
G Experiments Details
G.1
Datasets
We evaluate PreferGrow on five real-world benchmark datasets:
MoviesLens [67] is commonly used movie recommendation dataset that contains user ratings,
movie titles, and movie genres.
Steam
58
encompasses user reviews for video games on the Steam Store.
Beauty
68
contains movie details and user reviews from Jun 1996 to Sep 2023.
Toys 668
includes user reviews and metadata for toys and games from Jun 1996 to Jul 2014.
Sports
68
comprises reviews and metadata for sports and outdoor products from 1996 2014.
Following prior works 9 we adopt the user-splitting strategy, which has been shown to effectively all sequences chronologically for
prevent information leakage test [69] Specifically, each dataset and then split the data into training, validation, and test sets with an 8:1:1 ratio, while preserving the last 10 interactions as the historical sequence. The statistical characteristics of the processed dataset are shown in Table As observed from the table, the recommendation datasets face significant challenge severe data sparsity.
G.2 Baselines
We compare PreferGrow with both traditional discriminative recommenders using negative sampling and diffusion-basee generative recommenders, including classical recommenders (SASRec (58),
Caser (60), GRURec [61],, item-level diffusion-based recommenders (DreamRec [9) PreferDiff
27 and preference score-level diffusion-based recommenders (DiftRec [28]. DDSR (35)):
leverages the self-attention mechanism in Transformer to model user preference scores from interaction histories, addressing data sparsity through negative sampling.
SASRec [58]
Caser 660 utilizes horizontal and vertical convolutional filters capture sequential patterns the point-level and union-level, allowing for skip behaviors, and models user preference scores while addressing data sparsity through negative sampling.
GRURee [61] adopts RNNs to model user preference scores from interaction histories, mitigating data sparsity through negative sampling.
DreamRee 9 reshapes sequential recommendation oracle item generation, addressing data sparsity by adding Gaussian noise to dense item embeddings.
PreferDiff 27 introduces an optimization objective specifically designed for item-level DM-based recommenders, which can integrate multiple negative samples, addressing data sparsity by adding noise to dense item embeddings and using negative sampling both.
iiftusion-based generative recommender assuming Gaussian prior, addressing data sparsity by adding Gaussian noise preference scores, without considering the constraints of the probability simplex.
DiffRec 28 is preference score-level
preference score-level diffusion-based generative recommender assuming categor- ical prior, addressing data sparsity by adding discrete noise preference scores while respecting the constraints of the probability simplex.
DDSR 35
36

---

G.3 Implementation Details
Training settings: We implement all models using Python 3.7 and PyTorch 1.12.1 on an Nvidia GeForce RTX 3090. During training, all methods are trained with fixed batch size of 256 using the Adam optimizer. Additionally, we apply early stopping based on the model's performance on validation set. To ensure reproducibility, a randomly chosen value. For the classic recommenders with negative sampling. we employ the binary cross-entropy (BCE) loss. PreferGrow uses SASRec the encoding model for the user's s historical sequence, and we adopt both Hybrid-Win Wise and Adaptive settings for the fading matrix. For all SASRec modules, we apply RoPE position encoding [70]. The search space of hyperparameters for the baselines is shown in Table 5] and the optimal parameters for our PreferGrow under both hybrid and adaptive settings are presented in T To ensure the reliability of our findings, have conducted a comprehensive re-evaluation of our experiments under multiple random seeds {100, 200, 300, 400, 500) and statistical significance testing (p 0.001). The results shown in Table consistently confirm the significant performance gains of PreferGrow over baselines.
fix all random seeds to 100 in main experiments,
Table Re-evaluation results (NDCG@5) under multiple random seeds.
Dataset
Beauty
Toys
Sports
Steam
MovieLens
SASRec
0.0229=.0020
0,0258:00023
0.0104=00016
0.0193#.0002
0,0507= 0004
PreferDiff
0.0224+.0031 0.0312+40009 0007
0.01044.0004
0.0348
PreferGrow
0.0315++.0010
0.0326+.0007
4.0466.00044
0.0399:1.0004
1.0913+.0003
Evaluation Protocols and Metrics: To ensure comprehensive evaluation and mitigate potential biases, we adopt the all-rank protocol 771 9 27 which evaluates recommendations across all items. We employ two widely used ranking-based metrics: Normalized Discounted Cumulative Gain (N@K) and Mean Reciprocal Rank (M@K), to assess the effectiveness of the models.
G.4 Hlppe-pparameter Analysis
The personalization strength is locally stable within a reasonable range and highly consistent across data splits, thus posing no practical challenge for tuning. As shown in Figure our paper, PreferGrow is sensitive to large-man magnitude changes 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 [5, 15)), but drops notably when w 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 E {0, 2, 5, 10) and evaluated the mean absolute error between the optimal values selected 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 H R@55 + H R 10 DDG@@5 + NDCG@10 10. Our findings in Table 7 show that the optimal w is highly consistent across data splits, indicating that can be tuned on validation (or even training) set like other standard hyperparameters. This ensures that tuning w does not pose a practical challenge.
G.5 Efficiency Analysis
As summarized in Table we report each model's trainable parameters, the number of training epochs. GFLOPs per model output (recall that diffusion models require multiple outputs for denoising), and the total number of outputs.
Quantitative scalability analysis. We also provide quantitative analysis of PreferGrow and clarify the solution outlined in the Limitations. PreferGrow introduces only O(N) complexity in the network. loss, and inference paths - comparable to contemporary diffusion- recommenders. Table g contrasts model, loss, and inference complexities with representative baselines, where L is the history length, N the item set size, B the number of negatives, d the hidden dimension, and T the number of diffusion steps.
Model parameters. We never materialize the full N2 preference-ratio matrix. Equation requires only N scores; the additional computation cost is thus ((Nd).
12
37

---

Table Hyperparameters Search Space for Baselines.
Method
Hyperparameter Search Space
Ir~ {le-2, le-3, le-4, le-5) with decay 0, embedding size d ~ | [128, 256, 512)
Shared
the number of negative sampling (if using) ~ [64, 128, 256], bath size 256
DreamRee
[0, 2, 10), 7 (500, 1000, 2000,3000), p {0.05, 0.1, 0.15, 0.2, 0.25, 0.3)
PreferDiff
À ~ [0.2. 0.4, 0.6, 0.8), ~ (0, I, 2. 10), ~~ (500, 1000. 2000,3000]
DiffRec
noise scale (1e-1, 1e-2, Ic-3, Ic-4, 1e- 7 (, 20, 50, 100)
DDSR
T ~ [500, 1000, 2000,3000
T ~ (5, 10, 20, 30, 40), / ~ (0.05, 0.1, 0.150 0.2, 0.25, 0.3) w [0, 1, 5, 10), A ~ [0.9, 0.99, 0.999, 0.9999, 0.99999)
PreferGrow
Table Best Hyperparameters for PreferGrow on five datasets.
Variant
Dataset
Ir
d
p
T
A
MovieLens
lc-4
256
0.1
20
10
0.9999
Steam
Ic-3
256
0.1
20
10
0.99999
Hybrid
Beauty
lc-4
256
0.1
20
0.999
Toys
le-3
256
0.2
20
10
0.9999
Sports
le-3
256
0.2
20
0.9999
Variant
Method
P
T
w
Æ-11
Dataset
lc-4
256
0.2
20
10
True
Steam
le-3
256
0.05
20
10
True
Adaptive
Beauty
le-4
256
0.1
20
2
True
Toys
le-4
256
0.2
20
5
True
Sports
le-4
256
0.2
20
True
Table 7: Mean absolute error of w across data splits on five datasets.
Dataset
MovieLens Steam
Beauty
Sports
Toys
Mean abs. error of w
0.000 0.000
0.778
0.050
0.775
Loss computation. With the idempotent fading matrix E, the objective in Eq. (4) simplifies (Appendix C. C.4) to element wiie means, indexing, and a precomputable -vesralt-overall ONN). The score-entropy loss contains three parts: (i) positive term and (ii) a negative term (both simple mean/index operations, O(N)), and (iii) a constant term ensuring non-negativity that depends only on and can be precomputed in O(T) with 7020.
Inference cost. Idempotency yields
(P 1,
E Aoe, L,
pr
reducing Equation 19 from O(N2) O(N).
Where (NV2) comes from. The apparent O(N2) arises only from the modeling target: preference ratios expressive than conventional logits, which raises PreferGrow's performance ceiling but does not inflate the network/aloorihmic path beyond O(N).
Scaling to industry scale. When N is extremely large (e.g., billions of items), even O(N) becomes impractical. As discussed in the Limitations, Semantic IDs (SIDs) offer principled route to reduce complexity. SIDs encode each item with m codebooks of size enabling up to cm distinct items while keeping computation at O(me); since T > N > me, this yields a compact yet expressive representation space and allows PrefenGrow-on-SID: to reduce cost from O(N) to O (mc). practical blocker is that current SID pipelines (e.g., Tiger 774, ActionPiece (75). DDSR (35). OneRee (76) are not soun-soureed. Truly deploying PreferGrow industrial scale therefore requires
38

---

Table 8: Comparison of model complexities.
Complexity
Model Parameters
Lass Computation
Modeling Target
Inference
SASRec
O(Ld + L2d)
((Bd)
O(N)
OLL + + Nd)
DreamRec
+3(d LPd)
O(d)
O(T4)
O + 3)d + +TL' Nd)
PreferDift
((( + 33d L²d)
(
O(T4)
O(T(L + +3dd + +TL'd Nd)
Diffitee
O(LN2)
O(N)
O(TN)
O[TLN*)
DDSR
O(Ld² L2d)
((N)
O(TN)
O(TL + T TNd)
ProferGrow
O(LL +338 d2)
O(N + T)
O(TN2)
O + 336 TL TN 3)
Table 9: Comparison of efficiency on Steam.
Models
# Trainable Parameters
# training epochs
Inference GFLOPs
Inference steps
SASRec
2.70M
61
0.85
Caser
2.49M
58
0.38
GRURec
2.77M
4.06
DreamRec
7.25M
0.85
20
PreferDiff
7.25M
55
0.85
20
DiffRec
18.55M
1000
/
20
DDSR
3.03M
142
1.77
20
PreferGrow
3.03M
0.93
20
addressing the open challenge of large-scale multimodal SID pretraining. Extending PreferGrow to operate over SIDs our ongoing work.
39


Reviews:
[{"review_id": "ma56HZmYP0", "reviewer": "Reviewer_3oh1", "summary": "This paper proposes a discrete diffusion-based recommendation method that models relative preference ratios by fading and growing user preferences over the discrete item corpus, directly aligning with the ranking-oriented, discrete nature of recommendation tasks. It replaces the preferred item with alternatives, physically akin to negative sampling. During backward generation, it iteratively grows the preference signals from the estimated ratios.", "questions": "Please see Weakness. If there is any misunderstanding in my interpretation, please feel free to clarify it in the rebuttal. I would be happy to revise my score if my concerns are adequately addressed.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Advantage:\n1. The discrete modeling of preference ratios that aligns with the ranking-oriented nature of recommendation tasks. Noise is added by replacing the preferred item with alternatives, enabling explicit negative sampling and removing prior noise assumptions. \n2. Idempotent fading matrix with closed-form solution whose idempotent property is critical for ensuring both the Markov property and reversibility. Flexible architecture supports point-wise, pair-wise, and hybrid preference modeling. \n3. Extensive experiments on five datasets showing consistent outperformance of existing diffusion-based recommendation method. \n\nMy concern:\nIn this paper, the author adopts an idempotent matrix E as the fading matrix, which provides desirable properties for theoretical analysis (e.g., the Markov property and invertibility). However, at the implementation level, the proposed method uses a unified setting for the fading matrix, as shown in Appendices C.1, C.2, and C.3. In this setting, forward denoising either results in “staying at the current item” or “randomly replacing the current item with others uniformly.” I am concerned that such a design might degrade the model into a pure weighted negative sampling method, thereby eliminating the advantages of employing a diffusion model. Let me elaborate further. In item-level and preference score-level diffusion methods, the representations of items and preference scores are typically correlated during the forward process. However, in this paper, due to the uniform design of the fading matrix, the reference items at each step appear to be mostly independent of one another (e.g., “staying at the current item” or “randomly replacing the current item with others uniformly”  between two steps). As a result, the proposed method may degrade into a simple negative sampling scheme, as illustrated in Appendices C.1–C.3, thus undermining the core motivation and benefits of the diffusion model. In my opinion, 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—for example, assigning a higher probability to replacing the current item with a similar one.", "was_revised": true, "final_justification": "The authors' response addresses some of my concerns. I'd like to raise my score.", "rebuttal": "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. \n\n> Comment 1: the reference items at each step appear to be mostly independent of one another\n\n**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.\n\n\n> Comment 2: PreferGrow degenerates into weighted negative sampling under idempotent fading, losing the key strengths of diffusion.\n\n**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]. \n\n$$\n\\nabla \\log p_t(x_t) = \\frac{p_t(y\\in\\mathbb{R}^d) - p_t(x_t)}{p_t(x_t)} \\sim \\frac{p_t(y\\in\\{1,\\cdots,N\\}) - p_t(x_t)}{p_t(x_t)} = \\frac{p_t(y)}{p_t(x_t)} - 1.\n$$\n\nAs 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.\n\n> 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.\n\nIn 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.\n\nThank 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. \n\n> 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\n\n**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.\n\n**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.\n\n**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.\n\n**References:**\n\n[1] Ho, Jonathan, Ajay Jain, and Pieter Abbeel. \"Denoising diffusion probabilistic models.\" NeurIPS 2020.\n\n[2] Song, Yang, et al. \"Score-based generative modeling through stochastic differential equations.\" ICLR 2021.\n\n[3] Lou, Aaron, Chenlin Meng, and Stefano Ermon. \"Discrete diffusion modeling by estimating the ratios of the data distribution.\" ICML 2024."}, {"review_id": "SJnqgFaRyd", "reviewer": "Reviewer_1sA3", "summary": "This paper introduces a novel recommendation system model named PreferGrow, which is based on a discrete diffusion process to address the core challenge of user preference data sparsity. Unlike previous diffusion recommendation models that primarily rely on continuous Gaussian noise, PreferGrow operates directly on the discrete item space. Its core contributions are threefold:\n\n1.  **Discrete Preference Modeling**: The model directly models \"preference ratios\" between pairs of items, which is more aligned with the ranking-oriented nature of recommendation tasks.\n\n2.  **Preference Fading**: In the forward diffusion process, instead of injecting noise, the model employs a \"preference fading\" mechanism (analogous to negative sampling), where a user's liked item is replaced with other items with a certain probability. This avoids assumptions about a prior noise distribution.\n\n3.  **Preference Growing**: In the reverse generation process, the model iteratively \"grows\" and reconstructs user preferences from the estimated preference ratios.\n\nThe paper provides a solid theoretical foundation for PreferGrow, including a mathematical formulation based on an idempotent fading matrix, and theoretically proves the Markovian nature and reversibility of the process. Experimental results on five benchmark datasets show that PreferGrow consistently outperforms existing state-of-the-art recommendation algorithms, including other diffusion-based models.", "questions": "1. Please answer other questions in the weaknesses section.\n\n2.  **Further Discussion on Scalability**: The paper candidly admits that scalability is a major limitation. Could the authors provide specific empirical data on how training/inference time varies with the number of items `|X|` on the datasets used in the experiments? Additionally, regarding the \"sparse fading matrix\" mentioned in future work, how can the critical \"idempotency\" property be guaranteed when designing such a matrix? This seems to be a non-trivial challenge.\n\n3.  **The Deeper Connection Between \"Preference Fading\" and Negative Sampling**: The paper analogizes \"preference fading\" to negative sampling. Traditional negative sampling strategies (e.g., sampling based on item popularity) have been proven very effective. How do the authors view the relationship between the uniform replacement strategy in the \"Pair-Wise\" setting and these mature negative sampling techniques? Could more complex sampling strategies (like popularity bias) be incorporated into the design of the non-preferred state to further enhance performance?\n\n4.  **Sensitivity Issue of Personalization Strength `w`**: According to the hyperparameter analysis in the paper, model performance is highly sensitive to the personalization strength `w`, which is selected during the inference phase. Does this imply that for a new deployment scenario, a validation set must be reserved specifically to tune `w`? This would introduce additional overhead and could lead to `w` overfitting to the validation set. Have the authors considered enabling the model to adaptively learn `w`, or are there more robust strategies for selecting `w`?\n\n5.  **Explanation of $x_{-1}$ in the Point-Wise Setting**: In the paper's Point-Wise setting, the authors introduce a \"universal hard negative item.\" How is this item actually represented? Is it a learnable embedding vector? What is the essential difference between this and simply selecting a fixed item from the item pool as a global negative?", "limitations": "Yes, the authors have adequately discussed the limitations of their work. They pointed out two key issues: long training times and poor scalability on very large item sets. The authors not only identified these problems but also proposed constructive potential solutions as future research directions.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n1.  **Originality:** The core idea of this paper is highly original. Applying a discrete diffusion model to recommendation systems and proposing \"preference fading\" as the forward process is a significant breakthrough from the existing paradigm that relies on continuous Gaussian noise. The modeling objective of \"preference ratios\" is also insightful, drawing inspiration from the success of ideas like DPO (Direct Preference Optimization) in language models, making it naturally suited for ranking tasks.\n\n2.  **Quality:** The paper's theoretical foundation is very solid. The authors present a clear, matrix-based mathematical framework, particularly introducing the key concept of an \"idempotent fading matrix\" and providing complete proofs of the model's Markovian nature and reversibility around it. This provides strong theoretical support for the model's effectiveness.\n\n3.  **Clarity:** The paper is well-organized and clearly written. The introduction clearly articulates the research motivation and effectively distinguishes this work from existing methods. Figure 1 intuitively contrasts different diffusion recommendation paradigms, and Figure 2 clearly illustrates the PreferGrow pipeline, greatly enhancing the paper's comprehensibility.\n\n4.  **Significance:** This work is of significant importance to the subfield of diffusion-based recommendation systems. The experimental section comprehensively compares PreferGrow with a variety of baselines (including classic, item-level, and score-level diffusion models) and demonstrates its consistent and significant performance advantages across multiple datasets. If these results can be widely replicated, this model could become an important benchmark in the field. Furthermore, the flexibility of the model design (supporting Point-Wise, Pair-Wise, Hybrid modes) also increases its potential application value.\n\n**Weaknesses:**\n\n1.  **Fundamental Scalability Flaw:** This is the most critical weakness of the paper. The authors acknowledge this issue in Section 5, but its severity is somewhat underestimated. The model's core is to model the \"preference ratios\" of all item pairs, which leads to a computational/memory complexity of at least the square of the number of items, $|X|^2$. This is completely infeasible in real-world recommendation scenarios with millions of items. Compared to contemporary work, this is a step backward. For example, models based on negative sampling (like its encoder base, SASRec) and embedding diffusion models (like PreferDiff) have much better scalability. If the core algorithm of a recommendation system paper cannot be applied to large-scale datasets, its practical significance and impact will be considerably diminished. The proposed method feels more like a \"toy\" model validated on small datasets, lacking consideration for real-world application challenges.\n\n2.  **Theoretical \"Elegance\" at the Cost of Realism:** The paper's theoretical derivations heavily rely on a core assumption: the fading matrix $E$ is idempotent ($E^2=E$). To satisfy this condition, the paper further derives that all column vectors of $E$ must be identical ( $ E=\\frac{\\vec{p} _{T}\\cdot\\vec{1}^{\\top}}{\\vec{1}^{\\top}\\vec{p} _{T}}$ ). This means that during the forward \"fading\" process, regardless of the original preferred item $x_0$, it will be replaced according to the **same** target distribution $\\vec{p}_T$. This is a strong simplifying assumption that somewhat ignores the heterogeneity among items. For example, a reasonable substitute for a \"faded\" shirt item should be other clothing, not a refrigerator. The model cannot capture such fine-grained relationships, which calls its physical intuition and modeling capability into question. In contrast, although models like DDSR operate on simple categorical distributions, they at least do not introduce such a strong and counter-intuitive global structural assumption.\n\n3.  **Questionable Reliability of Experimental Evaluation:** The paper reports performance surpassing all baselines on five datasets. However, the reliability of these results is questionable. The authors admit that due to computational resource constraints, all experiments were run once with a fixed random seed, without reporting error bars or conducting statistical significance tests. In machine learning, especially in recommendation systems, model performance is highly sensitive to random factors like data splits and parameter initialization. Without statistical tests, we cannot determine whether the reported performance gains are genuinely effective or merely due to random chance or specific hyperparameter tuning. For a paper seeking to advance the state-of-the-art, this is unacceptable.\n\n4.  **Marginal Contribution vs. Disproportionate Complexity:** The core idea of this paper can be seen as \"grafting\" recent work in discrete diffusion and alignment (e.g., DPO) onto recommendation systems. While the combination is innovative, the resulting model complexity (e.g., the extremely tedious loss function derivation in Appendix) may be disproportionate to the actual, statistically unverified performance gains it brings. Compared to contemporary works, for instance, DDSR, which directly performs diffusion on a discrete state space, is conceptually more straightforward; PreferDiff, although a continuous model, has a more intuitive optimization objective designed for negative sampling. It is debatable whether the series of complex concepts introduced by PreferGrow, such as \"preference ratios\" and \"idempotent fading matrix,\" is the right path to solving the problem.", "was_revised": true, "final_justification": "The authors have resolved most of my concerns. I decided to raise my score to 4.", "rebuttal": "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.\n\n> Comment 1: **Fundamental Scalability Flaw:**\n\n**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\n\n|   Complexity   |          Model Parameters           |   Loss Computation    |    Modeling Target    |                 Inference                  |\n| :------------: | :---------------------------------: | :-------------------: | :-------------------: | :----------------------------------------: |\n| **PreferDiff** |  $\\mathcal{O}((L+3)d^{2}+L^{2}d)$   | $\\mathcal{O}((B+1)d)$ |   $\\mathcal{O}(Td)$   |   $\\mathcal{O}(T(L+3)d^{2}+TL^{2}d+Nd)$    |\n|    **DDSR**    |   $\\mathcal{O}(Ld^{2}+L^{2}d+Nd)$   |   $\\mathcal{O}(N)$    |   $\\mathcal{O}(TN)$   |     $\\mathcal{O}(TLd^{2}+TL^{2}d+TNd)$     |\n| **PreferGrow** | $\\mathcal{O}((L+3)d^{2}+L^{2}d+Nd)$ |  $\\mathcal{O}(N+T)$   | $\\mathcal{O}(TN^{2})$ | $\\mathcal{O}(T(L+3)d^{2}+TL^{2}d+TN(d+3))$ |\n| **PreferGrow on SIDs** | $\\mathcal{O}(m(L{+}3)d^2 {+} mL^2d {+} mcd)$ | $\\mathcal{O}(mc+T)$ | $\\mathcal{O}(Tmc^2)$ | $\\mathcal{O}(Tm(L{+}3)d^2 {+} TmL^2d {+} Tmc(d{+}3))$ |\n\n* **Model Parameters:** We never materialise the full $N^{2}$ preference‑ratio matrix; Equation (14) only requires $N$ scores, so the extra cost is only $\\mathcal O(Nd)$. \n* **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).\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.\n* **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 $c$. \n\nTo 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.\n\n**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.\n\n> Comment 2: **Theoretical \"Elegance\" at the Cost of Realism:**\n\n**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.\n\n**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.\n\n**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.\n\n> Comment 3: **Questionable Reliability of Experimental Evaluation:**\n\n**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.\n\n|  Dataset   |      Beauty      |       Toys       |      Sports      |      Steam       |    MovieLens     |\n| :--------: | :--------------: | :--------------: | :--------------: | :--------------: | :--------------: |\n|   SASRec   |   0.0229±.0020   |   0.0258±.0023   |   0.0104±.0016   |   0.0193±.0002   |   0.0507±.0004   |\n| PreferDiff |   0.0224±.0031   |   0.0312±.0009   |   0.0122±.0007   |   0.0104±.0004   |   0.0348±.0005   |\n| PreferGrow | **0.0315±.0010** | **0.0326±.0007** | **0.0146±.0004** | **0.0399±.0004** | **0.0913±.0003** |\n\n> Comment 4: **Marginal Contribution vs. Disproportionate Complexity:**\n\n**I. PreferGrow’s overall complexity is $\\mathcal{O}(N)$ and its performance gains have been statistically verified.** Please refer to the earlier discussion.\n\n**II. PreferGrow’s another contribution lies in its unified modeling objective that serves both generative modeling and ranking.**\n\n|   Model    | Generative Obj. | Ranking Obj. |      Loss      |\n| :--------: | :-------------: | :----------: | :------------: |\n|    DDSR    |  ✗ (discarded)  |    ✓ (CE)    | Merely Ranking |\n| PreferDiff |     ✓ (MSE)     |   ✓ (BPR)    |  Weighted sum  |\n| PreferGrow |   ✓ (ratios)    |  ✓ (ratios)  |  Unified one   |\n\nSpecifically, 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:\n$$\n\\nabla \\log p_t(x_t) = \\frac{p_t(y\\in\\mathbb{R}^d) - p_t(x_t)}{p_t(x_t)} \\sim \\frac{p_t(y\\in[1,\\cdots,N]) - p_t(x_t)}{p_t(x_t)} = \\frac{p_t(y)}{p_t(x_t)} - 1.\n$$\nOptimizing 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**.\n\n> Comment 5: **Further Discussion on Scalability**:\n\n**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. \n\n|    #Items    | 3883  | 9265  | 12101 | 11924 | 18357 |\n| :----------: | :---: | :---: | :---: | :---: | :---: |\n| Our Train B  | 1.90M | 1.85M | 0.50M | 0.50M | 0.50M |\n|  Our GFLOPs  | 18.90 | 19.31 | 19.53 | 19.52 | 20.00 |\n| DDSR Train B | 0.50M | 0.35M | 0.10M | 0.10M | 0.10M |\n| DDSR GFLOPs  | 18.30 | 18.44 | 18.51 | 18.51 | 18.67 |\n\nOnly 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.\n\n> Comment 6: **The Deeper Connection Between \"Preference Fading\" and Negative Sampling**:\n\n**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). \n\n> Comment 7: **Sensitivity Issue of Personalization Strength `w`**\n\n**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.\n\n|           Dataset           | MovieLens | Steam | Beauty | Sports | Toys  |\n| :-------------------------: | :-------: | :---: | :----: | :----: | :---: |\n| Mean abs. error of opt. $w$ |     0     |   0   | 0.778  |  0.05  | 0.775 |\n\n> Comment 8: **Explanation of $x\\_{-1}$ in the Point-Wise Setting**: \n\nYes, $x_{-1}$ 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.\n\nThanks again for insightful feedbacks."}, {"review_id": "bi8vUJ87Wp", "reviewer": "Reviewer_8EAw", "summary": "This paper proposes PreferGrow, a recommender systems to learn item representations through discrete diffusion models. Existing works based on diffusion models in recommendation systems suffer from lack of negative sampling as well as misalignment in the continuous probability modeling space vs. the discrete nature of user interaction. To tackle these two problems, PreferGrow introduces a preference fading discrete diffusion model tailored for recommendation. Specifically, during the forward pass of diffusion process, the target item embedding corruption is guided by fading matrices; and during the backward pass, user embedding generated by SASRec, as well as the corrupted item embedding are concatenated to predict the preference ratio.", "questions": "please refer to the weakness. \n\nfurthermore, I think the authors should comparatively discuss efficiency of the proposed framework.", "limitations": "limitations are properly discussed", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength: \n1. this paper is technically well-motivated -- it identifies misalignment between the continuous probability modeling space in diffusion models vs. the discrete nature of user interaction.\n\nWeakness:\n1. the clarity and writing needs to be significantly improved. the technical part of this paper is extremely difficult to understand. \n2. i'm confused by the incorporation of user embedding, which other diffusion-based baseline does not explore. The incorporation of user embedding seems agnostic of the idea proposed in this paper.\n3. while the authors try to provide theoretical proofs, the provided proofs are mostly general diffusion derivations and the added value is very marginal.", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n> Summary: ... to learn item representations ... the target item embedding corruption ...  the corrupted item embedding ...\n\nFrom 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:\n\n1. **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. \n\n2. **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.\n\n3. **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.\n\nWe hope that our detailed explanation helps you gain a clear understanding of PreferGrow.\n\n> Comment 1: the clarity and writing \n\nWe 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.\n\n>  Comment 2: incorporation of user embedding\n\n**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.\n\n>  Comment 3: theoretical contributions\n\n**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.\n\n> Comment 4: efficiency comparison.\n\nThank 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) \n\n|   Complexity   |                Model Parameters                 |         Loss Computation          |    Modeling Target    |                       Inference                        |\n| :------------: | :---------------------------------------------: | :-------------------------------: | :-------------------: | :----------------------------------------------------: |\n|   **SASRec**   |    $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d\\bigr)$     | $\\mathcal{O}\\!\\bigl((B+1)d\\bigr)$ |   $\\mathcal{O}(N)$    |      $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d+Nd\\bigr)$       |\n|  **DreamRec**  |  $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d\\bigr)$   |         $\\mathcal{O}(d)$          |   $\\mathcal{O}(Td)$   |   $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+Nd\\bigr)$    |\n| **PreferDiff** |  $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d\\bigr)$   | $\\mathcal{O}\\bigl((B+1)d\\bigr)$ |   $\\mathcal{O}(Td)$   |   $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+Nd\\bigr)$    |\n|  **DiffRec**   |              $\\mathcal{O}(LN^{2})$              |         $\\mathcal{O}(N)$          |   $\\mathcal{O}(TN)$   |                 $\\mathcal{O}(TLN^{2})$                 |\n|    **DDSR**    |   $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d+Nd\\bigr)$   |         $\\mathcal{O}(N)$          |   $\\mathcal{O}(TN)$   |     $\\mathcal{O}\\bigl(TLd^{2}+TL^{2}d+TNd\\bigr)$     |\n| **PreferGrow** | $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d+Nd\\bigr)$ |        $\\mathcal{O}(N+T)$         | $\\mathcal{O}(TN^{2})$ | $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+TN(d+3)\\bigr)$ |\n| **PreferGrow on SIDs** | $\\mathcal{O}(m(L{+}3)d^2 {+} mL^2d {+} mcd)$ | $\\mathcal{O}(mc+T)$ | $\\mathcal{O}(Tmc^2)$ | $\\mathcal{O}(Tm(L{+}3)d^2 {+} TmL^2d {+} Tmc(d{+}3))$ |\n\n* **Model Parameters:** We never materialise the full $N^{2}$ preference‑ratio matrix; Equation (14) only requires $N$ scores, so the extra cost is $\\mathcal O(Nd)$. \n* **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, $T=20$).\n* **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 $\\mathcal O(N)$.  \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.\n* **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 $c$. \n\nTo 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.\n\nWe 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. \n\n|    #Items    | 3883  | 9265  | 12101 | 11924 | 18357 |\n| :----------: | :---: | :---: | :---: | :---: | :---: |\n| Our Train B  | 1.90M | 1.85M | 0.50M | 0.50M | 0.50M |\n|  Our GFLOPs  | 18.90 | 19.31 | 19.53 | 19.52 | 20.00 |\n| DDSR Train B | 0.50M | 0.35M | 0.10M | 0.10M | 0.10M |\n| DDSR GFLOPs  | 18.30 | 18.44 | 18.51 | 18.51 | 18.67 |\n\nOverall, 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.** \n\nWe 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**.\n\n**References:**\n\n[1] Ho, Jonathan, Ajay Jain, and Pieter Abbeel. \"Denoising diffusion probabilistic models.\" NeurIPS 2020.\n\n[2] Rafailov, Rafael, et al. \"Direct preference optimization: Your language model is secretly a reward model.\" NeurIPS 2023."}, {"review_id": "Qr24x2Udu5", "reviewer": "Reviewer_XkWC", "summary": "This paper introduces PreferGrow, a discrete diffusion-based recommender system that models preference ratios by a novel process of preference fading and growing, specifically tailored for the discrete nature of user preference data. Unlike prior diffusion-based recommenders that rely on continuous Gaussian noise, PreferGrow directly operates on discrete items using a theoretically grounded matrix-based framework. The authors claim that this approach not only better captures the discrete nature of recommendation tasks but also improves empirical performance across multiple datasets.", "questions": "NA", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength\n\na) The paper offers solid theoretical backing, proving Markov properties, reversibility, and the validity of the fading process using an idempotent fading matrix.\n\nb) Extensive experiments across five benchmark datasets consistently show PreferGrow outperforming classical, item-level, and score-level diffusion-based recommenders.\n\nWeakness\nThe approach could face significant computational challenges when scaling to very large item sets due to the all-pair preference ratio modeling.", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n> Comment 1: significant computational challenges when scaling to very large item sets due to the all-pair preference ratio modeling.\n\nWe 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. \n\n**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\n\n|   Complexity   |               Model Parameters                |        Loss Computation         |    Modeling Target    |                      Inference                       |\n| :------------: | :-------------------------------------------: | :-----------------------------: | :-------------------: | :--------------------------------------------------: |\n|   **SASRec**   |    $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d\\bigr)$     | $\\mathcal{O}\\bigl((B+1)d\\bigr)$ |   $\\mathcal{O}(N)$    |      $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d+Nd\\bigr)$       |\n|  **DreamRec**  |  $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d\\bigr)$   |        $\\mathcal{O}(d)$         |   $\\mathcal{O}(Td)$   |   $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+Nd\\bigr)$    |\n| **PreferDiff** |  $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d\\bigr)$   | $\\mathcal{O}\\bigl((B+1)d\\bigr)$ |   $\\mathcal{O}(Td)$   |   $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+Nd\\bigr)$    |\n|  **DiffRec**   |             $\\mathcal{O}(LN^{2})$             |        $\\mathcal{O}(N)$         |   $\\mathcal{O}(TN)$   |                $\\mathcal{O}(TLN^{2})$                |\n|    **DDSR**    |   $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d+Nd\\bigr)$   |        $\\mathcal{O}(N)$         |   $\\mathcal{O}(TN)$   |     $\\mathcal{O}\\bigl(TLd^{2}+TL^{2}d+TNd\\bigr)$     |\n| **PreferGrow** | $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d+Nd\\bigr)$ |       $\\mathcal{O}(N+T)$        | $\\mathcal{O}(TN^{2})$ | $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+TN(d+3)\\bigr)$ |\n| **PreferGrow on SIDs** | $\\mathcal{O}(m(L{+}3)d^2 {+} mL^2d {+} mcd)$ | $\\mathcal{O}(mc+T)$ | $\\mathcal{O}(Tmc^2)$ | $\\mathcal{O}(Tm(L{+}3)d^2 {+} TmL^2d {+} Tmc(d{+}3))$ |\n\n\n* **Model Parameters:** We never materialise the full $N^{2}$ preference‑ratio matrix; Equation (14) only requires $N$ scores, so the extra cost is $\\mathcal O(Nd)$. \n* **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, $T=20$).\n* **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 $\\mathcal O(N)$.  \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.\n* **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.\n\nTo 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.\n\n**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 $\\mathcal{O}(mc)$ .\n\nHowever, 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.\n\nWe 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.\n\nThank you once again for your valuable feedback.  We remain receptive and ready to engage with any further questions or suggestions you may have.\n\n**References**\n\n[1] Rajput, Shashank, et al. \"Recommender systems with generative retrieval.\" NeurIPS 2023.\n\n[2] Xie, Wenjia, et al. \"Breaking determinism: Fuzzy modeling of sequential recommendation using discrete state space diffusion model.\" NeurIPS 2024.\n\n[3] Hou, Yupeng, et al. \"ActionPiece: Contextually Tokenizing Action Sequences for Generative Recommendation.\" ICML 2025.\n\n[4] Deng, Jiaxin, et al. \"Onerec: Unifying retrieve and rank with generative recommender and iterative preference alignment.\" ArXiv 2025."}, {"review_id": "p1cu34NTGq", "reviewer": "Reviewer_rfEc", "summary": "This paper introduces PreferGrow, a novel recommender system designed to overcome data sparsity by using a discrete diffusion model. Unlike other diffusion-based recommenders that rely on continuous noise, PreferGrow operates directly on the discrete set of items. It perturbs data by \"fading\" user preferences—replacing a liked item with alternatives in a process similar to negative sampling. The model then learns to reconstruct user preferences by \"growing\" them from these faded states, focusing on relative preference ratios between items rather than absolute scores. This approach, which is theoretically grounded with proofs of its key properties, demonstrates superior performance over existing methods across five datasets", "questions": "1. Your analysis shows high sensitivity to the personalization parameter $w$. Could you provide a qualitative analysis showing how the actual recommendations change for a user as $w$ is adjusted? \n\n2. Do you have depolyed your method into real world industrial ?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 1, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n1. The paper introduces a novel and significant approach to recommendation systems by leveraging discrete diffusion models. It diverges from the predominant trend of using continuous Gaussian noise in diffusion-based recommenders, which is often a mismatched assumption for discrete item data. \n\n2. The authors conduct extensive experiments on five benchmark datasets, comparing PreferGrow against a variety of classic and state-of-the-art diffusion-based recommenders.\n\n3. The paper is of high quality, grounded in solid theoretical analysis. A key contribution is the development of the  idempotent fading matrix and the formal proof that its idempotence is crucial for ensuring the Markov property and reversibility of the diffusion process.\n\nWeakenesses\n1. The most significant weakness, which the authors openly acknowledge, is the model's scalability. The pair-wise setting, which proves most effective, requires modeling preference ratios across all item pairs. This becomes computationally infeasible for recommendation scenarios with extremely large item corpora.\n\n2. The paper notes that PreferGrow incurs longer training times than prior diffusion-based recommenders. This is attributed to the uniform sampling of timesteps during training, where modeling preference ratios at different stages of \"fading\" can have varying levels of difficulty.\n\n3. The hyperparameter analysis reveals that the model's performance is highly sensitive to the personalization strength parameter $w$. Although the authors frame this positively by noting that $w$ can be tuned during inference without retraining, high sensitivity can still be a practical challenge, potentially requiring careful and costly tuning for each specific application or dataset to achieve optimal results.", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n> Comment 1: model’s scalability\n\nWe 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. \n\n**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\n\n|   Complexity   |               Model Parameters                |         Loss Computation          |    Modeling Target    |                       Inference                        |\n| :------------: | :-------------------------------------------: | :-------------------------------: | :-------------------: | :----------------------------------------------------: |\n|   **SASRec**   |    $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d\\bigr)$     | $\\mathcal{O}\\bigl((B+1)d\\bigr)$ |   $\\mathcal{O}(N)$    |      $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d+Nd\\bigr)$       |\n|  **DreamRec**  |  $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d\\bigr)$   |         $\\mathcal{O}(d)$          |   $\\mathcal{O}(Td)$   |   $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+Nd\\bigr)$    |\n| **PreferDiff** |  $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d\\bigr)$   | $\\mathcal{O}\\bigl((B+1)d\\bigr)$ |   $\\mathcal{O}(Td)$   |   $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+Nd\\bigr)$    |\n|  **DiffRec**   |             $\\mathcal{O}(LN^{2})$             |         $\\mathcal{O}(N)$          |   $\\mathcal{O}(TN)$   |                 $\\mathcal{O}(TLN^{2})$                 |\n|    **DDSR**    |   $\\mathcal{O}\\bigl(Ld^{2}+L^{2}d+Nd\\bigr)$   |         $\\mathcal{O}(N)$          |   $\\mathcal{O}(TN)$   |     $\\mathcal{O}\\bigl(TLd^{2}+TL^{2}d+TNd\\bigr)$     |\n| **PreferGrow** | $\\mathcal{O}\\bigl((L+3)d^{2}+L^{2}d+Nd\\bigr)$ |        $\\mathcal{O}(N+T)$         | $\\mathcal{O}(TN^{2})$ | $\\mathcal{O}\\bigl(T(L+3)d^{2}+TL^{2}d+TN(d+3)\\bigr)$ |\n| **PreferGrow on SIDs** | $\\mathcal{O}(m(L{+}3)d^2 {+} mL^2d {+} mcd)$ | $\\mathcal{O}(mc+T)$ | $\\mathcal{O}(Tmc^2)$ | $\\mathcal{O}(Tm(L{+}3)d^2 {+} TmL^2d {+} Tmc(d{+}3))$ |\n\n* **Model Parameters:** We never materialise the full $N^{2}$ preference‑ratio matrix; Equation (14) only requires $N$ scores, so the extra cost is $\\mathcal O(Nd)$. \n* **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, $T=20$).\n* **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 $\\mathcal O(N)$.  \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.\n* **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 $c$. \n\nTo 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.\n\n**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 $\\mathcal{O}(mc)$ :\n\nHowever, 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. \n\nWe 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.\n\n> Comment 2: training efficiency\n\n**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.\n\n|  Dataset  |  25%   |    50%     |  60%   |  70%   |  80%   |  90%   |    100%    |\n| :-------: | :----: | :--------: | :----: | :----: | :----: | :----: | :--------: |\n| MovieLens | 0.0791 |   0.0881   | 0.0895 | 0.0905 | 0.0907 | 0.0906 | **0.0911** |\n|   Steam   | 0.0349 |   0.0389   | 0.0392 | 0.0394 | 0.0397 | 0.0400 | **0.0406** |\n|  Beauty   | 0.0301 |   0.0306   | 0.0301 | 0.0306 | 0.0312 | 0.0321 | **0.0322** |\n|   Toys    | 0.0323 | **0.0331** | 0.0316 | 0.0305 | 0.0314 | 0.0303 |   0.0311   |\n|  Sports   | 0.0124 |   0.0122   | 0.0135 | 0.0134 | 0.0133 | 0.0133 | **0.0142** |\n\n**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.\n\n> Comment 3: sensitivity to the personalization strength parameter $w$\n\n**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.\n\n|        Dataset         | MovieLens | Steam | Beauty | Sports | Toys  |\n| :--------------------: | :-------: | :---: | :----: | :----: | :---: |\n| Mean abs. error of opt. $w$ |   0.000   | 0.000 | 0.778  | 0.050  | 0.775 |\n\n> Comment 4: Do you have depolyed your method into real world industrial ?\n\nWe 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.\n\nThank 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. \n\n**References**\n\n[1] Rajput, Shashank, et al. \"Recommender systems with generative retrieval.\" NeurIPS 2023.\n\n[2] Xie, Wenjia, et al. \"Breaking determinism: Fuzzy modeling of sequential recommendation using discrete state space diffusion model.\" NeurIPS 2024.\n\n[3] Hou, Yupeng, et al. \"ActionPiece: Contextually Tokenizing Action Sequences for Generative Recommendation.\" ICML 2025.\n\n[4] Deng, Jiaxin, et al. \"Onerec: Unifying retrieve and rank with generative recommender and iterative preference alignment.\" ArXiv 2025."}]

Author rebuttal:

Meta-review: This paper proposes PreferGrow, a discrete diffusion-based recommender that models preference ratios by fading and growing user preferences over the discrete items. PreferGrow possesses three distinct aspects compared to existing diffusion-based recommenders: 1) Discrete modeling of preference ratios, 2) Perturbing via preference fading, and 3) Preference reconstruction via growing. The authors showed the performance gains of PreferGrow compared to SOTA diffusion-based recommenders across five benchmark datasets.

Strengths:
+ Discrete modeling of preference ratios aligns well with the ranking-oriented nature of recommender systems.
+ A flexible architecture that supports  point-wise, pair-wise, and hybrid preference modeling
+ Thorough experiments on five datasets showing consistent performance compared to SOTA diffusion-based recommenders

Weaknesses:
- The authors addressed major concerns from the reviewers during rebuttal
- Minor limitations in terms of longer training time, as mentioned by the authors.

Overall, PreferGrow is a novel framework with an original core idea, solid theoretical analysis, and thorough experiments. I support this paper to be accepted.
Decision: Accept (poster)