1. Bibliographic Information

1.1. Title

DiffuRec: A Diffusion Model for Sequential Recommendation

1.2. Authors

ZIHAO LI and CHENLIANG LI (Key Laboratory of Aerospace Information Security and Trusted Computing, Ministry of Education, School of Cyber Science and Engineering, Wuhan University, China), AIXIN SUN (Nanyang Technological University, Singapore).

1.3. Journal/Conference

ACM Transactions on Information Systems (ACM Trans. Inf. Syst. 37, 4, Article 111). This is a highly reputable journal in the field of information systems and computer science, known for publishing high-quality research in areas like information retrieval, recommender systems, and data mining.

1.4. Publication Year

2023

1.5. Abstract

Mainstream solutions for Sequential Recommendation (SR) represent items using fixed vectors, which have limited capability in capturing items' latent aspects and users' diverse preferences. This paper proposes DiffuRec, a novel approach that adapts the Diffusion model to SR for the first time. DiffuRec addresses the limitations of fixed vectors by representing item embeddings as distributions, reflecting multiple user interests and item aspects adaptively. In the diffusion phase, the target item embedding is corrupted into a Gaussian distribution through noise addition, which then guides the generation of sequential item distribution representations and injects uncertainty. An Approximator (a Transformer model) is used to reconstruct the target item representation. In the reverse phase, based on historical interactions, a Gaussian noise is iteratively denoise into the target item representation, followed by a rounding operation for target item prediction. Experimental results on four datasets demonstrate that DiffuRec significantly outperforms strong baselines.

1.6. Original Source Link

• Original Source: https://arxiv.org/abs/2304.00686 (Preprint)
• PDF Link: https://arxiv.org/pdf/2304.00686v4.pdf (Preprint, version 4)
• Publication Status: The paper is officially published in ACM Trans. Inf. Syst., as indicated by the ACM Reference Format in the paper: "ACM Trans. Inf. Syst. 37, 4, Article 111 (August 2023)". The arXiv link is likely a preprint version that was updated prior to formal publication.

2. Executive Summary

2.1. Background & Motivation

The core problem the paper aims to solve is the inherent limitation of fixed vector representations for items in Sequential Recommendation (SR). Traditional SR models, including Transformer-based and GNN-based approaches, typically represent items as static embedding vectors.

This problem is important because:

• Multiple Latent Aspects: Real-world items often possess multiple intrinsic aspects (e.g., a movie having themes of 'Love' and 'War'). A single fixed vector struggles to encode this multi-faceted nature comprehensively, especially when these aspects are difficult to pre-define or annotate.

• Multiple User Interests: User preferences are dynamic, diverse, and context-dependent. A user's interest in an item's aspects can shift over time or based on their current context. A static item representation across all users and contexts is insufficient to capture these diverse and evolving interests.

• Uncertainty: User intentions and preferences are inherently uncertain. Current recommender systems are also expected to offer diversity, novelty, and serendipity, which often involves modeling this uncertainty. Modeling user interests as a fixed point in an embedding space does not account for this uncertainty.

• Guidance of Target Item: The target item (the next item a user will interact with) contains crucial information about the user's current intention. Incorporating this signal effectively during representation learning can significantly improve recommendation quality, but it's computationally challenging and often only applicable at the ranking stage.

  Prior research has attempted to address some of these issues through attention mechanisms, dynamic routing for multi-interest modeling, or Variational Auto-Encoders (VAEs) for distribution representation and uncertainty injection. However, multi-interest models often require heuristically pre-defined numbers of interests, limiting flexibility, while VAEs suffer from representation capacity issues and posterior distribution approximation collapse.

The paper's entry point or innovative idea is to leverage the unique merit of Diffusion models – their strong capability in distribution generation and diversity representation – and adapt them for the first time to the Sequential Recommendation problem. The authors posit that Diffusion models are a natural fit for modeling item representations as distributions, thereby addressing the limitations of fixed vectors and existing probabilistic models in a unified framework.

2.2. Main Contributions / Findings

The paper makes several primary contributions:

• First Application of Diffusion Models to SR: It presents the first successful attempt to adapt Diffusion models for Sequential Recommendation, bridging a significant gap between Diffusion models (predominantly used in CV and NLP) and recommender systems.

• Distributional Item Representation: Instead of fixed vectors, DiffuRec models items and user preferences as distributions. This allows for adaptive representation of items' multi-latent aspects and users' diverse intentions, inherently capturing more complex information.

• Target Item Guidance and Uncertainty Injection: The proposed diffusion process inherently fuses the target item embedding for historical interacted item representation generation. This introduces the user's current preference as an auxiliary signal, while also injecting uncertainty through noise, leading to more robust and discriminative item embedding learning.

• Novel Model Architecture: DiffuRec introduces a Transformer-based Approximator for reconstructing target item representations during training and iteratively denoising during inference. It also devises a rounding operation to map the continuous reversed representation back to discrete item indices for prediction.

• Superior Performance: Extensive experiments on four real-world datasets demonstrate that DiffuRec consistently and significantly outperforms nine strong baselines across various metrics, showcasing its effectiveness.

• Empirical Analysis: The paper provides in-depth ablation studies, hyperparameter analysis, comparisons with adversarial training, performance analysis on sequences of varying lengths and items of different popularity, and visualization of uncertainty and diversity, elucidating the efficacy of its components and approach.

  The key finding is that Diffusion models are highly effective for Sequential Recommendation by enabling the modeling of item representations and user preferences as distributions, injecting beneficial uncertainty, and leveraging target item guidance. This approach leads to substantial performance gains and offers greater flexibility and diversity in recommendations.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

To fully understand DiffuRec, a novice reader should be familiar with the following concepts:

3.1.1. Sequential Recommendation (SR)

Sequential Recommendation is a task in recommender systems where the goal is to predict the next item a user will interact with, given their historical sequence of interactions. It focuses on modeling the user's interest evolution over time.

• Example: If a user watched movies A, B, and C in that order, an SR model would predict movie D as their next likely watch.

3.1.2. Item Embeddings

In recommender systems and natural language processing (NLP), items (like movies, products, words) are often represented as dense, low-dimensional vectors called embeddings. These vectors capture the semantic and latent features of the items, allowing for mathematical operations (like dot products or cosine similarity) to measure their relatedness.

• Example: Movies with similar genres or actors would have embedding vectors that are close to each other in the embedding space.

3.1.3. Gaussian Distribution

A Gaussian distribution, also known as a normal distribution or bell curve, is a common probability distribution that is symmetric around its mean. It's defined by two parameters: its mean ($\mu$) and variance ($\sigma^2$).

• Formula: The probability density function of a univariate Gaussian distribution is: $ f(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $ Where:
  • $x$: the random variable
  • $\mu$: the mean (average value)
  • $\sigma^2$: the variance (spread of the distribution)
  • $\pi$: mathematical constant (approximately 3.14159)
  • $e$: Euler's number (approximately 2.71828)
• Importance: Gaussian noise is often used in machine learning for adding randomness or modeling uncertainty, particularly in diffusion models and VAEs.

3.1.4. Diffusion Models (Denoising Diffusion Probabilistic Models - DDPM)

Diffusion models are a class of generative models that learn to generate data by reversing a diffusion process. This process gradually adds Gaussian noise to data until it becomes pure noise, then the model learns to reverse this process, step-by-step, to reconstruct the original data from noise.

• Forward (Diffusion) Process: This is a fixed Markov chain that gradually adds Gaussian noise to an input data point $\mathbf{x}_0$ over $T$ time steps. $ q(\mathbf{x}t | \mathbf{x}{t-1}) = \mathcal{N}(\mathbf{x}t; \sqrt{1 - \beta_t}\mathbf{x}{t-1}, \beta_t\mathbf{I}) $ Where:
  • $\mathbf{x}_0$: the original data point.
  • $\mathbf{x}_t$: the noisy data point at time step $t$.
  • $\beta_t$: the variance schedule (a small, increasing value that controls the amount of noise added at each step).
  • $\mathbf{I}$: identity matrix.
  • $\mathcal{N}(\cdot; \mu, \Sigma)$: Gaussian distribution with mean $\mu$ and covariance $\Sigma$. A key property is that $\mathbf{x}_t$ can be sampled directly from $\mathbf{x}_0$ at any step $t$: $ q(\mathbf{x}_t | \mathbf{x}_0) = \mathcal{N}(\mathbf{x}_t; \sqrt{\bar{\alpha}_t}\mathbf{x}_0, (1 - \bar{\alpha}_t)\mathbf{I}) $ Where $\alpha_t = 1 - \beta_t$ and $\bar{\alpha}_t = \prod_{s=1}^t \alpha_s$.
• Reverse (Denoising) Process: This is a learned Markov chain that starts from pure Gaussian noise $\mathbf{x}_T \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ and iteratively removes noise to generate a sample from the data distribution. The goal is to learn $p_\theta(\mathbf{x}_{t-1} | \mathbf{x}_t)$. $ p_\theta(\mathbf{x}{t-1} | \mathbf{x}t) = \mathcal{N}(\mathbf{x}{t-1}; \mu\theta(\mathbf{x}t, t), \Sigma\theta(\mathbf{x}_t, t)) $ A neural network (often called an Approximator or denoiser) is trained to predict the noise $\epsilon$ added at each step, or directly predict $\mathbf{x}_0$ from $\mathbf{x}_t$.

3.1.5. Transformer Architecture

The Transformer is a neural network architecture introduced in "Attention Is All You Need" (Vaswani et al., 2017). It revolutionized sequence modeling tasks, particularly in NLP, by solely relying on self-attention mechanisms rather than recurrent neural networks (RNNs) or convolutional neural networks (CNNs).

• Self-Attention: A mechanism that allows a model to weigh the importance of different parts of the input sequence when processing each element. It computes attention scores between an element and all other elements in the sequence. $ \mathrm{Attention}(Q, K, V) = \mathrm{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V $ Where:
  • $Q$: Query matrix (derived from the input embedding).
  • $K$: Key matrix (derived from the input embedding).
  • $V$: Value matrix (derived from the input embedding).
  • $d_k$: dimension of keys.
  • $\mathrm{softmax}$: normalization function.
• Multi-Head Attention: Multiple self-attention layers are run in parallel, allowing the model to focus on different parts of the input sequence simultaneously and capture diverse relationships.
• Positional Encoding: Since Transformers do not have inherent sequential processing like RNNs, positional encodings are added to the input embeddings to provide information about the relative or absolute position of tokens in the sequence.
• Feed-Forward Networks: After attention, each position's output is passed through a separate, identical feed-forward neural network.
• Layer Normalization & Residual Connections: Used to stabilize training and enable deeper models.

3.1.6. Cross-Entropy Loss

Cross-entropy loss is commonly used in classification problems to measure the difference between two probability distributions: the true distribution (target labels) and the predicted distribution (model outputs). For next-item prediction, it often measures how well the model predicts the correct next item among all possible items.

• Formula (for a single sample in multi-class classification): $ L_{CE} = - \sum_{c=1}^{C} y_c \log(\hat{y}_c) $ Where:
  • $C$: number of classes (items).
  • $y_c$: a binary indicator (0 or 1) if class $c$ is the correct class (1 for the true target item, 0 otherwise).
  • $\hat{y}_c$: the predicted probability that sample belongs to class $c$.
• In recommender systems: The predicted probability $\hat{y}_c$ for an item $c$ is often derived from the softmax of the dot product (or similarity score) between the user's representation and the item's embedding.

3.1.7. Inner Product (Dot Product)

The inner product of two vectors is a scalar value that measures their similarity. In recommender systems, it's widely used to calculate the relevance or compatibility between a user's preference vector and an item's embedding vector. A higher inner product generally indicates higher predicted preference.

• Formula: For two vectors $\mathbf{a} = [a_1, a_2, \ldots, a_d]$ and $\mathbf{b} = [b_1, b_2, \ldots, b_d]$, their inner product is: $ \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^d a_i b_i $

3.1.8. t-SNE (t-Distributed Stochastic Neighbor Embedding)

t-SNE is a dimensionality reduction technique used for non-linear dimensionality reduction. It's particularly well-suited for visualizing high-dimensional data in a low-dimensional space (e.g., 2D or 3D) while preserving the local structure of the data points. Points that are close in the high-dimensional space tend to be close in the low-dimensional visualization.

3.2. Previous Works

The paper discusses previous work in three main categories:

3.2.1. Sequential Recommendation (SR)

• Early Approaches (Markov Chains): Markov Chains (e.g., [33, 38, 62]) modeled user interactions, assuming the next item depended only on the previous one.
  • Background: These models are simple but limited by the Markov assumption, which struggles with long-term dependencies.
• Deep Sequential Neural Networks:
  • GRU4Rec [19]: A pioneering work using Gated Recurrent Units (GRUs) to model sequential information, training with session-parallel mini-batches.
    • Background: GRUs are a type of Recurrent Neural Network (RNN) designed to capture dependencies in sequences, addressing the vanishing/exploding gradient problem of vanilla RNNs.
  • Caser [44]: Applied Convolutional Neural Networks (CNNs) to recent K adjacent items (treated as an "image") to capture sequential patterns.
    • Background: CNNs are effective at capturing local patterns and are typically used in image processing, but can be adapted for sequence data by treating subsequences as local windows.
  • SASRec [25]: A pioneering Transformer-based model using uni-directional masked self-attention to learn transition patterns.
    • Background: SASRec adapted the Transformer encoder for autoregressive sequence modeling, where masking prevents attending to future items.
  • BERT4Rec [42]: Adopted a bi-directional Transformer encoder and a cloze task (predicting masked items) to leverage both past and future context.
    • Background: Inspired by BERT in NLP, it uses a masked language model objective, allowing context from both directions to inform predictions.
• Graph Neural Networks (GNNs) for SR:
  • SR-GNN [51]: Introduced GNNs to model explicit relationships between items.
    • Background: GNNs are neural networks that operate on graph-structured data, enabling the capture of high-order relationships and information propagation between nodes (items).
  • Other GNN-based methods [8, 11, 14]: Incorporated time intervals, collaborative filtering signals, and different graph constructions (user-item, item-item transition graphs).

3.2.2. Multi-Interest Modeling

These methods acknowledge that users have multiple, dynamic interests.

• DIN (Deep Interest Network) [60]: Designed a local activation unit to adaptively learn user interest representations from historical sequences, allowing different parts of the sequence to contribute differently to predictions.
  • Background: Uses an attention mechanism to dynamically weigh historical items based on the candidate item, creating an interest representation specific to that candidate.
• MIND (Multi-Interest Network with Dynamic routing) [27]: Used a multi-interest extractor layer via capsule dynamic routing to extract multiple user interests.
  • Background: Capsule networks are designed to group neurons into "capsules" that represent different entities or properties, and dynamic routing allows these capsules to activate others.
• ComiRec [4]: Leveraged both dynamic routing and attention mechanisms for multi-interest modeling.
• TiMiRec [48]: A more recent multi-interest model that designs an auxiliary loss function incorporating the target item as a supervised signal for interest distribution generation.
  • Background: This is a key difference from ComiRec, explicitly guiding interest learning with the ground truth next item.
• Limitation: Many of these methods require a pre-defined number of interests (or capsules), which can be heuristic and constrain model flexibility.

3.2.3. Representation Distribution and Uncertainty Modeling

These approaches aim to model user preferences or item aspects as distributions rather than fixed points, often to inject uncertainty.

• Multi-VAE [29]: A pioneering work applying Variational Auto-Encoders (VAEs) to collaborative filtering.
  • Background: VAEs are generative models that learn a probabilistic latent space. They map input data to a distribution (typically Gaussian) in the latent space and then decode from that latent distribution back to data. They introduce uncertainty by sampling from these latent distributions.
• SVAE [36]: Combined recurrent neural networks with VAEs to capture temporal patterns for sequential recommendation.
• ACVAE [53]: Used an Adversarial Variational Bayes (AVB) framework to enhance the representation of latent variables and address posterior distribution approximation issues suffered by conventional VAEs.
  • Background: AVB uses an adversarial network to improve the quality of posterior approximation in VAEs, addressing issues like posterior collapse where the latent distribution becomes trivial.
• Distribution-based Transform [9]: Modeled item representations as Elliptical Gaussian distributions, using conventional item embeddings as means and introducing a stochastic vector for covariance representation and uncertainty injection. Used Wasserstein Distance as a loss function.
• STOSA (Stochastic Self-Attention) [10]: Replaced the inner product in the self-attention module with Wasserstein Distance to measure correlation between item distributions, injecting uncertainty via stochastic Gaussian distributions.
• Limitations: VAEs often struggle with representation degeneration and posterior collapse issues, limiting their effectiveness.

3.3. Technological Evolution

The evolution of Sequential Recommendation has mirrored general advancements in deep learning.

1. Early Statistical Models: Started with Markov Chains, simple but limited in capturing complex dependencies.
2. Recurrent Neural Networks (RNNs): GRU4Rec brought RNNs (specifically GRUs) to SR, enabling better capture of sequential patterns.
3. Convolutional Neural Networks (CNNs): Caser showed CNNs could capture local sequential patterns.
4. Attention Mechanisms and Transformers: SASRec and BERT4Rec adopted the Transformer architecture, demonstrating the power of self-attention for long-range dependencies and complex pattern learning, becoming state-of-the-art.
5. Graph Neural Networks (GNNs): SR-GNN and others leveraged GNNs to model high-order relationships and item transitions more explicitly.
6. Multi-Interest and Distributional Modeling: Recognizing the dynamic and diverse nature of user preferences, DIN, MIND, ComiRec, TiMiRec, and VAE-based methods (like Multi-VAE, ACVAE, STOSA) emerged to model multi-interests and uncertainty with probabilistic representations.
7. Diffusion Models (DiffuRec): DiffuRec marks the latest evolution by introducing Diffusion models from generative AI to SR, aiming to overcome the limitations of previous distributional models and fixed-vector representations by leveraging diffusion models' inherent strengths in distribution generation and diversity representation.

3.4. Differentiation Analysis

DiffuRec differentiates itself from previous methods primarily in its approach to item representation and uncertainty injection:

• Fixed Vectors vs. Distributions: Unlike mainstream SR methods (GRU4Rec, Caser, SASRec, BERT4Rec, GNN-based) that represent items as fixed vectors, DiffuRec models them as distributions. This allows it to adaptively capture multiple latent aspects of items and diverse user interests, which fixed vectors inherently struggle with.
• Superiority over VAEs: While VAE-based methods (Multi-VAE, SVAE, ACVAE) also model latent variables as distributions and inject uncertainty, they often suffer from representation degeneration and posterior collapse. DiffuRec leverages the more robust distribution generation capabilities of Diffusion models, which are known for producing high-quality samples and avoiding these VAE pitfalls. STOSA is closer as it also uses Gaussian distributions for items, but DiffuRec integrates diffusion over the target item directly.
• Enhanced Multi-Interest Modeling: Compared to multi-interest models (DIN, MIND, ComiRec, TiMiRec), DiffuRec doesn't require heuristically pre-defining a fixed number of interests. Instead, it implicitly models multi-latent aspects and diverse intentions through the distributional representations and the stochasticity of the diffusion process. TiMiRec uses the target item as a supervised signal, similar to DiffuRec's target item guidance, but DiffuRec's mechanism is integrated within a more powerful generative framework.
• Unique Uncertainty Injection and Target Item Guidance: DiffuRec's diffusion process not only injects uncertainty but also directly incorporates the target item embedding's information (even when noised) to guide the representation generation of historical items. This uncertainty is not just pure noise but target-item-aware, providing a more discriminative and robust learning signal. This is a novel way to fuse information compared to explicit adversarial perturbations or VAE sampling.
• Continuous to Discrete Mapping: DiffuRec introduces a rounding operation to map the reversed continuous representation back to discrete item indices, which is a practical consideration for recommendation tasks where the output must be a specific item from a catalog.

4. Methodology

4.1. Principles

The core idea of DiffuRec is to move beyond fixed vector representations for items in Sequential Recommendation (SR) and instead model them as distributions. This approach is motivated by the limitations of fixed vectors in capturing multiple latent aspects of items and diverse user interests, as well as the inherent uncertainty in user preferences. The theoretical basis lies in Diffusion Models, a class of generative models capable of generating diverse data samples by progressively denoising Gaussian noise.

The intuition is that if an item's representation is a distribution rather than a single point, it can inherently encompass various latent aspects. Similarly, a user's evolving preferences can be reflected by how these item distributions are interactively adjusted. The diffusion process provides a robust mechanism to:

1. Generate distributional representations: By gradually adding noise to a target item, DiffuRec learns to represent it as a Gaussian distribution, and this process is then leveraged to generate distributional representations for historical items.
2. Inject uncertainty: The stochastic nature of the diffusion and reverse processes naturally introduces uncertainty into the model, making it more robust and capable of generating diverse recommendations.
3. Incorporate target item guidance: The diffusion process is designed such that the noised target item embedding directly influences the representations of historical items, acting as an auxiliary signal to guide the model towards understanding the user's current intention.

4.2. Core Methodology In-depth (Layer by Layer)

DiffuRec is built upon the Diffusion Model framework, consisting of a diffusion (training) phase and a reverse (inference) phase, orchestrated by a Transformer-based Approximator.

4.2.1. Overview of DiffuRec

The overall architecture of DiffuRec is visualized in the figure below:

该图像是示意图，展示了DiffuRec模型在顺序推荐中的双阶段流程，包括扩散（训练）阶段和反向（推理）阶段。模型通过对目标项嵌入进行高斯噪声处理，生成分布表示，以适应用户的多样兴趣和项目的各方面特性。公式 $q(x_{s-1}|x_s)$ 表示在扩散过程中推断目标项表示。

Figure 2. Architecture of DiffuRec. The figure on the left is the Approximator, a Transformer backbone for the training and reverse phases, respectively.

At a high level, the process is as follows:

1. Item Embeddings: Each item $i_j$ is initially represented by a static embedding vector $\mathbf{e}_j$.
2. Diffusion (Training) Process:
   • The embedding of the target item $i_{n+1}$ (denoted as $\mathbf{e}_{n+1}$) is chosen as the starting point $\mathbf{x}_0$.
   • Gaussian noise is gradually added to $\mathbf{x}_0$ over $s$ steps to create a noised target item representation $\mathbf{x}_s$. This $\mathbf{x}_s$ is treated as a distributional representation sampled from $q(\cdot)$.
   • The historical item embeddings from the user's sequence $S = [i_1^u, i_2^u, \ldots, i_n^u]$ are then adjusted using this noised target item representation $\mathbf{x}_s$ and a step embedding $\mathbf{d}_s$. This adjustment creates a set of distributional representations $\mathbf{Z}_{x_s} = [\mathbf{z}_1, \mathbf{z}_2, \ldots, \mathbf{z}_n]$ for the historical items. This step is crucial for exploiting the guidance of the target item.
   • These adjusted historical item representations $\mathbf{Z}_{x_s}$ are fed into a Transformer-based Approximator $f_\theta(\cdot)$.
   • The Approximator is trained to reconstruct the original target item representation $\hat{\mathbf{x}}_0$ from $\mathbf{Z}_{x_s}$. The cross-entropy loss is used to minimize the difference between $\hat{\mathbf{x}}_0$ and the actual target item embedding $\mathbf{e}_{n+1}$.
3. Reverse (Inference) Process:
   • Starts with a pure Gaussian noise $\mathbf{x}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ (where $t$ is the maximum diffusion step), representing an uncertain initial state for the target item.
   • The model iteratively denoises $\mathbf{x}_t$ back to $\mathbf{x}_0$. In each step, the current noisy representation $\mathbf{x}_s$ is used to adjust the historical item representations $\mathbf{Z}_{x_s}$, which are then fed into the well-trained Approximator $f_\theta(\cdot)$ to estimate $\hat{\mathbf{x}}_0$.
   • This estimated $\hat{\mathbf{x}}_0$ and the current $\mathbf{x}_s$ are used to calculate the next less noisy representation $\mathbf{x}_{s-1}$. This process continues until $\mathbf{x}_0$ is reached.
   • A rounding function is then applied to map the continuous $\mathbf{x}_0$ into discrete item indices for final prediction.

4.2.2. Approximator

The Approximator $f_\theta(\cdot)$ is the core neural network responsible for reconstructing the target item representation. Following practices in NLP Diffusion models, DiffuRec utilizes a Transformer as the backbone for the Approximator, given its proven effectiveness in sequential dependency modeling.

The Approximator takes as input a sequence of adjusted historical item representations and outputs a reconstructed target item representation $\hat{\mathbf{x}}_0$. The input sequence to the Transformer consists of adjusted representations $\mathbf{z}_i$: $ \hat{\mathbf{x}}0 = f\theta(\mathbf{Z}_x) = \mathrm{Transformer}\left(\left[\mathbf{z}_1, \mathbf{z}_2, \ldots, \mathbf{z}_n\right]\right) $ Each adjusted historical item representation $\mathbf{z}_i$ is constructed as follows: $ \mathbf{z}_i = \mathbf{e}_i + \lambda_i \odot \left(\mathbf{x} + \mathbf{d}\right) $ Where:

• $\mathbf{e}_i$: The static embedding vector of the $i$-th historical item in the sequence.

• $\lambda_i$: A scalar sampled from a Gaussian distribution, $\lambda_i \sim \mathcal{N}(\delta, \delta)$, where $\delta$ is a hyperparameter defining both the mean and variance. This $\lambda_i$ controls the extent of noise injection for each historical item in the diffusion phase, and introduces stochasticity to model user interest evolution in the reverse phase.

• $\odot$: The element-wise product (Hadamard product).

• $\mathbf{x}$: This is the noised target item embedding $\mathbf{x}_s$ in the diffusion phase. In the reverse phase, it is the current reversed target item representation $\mathbf{x}_s$ (iterating from $t$ down to 1).

• $\mathbf{d}$: The embedding of the corresponding diffusion/reverse step, created following the standard Transformer positional encoding approach [47]. This step embedding injects information about the current diffusion or reverse step into the model.

  The Transformer itself maintains its standard architecture, including multi-head self-attention, feed-forward networks with ReLU activation, layer normalization, dropout, and residual connections. Multiple Transformer blocks are stacked to enhance representation ability. Finally, the representation $\mathbf{h}_n$ corresponding to the last item $i_n^u$ in the sequence (after passing through the Transformer layers) is used as the reconstructed target item representation $\hat{\mathbf{x}}_0$.

4.2.3. Diffusion Phase

The diffusion phase (or training phase) is where the model learns to reconstruct the original target item embedding from its noisy version.

The key component here is the noise schedule $\beta_s$, which dictates how much Gaussian noise is added at each step $s$. DiffuRec utilizes a truncated linear schedule for $\beta_s$ generation. The formula for the truncated linear schedule is: $ \beta_s = \begin{cases} \frac{a}{t}s + \frac{b}{s}, & \text{if } \beta_s \leq \tau \ \frac{1}{10}(\frac{a}{t}s + \frac{b}{s}), & \text{otherwise} \end{cases} $ Where:

• a, b: Hyperparameters controlling the range of $\beta$.

• $t$: Maximum diffusion step.

• $s$: Current diffusion step.

• $\tau$: A truncated threshold, set to 1 in the paper. If the calculated $\beta_s$ exceeds $\tau$, it is scaled down by a factor of 10.

  During training, for each target item, a diffusion step $s$ is randomly sampled from a uniform distribution over the range [1, t], i.e., s = \lfloor s'\rfloor, s' \sim U(0, t).

The noised target item representation $\mathbf{x}_s$ is then generated from the original target item embedding $\mathbf{e}_{n+1}$ (which is initially treated as $\mathbf{x}_0$) following the Markov chain property of diffusion models. Specifically, DiffuRec first derives $\mathbf{x}_0$ from the target item embedding $\mathbf{e}_{n+1}$ with one step of diffusion: $ q(\mathbf{x}0 | \mathbf{e}{n+1}) = \mathcal{N}(\mathbf{x}0; \sqrt{\alpha_0}\mathbf{e}{n+1}, (1 - \alpha_0)\mathbf{I}) $ Then, the noised target item representation $\mathbf{x}_s$ is generated from this $\mathbf{x}_0$ using the reparameterization trick: $ \mathbf{x}_s = \sqrt{\bar{\alpha}_s}\mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_s}\epsilon $ Where:

• $\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$: A random noise vector sampled from a standard Gaussian distribution.

• $\alpha_s = 1 - \beta_s$.

• $\bar{\alpha}_s = \prod_{k=1}^s \alpha_k$.

  The full diffusion (training) process is outlined in Algorithm 2:

Algorithm 2: Diffusion or Training 1: Input: 2: Historical Sequence: $(i_1, \ldots, i_n, i_{n+1})$; 3: Learning Epochs: $E$; 4: Maximum Diffusion Steps: $t$; 5: Schedule $\beta$: Linear; 6: Hyperparameter $\delta$ (mean and variance): $\lambda$ sampling; 7: Approximator: $f_\theta(\cdot)$; 8: Output: 9: Well-trained Approximator: $f_\theta(\cdot)$; 10: while $j < E$ do 11: s = \lfloor s'\rfloor, s' \sim U(0, t); // Diffusion Step Sampling 12: $\mathbf{x}_s \gets q(\mathbf{x}_s | \mathbf{x}_0)$; // Diffusion (where $\mathbf{x}_0$ is derived from $\mathbf{e}_{n+1}$ as above) 13: $(\mathbf{z}_1, \ldots, \mathbf{z}_n) \gets (\mathbf{e}_1 + \lambda_1(\mathbf{x}_s + \mathbf{d}_s), \ldots, \mathbf{e}_n + \lambda_n(\mathbf{x}_s + \mathbf{d}_s)), \lambda_i \sim \mathcal{N}(\delta, \delta)$; // Distribution Representation 14: $\hat{\mathbf{x}}_0 \gets f_\theta([\mathbf{z}_1, \ldots, \mathbf{z}_n])$; // Reconstruction 15: parameter update: $\mathcal{L}_{CE}(\hat{\mathbf{x}}_0, i_{n+1})$; 16: $j = j + 1$ 17: end while

4.2.4. Reverse Phase

The reverse phase (or inference phase) is where the trained DiffuRec model is used to predict the target item. It starts with a pure Gaussian noise and iteratively denoises it into the target item representation. The goal is to recover $\mathbf{x}_0$ iteratively from a standard Gaussian noise $\mathbf{x}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$.

At each reverse step, the well-trained Approximator $f_\theta(\cdot)$ is used to estimate $\hat{\mathbf{x}}_0$ from the current noisy representation $\mathbf{x}_s$ and the historical sequence. Then, this estimated $\hat{\mathbf{x}}_0$ is used to calculate the mean and variance for sampling $\mathbf{x}_{s-1}$. The mean $\tilde{\mu}_t(\mathbf{x}_t, \hat{\mathbf{x}}_0)$ and variance $\tilde{\beta}_t$ for sampling $\mathbf{x}_{t-1}$ are given by: $ \tilde{\mu}_t(\mathbf{x}_t, \hat{\mathbf{x}}0) = \frac{\sqrt{\bar{\alpha}{t-1}}\beta_t}{1 - \bar{\alpha}_t}\hat{\mathbf{x}}0 + \frac{\sqrt{\alpha_t}(1 - \bar{\alpha}{t-1})}{1 - \bar{\alpha}_t}\mathbf{x}_t $ $ \tilde{\beta}t = \frac{1 - \bar{\alpha}{t-1}}{1 - \bar{\alpha}t}\beta_t $ Using the reparameterization trick, $\mathbf{x}_{t-1}$ is sampled as: $ \mathbf{x}{t-1} = \tilde{\mu}_t(\mathbf{x}_t, \hat{\mathbf{x}}_0) + \tilde{\beta}_t \epsilon' $ Where $\epsilon' \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ is another standard Gaussian noise vector.

This process is repeated iteratively from $s=t$ down to $s=1$ until the final denoised representation $\mathbf{x}_0$ is obtained.

The full reverse (inference) process is outlined in Algorithm 1:

Algorithm 1: Reverse or Inference 1: Input: 2: Historical Sequence: $(i_1, \ldots, i_n)$; 3: Target item Representation $\mathbf{x}_t$: $\mathbf{x}_t \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$; 4: Total Reverse Steps: $t$; 5: Schedule $\beta$: Linear; 6: Hyperparameter $\delta$ (mean and variance): $\lambda$ sampling; 7: Approximator: $f_\theta(\mathbf{x})$; 8: Output: 9: Predicted Target Item: $i_{n+1}$; 10: $s = t$ 11: while $s > 0$ do 12: $(\mathbf{z}_1, \ldots, \mathbf{z}_n) \gets (\mathbf{e}_1 + \lambda_1(\mathbf{x}_s + \mathbf{d}_s), \ldots, \mathbf{e}_n + \lambda_n(\mathbf{x}_s + \mathbf{d}_s)), \lambda_i \sim \mathcal{N}(\delta, \delta)$; // Distribution Representation 13: $\hat{\mathbf{x}}_0 \gets f_\theta([\mathbf{z}_1, \ldots, \mathbf{z}_n])$; // Reconstruction 14: $\mathbf{x}_{s-1} \gets \tilde{\mu}(\hat{\mathbf{x}}_0, \mathbf{x}_s) + \tilde{\beta}_t * \epsilon' (\epsilon' \sim \mathcal{N}(\mathbf{0}, \mathbf{I}))$; // Reverse 15: $s = s - 1$; // Iteration 16: end while 17: $i_{n+1} \gets \mathrm{Rounding}(\mathbf{x}_0)$

4.2.5. Loss Function and Rounding

Loss Function: Unlike typical Diffusion models that often use mean-squared error (MSE) (e.g., to predict the noise $\epsilon$), DiffuRec employs cross-entropy loss for model optimization. This choice is motivated by two reasons:

1. Discrete Nature of Item Embeddings: Item embeddings in recommender systems are often static and represent discrete entities in a continuous space, making MSE less stable. Cross-entropy is more appropriate for predicting one out of many discrete items.

2. Relevance Calculation: In sequential recommendation, inner product is a common way to calculate relevance between item representations, which aligns well with cross-entropy loss used for ranking.

   The cross-entropy loss $\mathcal{L}_{CE}$ is calculated as follows: First, a softmax function is applied to the inner product between the reconstructed target item representation $\hat{\mathbf{x}}_0$ and all candidate item embeddings $\mathbf{e}_i$ to obtain predicted probabilities: $ \hat{y} = \frac{\exp\left(\hat{\mathbf{x}}0 \cdot \mathbf{e}{n+1}\right)}{\sum_{i \in J} \exp\left(\hat{\mathbf{x}}0 \cdot \mathbf{e}i\right)} $ Then, the cross-entropy loss is computed over all users: $ \mathcal{L}{CE} = \frac{1}{|\mathcal{U}|} \sum{u \in \mathcal{U}} - \log \hat{y}_u $ Where:

• $\hat{y}$: The predicted probability for the true target item $i_{n+1}$.
• $\hat{\mathbf{x}}_0$: The reconstructed target item representation from the Approximator.
• $\mathbf{e}_{n+1}$: The embedding of the actual target item.
• $J$: The set of all candidate items (including the target item and negative samples).
• $\mathbf{e}_i$: The embedding of a candidate item $i$.
• $\cdot$: Inner product operation.
• $\mathcal{U}$: The set of all users.

Rounding Operation: After the reverse phase produces the continuous representation $\mathbf{x}_0$, a rounding operation is needed to map it to a discrete item index for final recommendation. This is achieved by calculating the inner product between $\mathbf{x}_0$ and all candidate item embeddings $\mathbf{e}_i$, and selecting the item with the maximal inner product score. $ \mathrm{Rounding}(\mathbf{x}_0) = \underset{i \in \mathcal{I}}{\mathrm{arg}\max} \quad \mathbf{x}_0 \cdot \mathbf{e}_i^T $ Where $\mathcal{I}$ is the set of all items. This inner product is also used to rank the candidate items for the final recommendation list.

4.2.6. Discussion on Uncertainty and Adversarial Training

DiffuRec inherently injects uncertainty through two main mechanisms:

1. Corrupted/Reversed Target Item Representation: The diffusion process (forward) adds Gaussian noise to the target item, and the reverse process (inference) starts from Gaussian noise. This means $\mathbf{x}_s$ and $\mathbf{x}_{s-1}$ are inherently stochastic, representing distributions rather than fixed points.

2. Gaussian Sampling for $\lambda_i$: The parameter $\lambda_i$ used to adjust historical item embeddings (Equation 13) is sampled from a Gaussian distribution, $\lambda_i \sim \mathcal{N}(\delta, \delta)$. This explicitly introduces stochasticity and uncertainty into the item distribution representation generation.

   The paper discusses that this injected uncertainty (small perturbations) can improve model robustness and alleviate over-fitting. To analyze this, DiffuRec is compared against adversarial training, which also aims to improve robustness by adding adversarial perturbations.

Adversarial Training Formulation: Adversarial training transforms the optimization into a min-max problem. First, effective perturbations $\Delta$ are found that maximize the loss, making the model vulnerable. Then, the model is trained to minimize both the original loss and the additional loss with these adversarial perturbations. The min-max optimization problem is formalized as: $ \Theta^, \Delta^ = \underset{\Theta}{\arg\min} \ \underset{\Delta, ||\Delta||\leq\epsilon}{\max} \mathcal{L}(\mathcal{D} | \Theta) + \gamma \mathcal{L}(\mathcal{D} | \Theta + \Delta) $ Where:

• $\Theta$: Learnable parameters of the model.

• $\Delta$: Perturbations added to embeddings.

• $\epsilon$: Regulates the magnitude of perturbations.

• $\gamma$: Balances the adversarial regularizer and the main loss.

• $||\cdot||$: $L_2$ norm.

• $\mathcal{L}(\mathcal{D} | \Theta)$: The original loss function (e.g., cross-entropy) on dataset $\mathcal{D}$.

• $\mathcal{L}(\mathcal{D} | \Theta + \Delta)$: The loss function with adversarial perturbations.

  Finding the exact $\Delta$ is intractable. Following [13], adversarial perturbations $\Delta_{adv}$ are approximated by moving in the gradient direction of the objective function. $ \Delta_{adv} = \epsilon \frac{\Gamma}{||\Gamma||} \quad \mathrm{where} \quad \Gamma = \frac{\partial \mathcal{L}(\mathcal{D} | \Theta + \Delta^t)}{\partial \Delta^t} $ Here, $\Delta^t$ are perturbations in the $t$-th training epoch, initialized as zeros. $\Delta^{t+1}$ is updated using $\Delta_{adv}$. The final loss function for adversarial training becomes: $ \mathcal{L}^*(\mathcal{D} | \Theta) = \mathcal{L}(\mathcal{D} | \Theta) + \gamma \mathcal{L}(\mathcal{D} | \Theta + \Delta_{adv}) $ The paper argues that DiffuRec's injected uncertainty is superior because it's not just a pure noise for robustness but also incorporates auxiliary information from the target item representation as a supervised signal to capture user intentions, which adversarial training typically lacks.

5. Experimental Setup

5.1. Datasets

The experiments were conducted on four real-world datasets widely used for sequential recommendation tasks.

• Amazon Beauty: A subset of the Amazon review dataset, focusing on beauty products.
• Amazon Toys: Another subset of the Amazon review dataset, focusing on toys.
• Movielens-1M: A benchmark dataset containing one million movie ratings from 6,000 users on 4,000 movies.
• Steam: A dataset collected from a large online video game distribution platform, containing interactions with over 40,000 games and rich external information.

Data Preprocessing:

• All reviews/ratings are treated as implicit feedback (user-item interaction).

• Interactions are chronologically ordered by timestamps.

• Unpopular items and inactive users (fewer than 5 interactions) are filtered out.

• A leave-one-out strategy is used for evaluation: the most recent interaction is for testing, the penultimate for validation, and earlier ones for training.

• Maximum sequence length is set to 200 for Movielens-1M and 50 for the other three datasets.

  The following are the results from Table 1 of the original paper:

  Dataset	# Sequence	# items	# Actions	Avg_len	Sparsity
  Beauty	22,363	12,101	198,502	8.53	99.93%
  Toys	19,412	11,924	167,597	8.63	99.93%
  Movielens-1M	6,040	3,416	999,611	165.50	95.16%
  Steam	281,428	13,044	3,485,022	12.40	99.90%

These datasets were chosen because they represent a broad spectrum of real-world scenarios, varying in size, sequence length, and sparsity, which allows for a comprehensive validation of the method's performance under different conditions.

5.2. Evaluation Metrics

All models are evaluated using Hit Rate (HR@K) and Normalized Discounted Cumulative Gain (NDCG@K). The experiments report results for $K = \{5, 10, 20\}$. Evaluation is performed by ranking all candidate items (no sampling of negative items).

5.2.1. Hit Rate (HR@K)

• Conceptual Definition: HR@K measures the proportion of users for whom the target item (the item they actually interacted with next) is present within the top-$K$ items recommended by the system. It indicates whether the relevant item was found within the candidate list, regardless of its position.
• Mathematical Formula: $ \mathrm{HR@K} = \frac{\text{Number of users with hit in top-K}}{\text{Total number of users}} $ A hit occurs if the true next item is in the top-$K$ recommended list.
• Symbol Explanation:
  • $\text{Number of users with hit in top-K}$: The count of users for whom the ground truth next item is found among the top $K$ recommended items.
  • $\text{Total number of users}$: The total number of users in the evaluation set.

5.2.2. Normalized Discounted Cumulative Gain (NDCG@K)

• Conceptual Definition: NDCG@K is a ranking quality metric that considers not only whether a relevant item is in the top-$K$ list but also its position. Higher positions for relevant items contribute more to the score. It's especially useful for scenarios where the order of recommended items matters.

• Mathematical Formula: First, Discounted Cumulative Gain (DCG@K) is calculated: $ \mathrm{DCG@K} = \sum_{j=1}^{K} \frac{2^{rel_j} - 1}{\log_2(j+1)} $ Then, NDCG@K is calculated by normalizing DCG@K by the Ideal DCG (IDCG@K): $ \mathrm{NDCG@K} = \frac{\mathrm{DCG@K}}{\mathrm{IDCG@K}} $ Where:

  • $\mathrm{IDCG@K}$ is the maximum possible DCG@K if all relevant items were perfectly ranked at the top.
  • For implicit feedback (next-item prediction where relevance is binary), $rel_j$ is 1 if the item at rank $j$ is the true next item, and 0 otherwise. Thus, the formula simplifies to $\frac{1}{\log_2(j+1)}$ if the item at rank $j$ is the target, and 0 otherwise.

• Symbol Explanation:

  • $K$: The number of top items in the recommendation list being considered.
  • $j$: The rank of an item in the recommendation list.
  • $rel_j$: The relevance score of the item at rank $j$. For binary relevance (hit/no hit), $rel_j=1$ if the item at rank $j$ is the target item, and $rel_j=0$ otherwise.
  • $\log_2(j+1)$: The logarithmic discount factor, which reduces the contribution of relevant items as their rank increases.
  • $\mathrm{DCG@K}$: The Discounted Cumulative Gain up to rank $K$.
  • $\mathrm{IDCG@K}$: The Ideal Discounted Cumulative Gain up to rank $K$, which is the DCG achieved by a perfect ranking where all relevant items are placed at the top.

5.3. Baselines

DiffuRec is compared against nine baselines from three categories:

5.3.1. Conventional Sequential Neural Models

• GRU4Rec [19]: Uses GRU for sequential modeling.
• Caser [44]: Employs horizontal and vertical CNNs to capture recent sub-sequence behaviors.
• SASRec [25]: A Transformer-based model with uni-directional masking for self-attention.
• BERT4Rec [42]: Uses a bi-directional Transformer and a cloze task for sequential recommendation.

5.3.2. Multi-Interest Models

• ComiRec [4]: Adopts attention mechanism and dynamic routing for multi-interest extraction.
• TiMiRec [48]: A recent multi-interest model that uses an auxiliary loss function with the target item as a supervised signal.

5.3.3. VAE and Uncertainty Models

• SVAE [36]: Combines GRU with Variational Autoencoder for next item prediction.
• ACVAE [53]: Uses the Adversarial Variational Bayes (AVB) framework to enhance latent variable representations.
• STOSA [10]: Develops stochastic self-attention with Wasserstein Distance and Gaussian distributions for uncertainty injection.

5.4. Implementation Details

• Optimizer: Adam optimizer with an initial learning rate of 0.001.
• Parameter Initialization: Transformer parameters initialized with Xavier normalization distribution.
• Model Architecture: Transformer block numbers set to 4.
• Dimensions: Both embedding dimension and all hidden state sizes fixed at 128.
• Batch Size: 1024.
• Dropout Rate: 0.1 for Transformer blocks, 0.3 for item embedding across all datasets.
• $\lambda$ Parameter: Sampled from a Gaussian distribution with both mean and variance $\delta = 0.001$.
• Total Reverse Steps ($t$): Set to 32. Further analysis is conducted for $t \in \{2, 4, 8, 16, 32, 64, 128, 256, 512, 1024\}$.
• Noise Schedule ($\beta$): Truncated linear schedule (Equation 14) is primarily used, with comparisons against sqrt, cosine, and linear schedules.
• Experiment Repetitions: Each method's experiments are run five times, and averaged results are reported.
• Statistical Significance: Student t-test is used for statistical significance.

6. Results & Analysis

6.1. Core Results Analysis (RQ1)

The following are the results from Table 2 of the original paper:

• DiffuRec consistently outperforms all baselines across all four datasets (Amazon Beauty, Amazon Toys, Movielens-1M, and Steam) and all six metrics ($HR@5/10/20$, $NDCG@5/10/20$). This strong performance confirms the effectiveness of modeling item representations as distributions and incorporating uncertainty via a diffusion model.
• The performance gains are substantial, with improvements up to 57.26% / 56.72% (HR/NDCG) on Amazon Beauty and 22.76% / 34.34% on Amazon Toys. On larger datasets like Movielens-1M and Steam, improvements are still significant, around 10-11%. This suggests DiffuRec's unified framework effectively addresses the challenges of multiple latent aspects, multiple user interests, uncertainty, and target item guidance.
• Conventional SR Models: GRU4Rec and Caser show the lowest performance, likely due to their limited ability to capture long-term dependencies compared to Transformer-based models. SASRec and BERT4Rec perform better, especially SASRec, highlighting the effectiveness of Transformer architecture for sequential recommendation.
• Multi-Interest Models: ComiRec and TiMiRec do not consistently outperform SASRec and BERT4Rec. TiMiRec performs better than ComiRec on Movielens-1M and Steam datasets, which are more complex and have longer sequences. This is attributed to TiMiRec's use of the target item as an auxiliary supervised signal, a concept that DiffuRec also leverages in a more integrated manner.
• VAE and Uncertainty Models: SVAE shows the weakest performance among this group. ACVAE and STOSA achieve competitive results. STOSA, in particular, demonstrates the value of distributional representation learning and uncertainty injection, aligning with DiffuRec's core principles and providing strong support for the direction of this research. DiffuRec’s superior performance over STOSA suggests that the diffusion framework provides a more robust and effective way to achieve these benefits compared to Wasserstein distance-based self-attention.

6.2. Ablation Study (RQ2)

The following are the results from Table 3 of the original paper:

6.2.1. Approximator Type (w GRU vs. DiffuRec)

• When the Transformer Approximator is replaced with GRU (w GRU), DiffuRec's performance drops, but it still achieves strong results on Movielens-1M and Steam. This suggests that the core diffusion mechanism is effective even with a simpler backbone, and Transformer enhances its capabilities. The observation that GRU can still yield good results with diffusion is consistent with CV where simpler architectures like U-Net can work well with Diffusion models.

6.2.2. Rounding Operation ($w R*$ vs. DiffuRec)

• Replacing the inner product-based rounding operation (Equation 19) with the variant used in [12] ($w R*$) leads to a drastic performance drop (e.g., HR/NDCG reducing by 68-95% across datasets). This strongly validates that using inner product for both loss calculation and final ranking is crucial and aligns with the typical practices in sequential recommendation. The alternative rounding function might not be suitable for this domain.

6.2.3. Noise Schedule (w Cosine, w L, w Sqrt vs. DiffuRec)

The performance impact of different noise schedules is visualized in the following figure:

该图像是一个图表，展示了DiffuRec在不同数据集（Amazon Beauty、Amazon Toys、Movielens-1M、Steam）上的推荐性能，包括HR@k和NDCG@k指标。图表中使用了不同的线型标识，包括余弦相似度（Cosine）、线性（Linear）、平方根（Sqrt）和截断线性（Truncated Linear）。

Figure 3. Performance of DiffuRec with different schedule.

The plot showing $\bar{\alpha}_t$ values for different noise schedules is presented below:

该图像是一个图表，展示了不同扩散调度下的 ar{ eta }_t值随扩散步骤的变化。图中包含线性（蓝色）、余弦（橙色）、平方根（绿色）和截断线性（红色）四种曲线，帮助理解在扩散过程中不同调度对 ar{ eta }_t 的影响。

Figure 4. $\bar{\alpha}_t$ throughout diffusion in different schedule.

• Figure 3 shows that the Truncated Linear schedule generally performs best on Amazon Beauty and Amazon Toys, but Linear schedule can sometimes achieve sub-optimal performance on other datasets.
• Figure 4 illustrates that the Truncated Linear schedule provides a "near-linear sharp drop in the middle of the process and subtle changes near the extremes of $t=T$". The authors hypothesize that this sharp corruption (adding more noise) in training is beneficial for better denoising ability, especially for one-step estimation in the reverse process.
• The ablation indicates that while noise schedules have some impact, they don't cause huge fluctuations, suggesting that the core diffusion framework's benefits are relatively robust to this choice, as noted in previous diffusion model literature [17].

6.3. Impact of Hyper-parameter Setting (RQ3)

The sensitivity of DiffuRec's performance to the hyperparameter $\delta$ (controlling the mean and variance of $\lambda$ sampling distribution) is presented below:

该图像是一个图表，展示了在两个数据集（Amazon Beauty 和 Movielens-1M）上，随着参数 λ 的不同取值（0.001、0.01、0.1 和 1）对 HR@K 和 NDCG@K 的影响。图中包含了 HR@5、HR@10、HR@20 和 NDCG@5、NDCG@10、NDCG@20 的条形图。

Figure 5. Parameter sensitivity of $\lambda$.

• Figure 5 demonstrates that DiffuRec is sensitive to the value of $\delta$. A small $\lambda$ value (e.g., $\delta = 0.001$) generally yields the best performance on both Amazon Beauty and Movielens-1M datasets.
• Increasing $\lambda$ to 0.1 or 1.0 leads to a sharp drop in HR and NDCG, particularly noticeable on Movielens-1M.
• Analysis: This suggests that a very large $\lambda$ introduces excessive noise to the historical interaction sequences. This excessive noise corrupts the original information, making it difficult for the model to precisely understand the user's interests and hindering effective representation generation. A smaller $\lambda$ allows for controlled stochasticity and uncertainty injection without overwhelming the meaningful signals.

6.4. Compared with Adversarial Training (RQ4)

The comparison between DiffuRec and adversarial training is shown in the following figure:

该图像是一个图表，展示了 adversarial training 和 DiffuRec 在四个数据集（Amazon Beauty、Amazon Toys、Movielens-1M 和 Steam）上的表现。图表中，横坐标为不同的指标（HR@5、HR@10、HR@20 和 NDCG@5、NDCG@10、NDCG@20），纵坐标为对应的评分，蓝色条表示 adversarial，橙色条表示 DiffuRec。

Figure 6. Performance of adversarial training and DiffuRec.

• Figure 6 shows that an adversarial Transformer (using BERT4Rec as backbone) performs better than the standard BERT4Rec baseline. Specifically, adversarial Transformer improves BERT4Rec by at least 2.57% / 10.23% (HR/NDCG) on Amazon Beauty and 8.05% / 4.89% on Movielens-1M. This validates that injecting perturbations through adversarial training can indeed enhance model robustness and performance.
• However, DiffuRec significantly outperforms the adversarial Transformer across all datasets.
• Analysis: The authors explain this superiority by arguing that DiffuRec's uncertainty injection is more sophisticated. The perturbations in DiffuRec (from the diffusion and reverse phases) are not merely pure noise for robustness. Instead, they are infused with auxiliary information from the target item representation, which acts as a supervised signal to help the model capture the user's current intentions. This target-item-aware uncertainty injection is a key advantage that differentiates DiffuRec from generic adversarial training methods.

6.5. Performance on Varying Lengths of Sequences and Items with Different Popularity (RQ5)

The distribution of sequence length and item interaction times for Amazon Beauty and Movielens-1M datasets is shown below:

该图像是条形图，展示了两个数据集（Amazon Beauty 和 Movielens-1M）中序列长度和项目互动次数的分布情况。图左侧以蓝色表示序列长度，图右侧以橙色表示项目互动次数，反映了不同数据集中用户行为的特点。

Figure 7. Frequency histogram of sequence length and item interaction times for Amazon Beauty and Movielens-1M dataset. The maximum length and maximum interactions are 204/431 and 2341/3428 respectively for Amazon Beauty and Movielens-1M datasets.

6.5.1. Head and Long-tail Items

The performance on head (top 20% most frequent) and long-tail items is shown in the following figure:

该图像是一个图表，展示了在 Amazon Beauty 和 Movielens-1M 数据集上，DiffuRec 与其他方法在头部项和长尾项推荐性能的比较。图中提供了各方法在 HR@10 和 NDCG@10 指标下的表现，DiffuRec 方法的效果表现在不同类型的项中均优于其他对比方法。

Figure 8. Performance on the head and long-tail items.

• Figure 8 reveals that for all models, performance on head items is significantly better than on long-tail items. This is a common challenge in recommender systems due to data sparsity for less popular items.
• The performance gap between head and long-tail items is smaller on Movielens-1M compared to Amazon Beauty. This might be because Movielens-1M has fewer total items, leading to a less pronounced distinction between head and long-tail in terms of data density.
• Despite this challenge, DiffuRec consistently outperforms all baselines for both head and long-tail items across HR@10 and NDCG@10 metrics, demonstrating its robust performance across different item popularity levels. Its superiority in NDCG implies better ranking of relevant items, even for less popular ones.

6.5.2. Different Sequence Length

The performance of different baselines on varied sequence lengths is depicted in the following figure:

该图像是图表，展示了在不同序列长度下，DiffuRec与其他推荐方法（如SASRec、TimiRec、STOSA）的性能比较。横轴表示序列长度，纵轴分别为HR@10和NDCG@10，能够直观地反映各模型在Amazon Beauty和Movielens-1M数据集上的效果。

Figure 9. Performance on different length of sequence.

• Figure 9 shows contrasting trends between Amazon Beauty and Movielens-1M datasets regarding sequence length:
  • On Amazon Beauty, DiffuRec performs better on longer sequences.
  • On Movielens-1M, performance generally declines as sequence length increases for all models, including DiffuRec.
• Analysis: The authors point out that 80% of sequences in Amazon Beauty are shorter than 10, while for Movielens-1M, 80% are longer than 37. This suggests that very short sequences might not provide enough information for accurate recommendations, while very long sequences pose a challenge for all models to capture relevant long-term dependencies and filter out noise.
• Regardless of these trends, DiffuRec consistently maintains the best performance across all sequence length categories when compared to other baselines. This indicates that DiffuRec is robust to varying sequence lengths, adapting well to both shorter and longer user interaction histories.

6.6. Convergence and Efficiency (RQ6)

6.6.1. Convergence

The loss curves on the training process for Amazon Beauty and Movielens-1M datasets are shown below:

该图像是图表，展示了SASRec和DiffuRec在Amazon Beauty和Movielens-1M数据集上的训练损失曲线。左侧为SASRec模型，右侧为DiffuRec模型，横轴为训练轮数，纵轴为损失值。可以观察到，DiffuRec在训练过程中损失下降幅度较大，表现更优。

Figure 10. Curve of training loss on Amazon Beauty and Movielens-1M datasets.

• Figure 10 compares the convergence speed of DiffuRec with SASRec (a representative Transformer-based baseline).
• On Amazon Beauty, both DiffuRec and SASRec converge around 150 epochs.
• On Movielens-1M, DiffuRec demonstrates faster convergence (around 100 epochs) compared to SASRec (around 250 epochs).
• Analysis: This faster convergence on Movielens-1M (which has a much longer average sequence length, 165.50 vs. 8.53 for Amazon Beauty) is attributed to DiffuRec's ability to induce the user's current interest by extracting information from the target item representation as auxiliary information. This target item guidance accelerates model learning, especially when dealing with longer and more complex sequences where SASRec might struggle to efficiently capture long-term dependencies.

6.6.2. Efficiency

The inference time on Amazon Beauty dataset and average training time on Amazon Beauty dataset are shown below:

该图像是图表，展示了不同方法在扩散步骤下的时间表现。左侧的折线图比较了SASRec、BERT4Rec等方法随扩散步骤的运行时间，右侧的柱状图则展示了不同算法在特定扩散步骤下的总时间消耗，包括DiffuRec和其他对比方法。

Figure 11. Inference time (in seconds for one sample) on Amazon Beauty dataset (left). The points in the inset illustrate performance (Recall@K and NDCG@K) at different reverse steps. Average training time (in seconds for one batch) on Amazon Beauty dataset (right).

The following are the results from Table 4 of the original paper:

Where $d$ is the representation dimension and $n$ is the sequence length.

• Time Complexity (Table 4):

  • GRU4Rec and ACVAE (both GRU-based) have a complexity of $O(nd)$, linear with sequence length $n$.
  • SASRec, BERT4Rec, and STOSA (all Transformer-based) have a complexity of $O(d^2 + n^2)$, due to the self-attention mechanism's quadratic dependency on sequence length.
  • DiffuRec has a complexity of $O(n^2 + dn)$. This aligns with Transformer-based models, given its Approximator. The dn term likely comes from operations involving the item embeddings and step embeddings within the diffusion process.

• Inference Time (Figure 11, left):

  • At a very small total diffusion step ($t=2$), DiffuRec's inference time is comparable to other Transformer-based models (SASRec, BERT4Rec).
  • As the number of reverse steps increases, DiffuRec's inference time for a single sample grows exponentially. This is expected as the reverse phase is an iterative process.
  • The inset in Figure 11 (left) shows that performance (Recall@K and NDCG@K) fluctuates with reverse steps. The full analysis of this fluctuation is provided in Figure 12.
  • GRU4Rec and ACVAE have generally faster inference times due to their lower time complexity. STOSA is slightly slower than pure Transformer-based models due to Wasserstein distance calculations.
  • The paper suggests that a moderate number of reverse steps (e.g., 32 or 64) offers a good balance between representation quality and inference speed.

• Training Time (Figure 11, right):

  • GRU4Rec has significantly lower training time per batch due to its simpler and more lightweight architecture.

  • SASRec, ACVAE, STOSA, DiffuRec, and BERT4Rec have very similar training times, ranging from 14s to 18s per batch.

  • Analysis: This indicates that the training process of DiffuRec is not a bottleneck compared to other Transformer-based or VAE-based models, despite its sophisticated diffusion mechanism. The main computational overhead is concentrated in the iterative inference phase.

    The performance (Recall@K and NDCG@K) at different reverse steps on Amazon Beauty and Movielens-1M datasets is shown below:

    该图像是图表，展示了在不同反向步骤下，Amazon Beauty 和 Movielens-1M 数据集的 HR @ K 和 NDCG @ K 指标变化。上方左侧为 Amazon Beauty 的 HR，右侧为 NDCG；下方左侧为 Movielens-1M 的 HR，右侧为 NDCG。

Figure 12. Recall@\mathsf{K}and NDCG@$\mathsf{K}$ at different reverse steps on Amazon Beauty and Movielens-1M datasets.

• Figure 12 shows how Recall@K and NDCG@K metrics vary with the number of reverse steps.
• Generally, increasing the reverse steps initially improves performance as the denoising process becomes more refined, but beyond a certain point, the gains diminish or even slightly fluctuate.
• For Amazon Beauty, performance peaks around 64 steps for HR@10 and NDCG@10. For Movielens-1M, it's around 32-64 steps. This confirms the earlier point about balancing quality and speed.

6.7. Visualization of Uncertainty and Diversity (RQ7)

6.7.1. Uncertainty

A t-SNE plot of the reconstructed $\mathbf{x}_0$ representation from the reverse phase is shown below:

该图像是一个t-SNE图，展示了从反向阶段重构的 extbf{x}_0 表示。左侧是来自Amazon Beauty的数据，紫点表示通过100种不同高斯噪声反转的相同项，黄点表示随机序列的其他项；右侧是Movielens-1M的数据，图示相似。

Figure 13. A t-SNE plot of the reconstructed $\mathbf{x}_0$ representation from reverse phase. The purple points represent the same item but reversed by 100 different Gaussian noise for a specific historical sequence. The yellow points are the others each of which is reversed from a random sequence in Amazon Beauty and Movielens-1M datasets respectively.

• Figure 13 visualizes the uncertainty captured by DiffuRec using t-SNE.
• Yellow Points: The yellow points (reversed $\mathbf{x}_0$ from random sequences) are dispersed uniformly throughout the 2D space, indicating high diversity across different user contexts.
• Purple Points: The purple points represent 100 different reconstructed target item representations $\mathbf{x}_0$ for the same historical sequence, but each generated from a different Gaussian noise initialization in the reverse phase. These points are clustered relatively closely but still exhibit distinct deviations from each other.
• Analysis: This visualization confirms that DiffuRec does not produce a single deterministic output for a given sequence, unlike most traditional models. Instead, it generates a distribution of plausible target item representations for the same input sequence, reflecting the inherent uncertainty in user preferences. This stochasticity is a key feature, allowing for diversified retrieval results by running the reverse process multiple times in parallel, which can be beneficial in the retrieval stage of recommender systems. For instance, reversing 100 different $\mathbf{x}_0$ for an exemplar sequence resulted in 643 and 82 unique items in the top-20 for Amazon Beauty and Movielens-1M, respectively.

6.7.2. Diversity

The categories of recommended items for a sampled sequence from Amazon Toys dataset are shown below:

• Table 5 illustrates the diversity of recommendation results by showing the categories of top-5 recommended items for a single user sequence from Amazon Toys. The target item category is 'Electronic Toys'.
• SASRec and TiMiRec tend to recommend items from very similar and narrow categories (e.g., 'Electronic Toys', 'Vehicles'). This is especially true for TiMiRec, which explicitly leverages the target item as a supervised signal, potentially constraining its recommendations to highly similar items.
• In contrast, STOSA and DiffuRec recommend a wider range of categories, moving beyond just 'Electronic Toys'. For example, DiffuRec suggests 'Card Games' and 'Electronic Software & Books'.
• Analysis: This demonstrates that the uncertainty injection inherent in STOSA and, more effectively, in DiffuRec, leads to more diverse recommendation results. By modeling items and preferences as distributions and leveraging stochasticity, DiffuRec can explore a broader range of potentially relevant items, fostering greater flexibility and potentially increasing serendipity in recommendations, which is a desired characteristic of modern recommender systems.

7. Conclusion & Reflections

7.1. Conclusion Summary

This paper presents DiffuRec, a pioneering application of Diffusion models to Sequential Recommendation (SR). The core innovation is to replace the traditional fixed vector representation of items with distributional representations, which more effectively capture multiple latent aspects of items and diverse user interests. DiffuRec leverages the diffusion process to inject uncertainty and incorporate target item guidance into the representation learning. The model consists of a Transformer-based Approximator that learns to reconstruct target item representations during training and iteratively denoise Gaussian noise during inference, followed by a rounding operation for discrete item prediction. Extensive experiments on four real-world datasets demonstrate DiffuRec's significant superiority over strong baselines, confirming the effectiveness of its diffusion-based approach to SR. Ablation studies and detailed analyses further elucidate the importance of its components and its robustness across varying sequence lengths and item popularity.

7.2. Limitations & Future Work

The authors highlight several potential directions for future research:

• Other Diffusion Model Paradigms: Exploring different ways of adapting Diffusion models to recommendation tasks beyond the current DiffuRec framework.
• Alternative Recommendation Scenarios: Applying Diffusion models to other recommendation settings such as session-based recommendation (where user identity might be implicit or shorter sequences are common) or click-through rate (CTR) prediction.

7.3. Personal Insights & Critique

DiffuRec presents a highly innovative and promising direction for recommender systems. The core insight that Diffusion models are well-suited for distribution generation and diversity representation in SR is powerful.

Inspirations:

• Bridging Generative AI and Recommender Systems: This paper successfully imports a state-of-the-art generative paradigm (Diffusion models) from CV/NLP into recommender systems. This opens up a vast new area of research, suggesting that many other advanced generative models could be adapted to recommendation to address long-standing challenges like cold-start, data sparsity, and diversity.
• Unified Uncertainty and Multi-Interest Modeling: DiffuRec provides a more elegant and unified framework for handling user interest diversity and uncertainty compared to prior piecemeal solutions (e.g., explicit multi-interest networks or VAEs with their known issues). The idea of target-item-aware uncertainty injection is particularly compelling, as it grounds the stochasticity in a meaningful, supervised signal.
• Robustness and Serendipity: The demonstrated robustness across datasets and the increased diversity in recommendations (as shown in the t-SNE and category analysis) are highly desirable qualities for modern recommender systems, moving beyond simple accuracy to user satisfaction.

Potential Issues/Areas for Improvement:

• Inference Speed: While training time is comparable, the iterative reverse process makes inference exponentially slower with more steps. Though the authors suggest a moderate number of steps for balance, for real-time recommendation systems with extremely low latency requirements, this could still be a bottleneck. Further research into faster diffusion sampling techniques (e.g., DDIM, ODE solvers) is critical for practical deployment.

• Complexity of Model Interpretation: Diffusion models, like many deep generative models, are complex. Interpreting why certain recommendations are made or how latent aspects are modeled within the distributions might be challenging. Future work could focus on explainable AI (XAI) for diffusion-based recommender systems.

• Hyperparameter Sensitivity: The sensitivity to $\lambda$ indicates that careful hyperparameter tuning is necessary, and large values can degrade performance significantly. Robustness to hyperparameters could be an area of improvement.

• Scalability to Extremely Large Item Sets: While impressive on the tested datasets, sequential recommendation with cross-entropy loss and inner product ranking over all candidate items (as done in this paper) can be computationally intensive for systems with millions of items. Exploring sampling strategies or more efficient retrieval mechanisms for the rounding operation might be necessary for industrial-scale applications.

• Generalization to Other Data Types: The current work focuses on implicit feedback. It would be interesting to see how DiffuRec performs with explicit feedback (ratings) or side information (item features, user demographics).

  Overall, DiffuRec represents a significant advancement, demonstrating the untapped potential of Diffusion models in recommender systems and setting a new benchmark for modeling dynamic user preferences and uncertainty.