1. Bibliographic Information

1.1. Title

InfoDCL: Informative Noise Enhanced Diffusion Based Contrastive Learning

1.2. Authors

• Xufeng Liang (xfliang@bit.edu.cn) - Beijing Institute of Technology, Beijing, China
• Zhida Qin (zanderqin@bit.edu.cn) - Beijing Institute of Technology, Beijing, China
• Chong Zhang (zhangchong@xjtu.edu.cn) - Xi'an Jiaotong University, Xi'an, China
• Tianyu Huang (huangtianyu@bit.edu.cn) - Beijing Institute of Technology, Beijing, China
• Gangyi Ding (dgy@bit.edu.cn) - Beijing Institute of Technology, Beijing, China

1.3. Journal/Conference

Proceedings of the 32nd ACM SIGKDD Conference on Knowledge Discovery and Data Mining V.1 (KDD '26), August 09–13, 2026, Jeju Island, Republic of Korea.

Comment on Venue: KDD (ACM SIGKDD Conference on Knowledge Discovery and Data Mining) is one of the premier international conferences in the fields of data mining, data science, and big data. It is highly reputable and influential, typically publishing cutting-edge research with significant impact. Publication at KDD '26 signifies that the work has undergone rigorous peer review and is considered a significant contribution to the field.

1.4. Publication Year

2026

1.5. Abstract

Contrastive learning has demonstrated promising potential in recommender systems. Existing methods typically construct sparser views by randomly perturbing the original interaction graph, as they have no idea about the authentic user preferences. Owing to the sparse nature of recommendation data, this paradigm can only capture insufficient semantic information. To address this issue, the authors propose InfoDCL, a novel diffusion-based contrastive learning framework for recommendation. Rather than injecting randomly sampled Gaussian noise, InfoDCL employs a single-step diffusion process that integrates noise with auxiliary semantic information to generate signals and feed them to the standard diffusion process to generate authentic user preferences as contrastive views. Besides, based on a comprehensive analysis of the mutual influence between generation and preference learning in InfoDCL, a collaborative training objective strategy is built to transform the interference between them into mutual collaboration. Additionally, InfoDCL employs multiple GCN layers only during the inference stage to incorporate higher-order co-occurrence information while maintaining training efficiency. Extensive experiments on five real-world datasets demonstrate that InfoDCL significantly outperforms state-of-the-art methods. The proposed InfoDCL offers an effective solution for enhancing recommendation performance and suggests a novel paradigm for applying diffusion methods in contrastive learning frameworks.

1.6. Original Source Link

• Official Source Link: https://arxiv.org/abs/2512.16576v1
• PDF Link: https://arxiv.org/pdf/2512.16576v1.pdf
• Publication Status: This paper is a preprint published on arXiv (v1) with a planned publication in KDD '26, indicating it is currently under review or accepted for a future conference.

2. Executive Summary

2.1. Background & Motivation

Core Problem: The core problem InfoDCL aims to solve is the limitation of existing contrastive learning (CL) methods in recommender systems, particularly their inability to capture sufficient semantic information due to the inherent sparsity of recommendation data.

Why this problem is important:

• Sparsity of Recommendation Data: User-item interaction data is notoriously sparse. Users only interact with a tiny fraction of available items.
• Limitations of Current CL: Traditional CL methods in recommendation generate "views" (augmented versions of data) by randomly perturbing the original interaction graph or adding noise to embeddings. These perturbations, when applied to sparse data, often create even sparser or less meaningful views, leading to an insufficient capture of authentic user preferences and limited semantic information. They essentially "perturb" existing knowledge rather than "generating" new, richer insights.
• Insufficient Semantic Information: Without understanding authentic user preferences, these methods struggle to learn robust and discriminative representations that truly reflect what a user might like, especially for items they haven't interacted with yet.
• Challenges with Auxiliary Data: While some methods incorporate auxiliary data (e.g., multimodal features), they often face issues with distributional gaps and noise between different modalities, making it difficult to align them effectively for accurate preference learning.

Paper's Entry Point/Innovative Idea: The paper's innovative idea is to leverage the powerful generative capabilities of diffusion models to create preference-aware views for contrastive learning. Instead of just perturbing existing data, InfoDCL aims to generate views that authentically reflect user preferences by integrating auxiliary semantic information into the noise generation process of diffusion models. This "informative noise" is designed to enrich the embeddings with semantics, enabling the diffusion model to synthesize user preferences more accurately.

2.2. Main Contributions / Findings

The paper's primary contributions and key findings are:

• Novel Diffusion-based Contrastive Learning Framework (InfoDCL): The authors propose a flexible framework that substantially modifies the construction of contrastive views. It addresses data sparsity by generating semantically rich views that reflect authentic user preferences, and it can be extended to incorporate various types of auxiliary information.
• Informative Noise Generation: A dedicated PsNet (Preference Signal Network) is designed to simulate a single-step diffusion process. This process efficiently injects auxiliary semantic information into Gaussian noise, producing "informative noise" that better enhances embedding generation to reflect genuine user preferences.
• Collaborative Training Objective Strategy: A novel training strategy is introduced that transforms the potential interference between the generative process (diffusion model) and preference learning (contrastive learning and BPR loss) into mutual collaboration. This strategy jointly optimizes reconstruction loss, contrastive loss, and BPR loss, along with a custom "collaboration balance loss," to significantly improve the model's learning ability and adaptability.
• Efficient Inference with GCN: To incorporate higher-order co-occurrence information while maintaining training efficiency, multiple GCN layers are employed only during the inference phase. This avoids the computational overhead of graph convolution operations during training.
• Empirical Superiority and Interpretability: Extensive experiments on five real-world datasets demonstrate that InfoDCL significantly outperforms state-of-the-art recommendation methods. The paper also provides theoretical analyses (Theorems A.1 and B.1) and empirical studies (e.g., SNR comparison, t-SNE visualization) to improve the interpretability and validate the effectiveness of the informative noise and the framework.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

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

Recommender Systems

Recommender systems are information filtering systems that predict what a user might like. They are widely used in e-commerce, streaming services, and social media to help users discover relevant items (e.g., products, movies, articles).

• Collaborative Filtering (CF): A common technique in recommender systems that makes predictions about a user's interests by collecting preferences from many users. The underlying assumption is that if two users have similar preferences for a set of items, they will have similar preferences for other items.
• User-Item Interaction Graph: In many recommender systems, user interactions with items (e.g., clicks, purchases, ratings) are modeled as a bipartite graph where one set of nodes represents users and the other represents items. An edge exists between a user and an item if an interaction has occurred. This graph is often very sparse, meaning most users have interacted with only a small fraction of available items.
• Embeddings: Low-dimensional vector representations of users and items that capture their characteristics and relationships. These embeddings are learned through various machine learning techniques and are used to compute similarity and make recommendations.

Contrastive Learning (CL)

Contrastive learning is a self-supervised learning paradigm where a model learns representations by contrasting positive pairs (similar samples) with negative pairs (dissimilar samples). The goal is to pull positive pairs closer together in the embedding space while pushing negative pairs further apart.

• Views/Augmentations: In contrastive learning, multiple "views" or augmented versions of an input sample are created. These views are considered positive pairs. For example, in image contrastive learning, two different crops or color jitters of the same image form a positive pair. In graph-based recommendation, this often involves perturbing the graph structure (e.g., node dropout, edge dropout).
• Self-Supervised Learning: A type of machine learning where the model learns from data that is automatically labeled, typically by designing pretext tasks. Contrastive learning is a prominent form of self-supervised learning.

Diffusion Models

Diffusion models are a class of generative models that learn to generate data by reversing a gradual diffusion (noising) process.

• Forward Diffusion Process: This process gradually adds random noise (typically Gaussian noise) to an input data sample over several time steps, transforming it into pure noise. If $\mathbf{x}_0$ is the original data, then at time step $t$, the noisy data $\mathbf{x}_t$ is generated by adding noise to $\mathbf{x}_{t-1}$.
• Reverse Diffusion Process (Denoising): This is the generative part. A neural network is trained to predict or estimate the noise added at each step, allowing it to reverse the forward process. Starting from pure noise, the model iteratively removes predicted noise to reconstruct a clean data sample.
• DDPM (Denoising Diffusion Probabilistic Models): A specific type of diffusion model that defines the forward process as a fixed Markov chain and trains a neural network to learn the reverse process.
• DDIM (Denoising Diffusion Implicit Models): An improvement over DDPMs that allows for faster sampling and deterministic generation by reformulating the reverse process to be non-Markovian.
• Noise Prediction Network: The neural network component of a diffusion model that learns to predict the noise component from a noisy input at a given time step.

Graph Neural Networks (GNNs)

GNNs are neural networks designed to operate directly on graph-structured data. They learn node embeddings by aggregating information from a node's neighbors.

• Graph Convolutional Networks (GCNs): A specific type of GNN that generalizes convolutional operations to irregular graph data. A GCN layer typically aggregates features from a node's direct neighbors and its own features to produce a new representation for that node.
• LightGCN: A simplified GCN model for recommendation that removes non-linear activation functions and feature transformation matrices, focusing purely on linear aggregation. It has been shown to be very effective while being lightweight.

Singular Value Decomposition (SVD)

SVD is a powerful matrix factorization technique that decomposes a matrix into three other matrices: $\mathbf{A} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^\top$.

• $\mathbf{U}$: Left singular vectors (orthogonal matrix).
• $\mathbf{\Sigma}$: Singular values (diagonal matrix with non-negative real numbers).
• $\mathbf{V}^\top$: Right singular vectors (orthogonal matrix). SVD is used for dimensionality reduction, noise reduction, and identifying latent features in data.

3.2. Previous Works

The paper builds upon and differentiates itself from several lines of prior research:

3.2.1. Contrastive Learning in Recommendation

• SGL [34]: One of the pioneering works incorporating contrastive learning into graph-based collaborative filtering. It generates multiple graph augmentations (e.g., node dropout, edge dropout) and applies a self-supervised objective to maximize agreement between different views of the same node.
• NCL [16]: Expands on SGL by forming contrastive pairs with both structural (graph neighbors) and semantic neighbors (nodes with similar attributes or learned representations). This aims to better utilize neighborhood information.
• SimGCL [41]: A model-agnostic framework that proposes injecting simple Gaussian noise into embeddings (rather than perturbing the graph structure) to facilitate robust learning. It argues that complex graph augmentations are often unnecessary.
• CoGCL [51]: Leverages discrete collaborative codes and virtual neighbors to create semantically meaningful contrastive views, aiming to strengthen collaborative signals beyond direct interactions.
• Limitations addressed by InfoDCL: These methods largely rely on perturbing existing interaction information, which is limited by the inherent sparsity of recommendation data. They struggle to generate novel, semantically rich views that genuinely reflect user preferences for unobserved items. InfoDCL aims to generate these views using diffusion models rather than just perturbing.

3.2.2. Generative Models and Diffusion in Recommendation

• DiffMM [11]: Integrates a modality-aware graph diffusion model with cross-modal contrastive learning. It aims to align multimodal item contexts with collaborative relations, generating modality-aware user-item graphs via a diffusion process.
• DiffRec [30]: One of the first to apply diffusion models to recommendation, framing the task as a denoising problem. It uses a standard diffusion process to learn user-item interaction distributions.
• DDRM [50]: Implements mutual conditioning between users and items in a diffusion model, allowing user and item representations to co-evolve during the reverse diffusion process through joint gradient updates.
• GiffCF [53]: Proposes a diffusion mechanism rooted in graph signal processing, simulating a heat diffusion process on the interaction graph via the Laplacian to propagate affinity signals.
• Limitations addressed by InfoDCL: While these methods introduce diffusion, they often use randomly sampled Gaussian noise as input to the diffusion process. For sparse recommendation data, this random noise can corrupt the already limited user preference signals, hindering the generation of truly authentic and semantically rich preferences. InfoDCL specifically addresses this by generating informative noise.

3.3. Technological Evolution

The field of recommender systems has evolved significantly:

1. Early Methods (e.g., MF): Started with matrix factorization (MF) techniques that decompose the user-item interaction matrix into lower-dimensional user and item latent factors. These methods were effective but struggled with sparsity and cold-start problems.

2. Graph-based Models (e.g., GCNs, LightGCN): The advent of Graph Neural Networks allowed for more sophisticated modeling of user-item interaction graphs, explicitly leveraging multi-hop relationships to enrich user and item representations. LightGCN simplified this by removing non-linearities, demonstrating that message passing is key.

3. Self-Supervised Learning (SSL) and Contrastive Learning (CL): Inspired by successes in computer vision and NLP, SSL (especially CL) was adopted to learn robust representations by creating augmented views of the interaction graph or embeddings and maximizing agreement between positive pairs. This helped mitigate sparsity by learning from the data's inherent structure.

4. Generative Models (e.g., VAEs, GANs, Diffusion Models): More recently, generative models have been explored to capture the underlying data distribution and generate new interaction patterns or item embeddings. Diffusion models, with their ability to generate high-quality samples, represent a significant step in this direction, particularly for complex data distributions.

   InfoDCL fits into this timeline by combining the strengths of Contrastive Learning and Diffusion Models. It addresses a key limitation of existing CL methods (insufficient semantic views from sparse data) by incorporating a sophisticated diffusion process that is enhanced by auxiliary semantic information, rather than being driven by generic random noise. This positions InfoDCL at the forefront of generative self-supervised learning for recommendation.

3.4. Differentiation Analysis

Compared to the main methods in related work, InfoDCL offers the following core differences and innovations:

• Novel View Generation (Diffusion-based, Informative Noise):

  • Vs. Traditional CL (SGL, NCL, SimGCL, CoGCL): Instead of perturbing existing interaction graphs or adding random noise to embeddings to create views, InfoDCL actively generates rich, preference-aware views. It leverages the generative power of diffusion models, which go beyond merely augmenting existing data.
  • Vs. Diffusion Models with Random Noise (DiffRec, DDRM, GiffCF): Unlike these methods that inject randomly sampled Gaussian noise into the diffusion process, InfoDCL introduces a PsNet to generate informative noise. This informative noise is enriched with auxiliary semantic information, making the diffusion process more effective at synthesizing authentic user preferences, especially in sparse data environments.

• Semantic Integration through PsNet:

  • Innovation: The PsNet explicitly integrates auxiliary semantic information (e.g., from user similarity graphs, item feature-similarity graphs) into the noise generation, which is a key differentiator. This ensures that the generated views are not just diverse but also semantically meaningful and aligned with user preferences.
  • Vs. Multimodal/Auxiliary Methods (DiffMM): While some methods like DiffMM align views from heterogeneous sources, InfoDCL directly injects semantic information into the noise itself during the diffusion process, aiming for a more fundamental integration rather than just alignment of distinct views.

• Collaborative Training Objective Strategy:

  • Innovation: InfoDCL proposes a unique collaborative training strategy that explicitly balances and harmonizes the objectives of generation (diffusion reconstruction loss), contrastive learning, and traditional recommendation (BPR loss). The introduction of a collaboration balance loss ($\mathcal{L}_c$) is novel in ensuring that the generative process does not diverge from the preference learning.
  • Vs. Other Methods: Most existing methods optimize these components more independently or with simpler weighting schemes, without an explicit mechanism to manage their mutual interference and foster collaboration during training.

• Efficient Inference with GCN:

  • Innovation: The strategy of using multiple GCN layers only during inference for higher-order co-occurrence information, while keeping training free of complex graph convolutions, is a smart design choice for efficiency.

  • Vs. GNN-based CL: Many GNN-based CL methods perform graph convolutions during training, which can be computationally intensive, especially with deep GNNs. InfoDCL decouples this to maintain efficiency.

    In essence, InfoDCL's innovation lies in its ability to proactively generate semantically rich, preference-aware views for contrastive learning via an intelligently informed diffusion process, coupled with a sophisticated optimization strategy that harmonizes diverse learning objectives for superior recommendation performance in sparse data settings.

4. Methodology

The InfoDCL framework addresses the challenge of sparse recommendation data by generating rich, preference-aware contrastive views using an informative noise-enhanced diffusion process. It integrates auxiliary semantic information into the diffusion model and optimizes it with a collaborative training strategy.

4.1. Principles

The core idea of InfoDCL is to overcome the limitations of traditional contrastive learning in recommendation, which often generates insufficient semantic information due to data sparsity. Instead of merely perturbing existing interaction graphs, InfoDCL leverages the powerful generative capacity of diffusion models to produce item embeddings that authentically reflect user preferences. The key principles are:

1. Informative Noise for Diffusion: Rather than using random Gaussian noise, the model generates "informative noise" by integrating auxiliary semantic information (e.g., user/item similarities) via a Preference Signal Network (PsNet). This noise guides the diffusion process to create semantically meaningful embeddings.
2. Semantics Enhanced View Generation: The informative noise is fed into a forward diffusion process to progressively corrupt an initial item embedding. The subsequent denoising process then generates a rich, preference-aware item embedding, which serves as a contrastive view. This ensures the views capture latent user preferences.
3. Collaborative Optimization: The training process collaboratively optimizes multiple objectives:
   • Reconstruction Loss: Ensures the diffusion model accurately reconstructs original embeddings from noisy versions.
   • Contrastive Loss: Aligns the generated preference-aware embeddings with the initial item embeddings, injecting semantic richness.
   • Bayesian Personalized Ranking (BPR) Loss: Incorporates traditional collaborative filtering signals based on user-item interactions.
   • Collaboration Balance Loss: Explicitly balances the generative and preference learning objectives to prevent divergence and ensure stable, effective training.
4. Efficient Higher-Order Information: To capture complex co-occurrence patterns without sacrificing training efficiency, higher-order information (via GCN layers) is incorporated only during the inference stage, not during training.

4.2. Core Methodology In-depth

The InfoDCL framework consists of several key components: Informative Noise Generation, Semantics Enhanced Contrastive Learning, and a Collaborative Training Objective Strategy, culminating in an efficient Inference Stage.

4.2.1. Informative Noise Generation

The goal of this module is to generate informative noise that is enriched with auxiliary semantic information, rather than relying on randomly sampled Gaussian noise. This informative noise will then be used in the diffusion process. This module is called Preference Signal Network (PsNet).

4.2.1.1. Auxiliary Metadata Synthesis

The process begins by synthesizing auxiliary metadata. Since raw noise lacks semantic content, InfoDCL constructs auxiliary metadata by creating user similarity graphs and item similarity graphs. These graphs are based solely on user-item interactions from the training set. LightGCN pre-trained embeddings are then aggregated over these similarity graphs to obtain the auxiliary metadata, denoted as $\mathbf{m}$. The paper notes that this auxiliary metadata is extensible; if a dataset provides actual social relations, knowledge graph links, or multimodal attributes, these can also be incorporated to enrich $\mathbf{m}$.

4.2.1.2. Spectral Rectification

The first core part of PsNet is spectral rectification, which approximates a single-step diffusion generation process. It does this by performing Singular Value Decomposition (SVD) on an input signal. The input signal, $\mathbf{y}$, is formed by combining randomly sampled Gaussian noise $\epsilon$ with the auxiliary metadata $\mathbf{m}$. The process is formulated as follows: $$\mathbf{y} = \epsilon + \mathbf{m}$$ Here, $\epsilon$ is a randomly sampled Gaussian noise vector, and $\mathbf{m}$ is the auxiliary metadata. Next, SVD is applied to $\mathbf{y}$: $$\mathbf{y} = \mathbf{U} \Sigma \mathbf{V}^\top$$ where:

• $\mathbf{U} \in \mathbb{R}^{1 \times d}$ represents the left singular vectors (though the notation $1 \times d$ seems to imply a single row vector, it's typically $D \times D$ or $D \times d$ where $D$ is the embedding dimension and $d$ is the number of singular values). Given $\mathbf{y}$ is a vector of dimension $D$, it's likely representing a transformation or a simplified view where $d \ll D$.

• $\mathbf{V} \in \mathbb{R}^{D \times d}$ represents the right singular vectors.

• $\Sigma \in \mathbb{R}^{d \times d}$ is a diagonal matrix containing the singular values, with $d \ll D$ indicating a dimensionality reduction.

  The singular vectors and values are then processed by distinct Multi-Layer Perceptron (MLP) layers: $$\tilde{\mathbf{U}} = \mathrm{MLP}(\mathbf{U}), \quad \tilde{\mathbf{V}} = \mathrm{MLP}(\mathbf{V}_{1:}^\top), \quad \tilde{\Sigma} = \mathrm{MLP}(\mathrm{diag}(\Sigma))$$ where:

• $\tilde{\mathbf{U}}, \tilde{\mathbf{V}}, \tilde{\Sigma} \in \mathbb{R}^{d}$ are the transformed representations.

• $\mathrm{diag}(\Sigma)$ extracts the singular values from the diagonal of the $\Sigma$ matrix.

  These transformed components are then concatenated and mapped back into the original embedding space dimension $D$ using a non-linear mapping $\Phi$: $$\mathbf{g} = \Phi ([\tilde{\mathbf{U}} || \tilde{\mathbf{V}} || \tilde{\Sigma}]) \in \mathbb{R}^D$$ where:

• $\Phi : \mathbb{R}^{3d} \to \mathbb{R}^D$ is a non-linear mapping (e.g., an MLP).

• $[ \cdot || \cdot || \cdot ]$ denotes concatenation.

  Finally, a residual connection is applied to obtain the output of spectral rectification: $$S(\epsilon, \mathbf{m}) = \mathbf{y} + \mathrm{diag}(\tanh(\alpha)) \cdot \mathbf{g}$$ where:

• $\mathbf{y} = \epsilon + \mathbf{m}$ is the initial combined signal.

• $\alpha \in \mathbb{R}^D$ is a learnable parameter vector that controls the scale of the residual term.

• $\mathrm{diag}(\tanh(\alpha))$ creates a diagonal matrix from the tanh-activated $\alpha$ vector, allowing element-wise scaling.

  This step approximates a single-step diffusion, incorporating semantics from $\mathbf{m}$ into the noise.

4.2.1.3. Contextual Re-encoding

To further bridge the discrepancy between Gaussian noise and the desired informative noise, a contextual re-encoding module is introduced. This module aims to learn the residual between the output of spectral rectification and the ideal informative noise. First, an augmented input $\epsilon^+$ is formed: $$\epsilon^+ = \epsilon + \varphi(\mathbf{m}) \in \mathbb{R}^{\mathbf{D}}$$ where:

• $\epsilon$ is the randomly sampled Gaussian noise.

• $\varphi(\mathbf{m}) \in \mathbb{R}^{\mathbf{D}}$ is an embedding of the auxiliary metadata.

  A linear transformation $\Psi$ and layer normalization $N$ are then applied: $$\mathbf{h} = N(\Psi(\epsilon^+)) \in \mathbb{R}^{\mathbf{D}}$$ where:

• $\Psi$ is a linear function.

• $N$ denotes layer normalization.

  A residual scaling factor $\rho$ regulates the residual strength. The output of contextual re-encoding is: $$C(\epsilon, \mathbf{m}) = \epsilon + \sigma(\rho) \cdot \mathbf{h}$$ where:

• $\sigma(\cdot)$ is the sigmoid function.

• $\rho$ is a learnable scalar.

4.2.1.4. Output

The final informative noise representation, $\epsilon^{(\mathbf{m})}$, is obtained by fusing the outputs of spectral rectification and contextual re-encoding with an additional residual connection involving the auxiliary metadata embedding: $$\epsilon^{(\mathbf{m})} = \underbrace{S(\epsilon, \mathbf{m})}_{\text{spectral rectification}} + \underbrace{\eta_0 C(\epsilon, \mathbf{m})}_{\text{contextual re-encoding}} + \underbrace{\sigma(\eta_1 - 1) \varphi(\mathbf{m})}_{\text{residual term}}$$ where:

• $S(\epsilon, \mathbf{m})$ is the output from spectral rectification.

• $C(\epsilon, \mathbf{m})$ is the output from contextual re-encoding.

• $\varphi(\mathbf{m}) \in \mathbb{R}^{\mathbf{D}}$ is the embedding of the auxiliary metadata.

• $\eta_0, \eta_1 \in \mathbb{R}$ are learnable scalars.

  This $\epsilon^{(\mathbf{m})}$ is the informative noise enriched with semantic information, ready to be used in the subsequent diffusion process.

4.2.2. Semantics Enhanced Contrastive Learning

This module uses the generated informative noise to create semantically enhanced item embeddings that serve as contrastive views.

The informative noise $\epsilon^{(\mathbf{m})}$ is injected into the forward process of a diffusion model. For an initialized item embedding $\mathbf{e}_i \in \mathbb{R}^{\mathbf{d}}$ (where $\mathbf{d}$ is the embedding dimension) and the informative noise $\epsilon^{(\mathbf{m})}$, the noisy embedding $\mathbf{z}_t^{(\mathbf{m})}$ at time step $t$ is defined as: $$\mathbf{z}_t^{(\mathbf{m})} = \sqrt{\bar{\alpha}_t} \mathbf{e}_i + \sqrt{1 - \bar{\alpha}_t} \epsilon_t^{(\mathbf{m})}$$ where:

• $\mathbf{e}_i$ is the initial clean embedding of item $i$.

• $\epsilon_t^{(\mathbf{m})}$ is the informative noise at time $t$ (derived from $\epsilon^{(\mathbf{m})}$).

• $\bar{\alpha}_t = \prod_{s=1}^{t} \alpha_s$ is the cumulative product of variance schedule parameters $\alpha_s$, with $0 < \alpha_t < 1$. These parameters control how much of the original signal and noise are present at each step.

• When $t = T$, $\mathbf{z}_T^{(\mathbf{m})}$ is close to pure noise.

  The diffusion model's goal is to learn to reverse this noising process, i.e., to denoise $\mathbf{z}_t^{(\mathbf{m})}$ back to the original $\mathbf{e}_i$. This denoising is driven by a reconstruction loss: $$\mathcal{L}_{\mathbf{r}} = \sum_{i=1}^{\mathrm{N}} \|\mathbf{e}_i - \mu_{\theta}(\mathbf{z}_t^{(\mathbf{m})}, t)\|^2$$ where:

• $\mathcal{L}_{\mathbf{r}}$ is the reconstruction loss, a Mean Squared Error (MSE) comparing the original item embedding $\mathbf{e}_i$ with the denoised output.

• $\mu_{\theta}(\mathbf{z}_t^{(\mathbf{m})}, t)$ is the predicted clean embedding from the noise prediction network (parameterized by $\theta$) given the noisy input $\mathbf{z}_t^{(\mathbf{m})}$ and time step $t$.

• $\mathrm{N}$ is the total number of items. This loss encourages the model to denoise the noisy embeddings to recover the original item embeddings, thereby learning the underlying preference distribution. The final generated output after denoising is denoted as $\hat{\mathbf{e}}_i^{(\mathbf{m})}$.

Subsequently, a view-level contrastive loss is applied to align these generated embeddings $\hat{\mathbf{e}}_i^{(\mathbf{m})}$ (which capture item semantics and user preferences) with the initialized item embeddings $\mathbf{e}_i$. This process injects authentic latent user preferences into the item embeddings. $$\mathcal{L}_{\mathrm{con}} = - \sum_{i=1}^{\mathrm{N}} \log \frac{\exp(\mathrm{sim}(\hat{\mathbf{e}}_i^{(\mathbf{m})}, \mathbf{e}_i) / \tau)}{\sum_{j=1}^{\mathrm{N}} \exp(\mathrm{sim}(\hat{\mathbf{e}}_i^{(\mathbf{m})}, \mathbf{e}_j) / \tau)}$$ where:

• $\mathcal{L}_{\mathrm{con}}$ is the contrastive loss, typically a InfoNCE-like loss.
• $\mathrm{sim}(\cdot, \cdot)$ denotes a similarity function, often cosine similarity.
• $\hat{\mathbf{e}}_i^{(\mathbf{m})}$ is the item embedding generated by the diffusion process for item $i$, enriched with semantic information.
• $\mathbf{e}_i$ is the initial item embedding for item $i$. This forms the positive pair.
• $\mathbf{e}_j$ for $j \neq i$ are initial item embeddings for other items, serving as negative samples.
• $\tau > 0$ is the temperature hyperparameter, which scales the similarity scores.

4.2.3. Collaborative Training Objective Strategy

To ensure that the generated embeddings reflect co-occurrence relationships and to harmonize the different learning objectives, a collaborative training strategy is employed.

4.2.3.1. Bayesian Personalized Ranking (BPR) Loss

The Bayesian Personalized Ranking (BPR) loss, a standard objective in collaborative filtering, is introduced. This pairwise ranking loss encourages the model to assign a higher prediction score to an interacted item (positive sample) than to a non-interacted item (negative sample) for a given user. $$\mathcal{L}_{bpr} = - \sum_{u=1}^{\mathbf{M}} \sum_{i \in \mathcal{N}_u} \sum_{j \notin \mathcal{N}_u} \ln \sigma(\hat{\mathbf{y}}_{ui} - \hat{\mathbf{y}}_{uj})$$ where:

• $\mathcal{L}_{bpr}$ is the BPR loss.
• $\mathbf{M}$ is the total number of users.
• $\mathcal{N}_u$ is the set of items interacted with by user $u$.
• $i \in \mathcal{N}_u$ represents a positive item (interacted).
• $j \notin \mathcal{N}_u$ represents a negative item (not interacted).
• $\sigma(\cdot)$ is the sigmoid function.
• $\hat{\mathbf{y}}_{ui}$ is the predicted score for user $u$ and item $i$, typically computed as the inner product of their embeddings ($\mathbf{u}^\top \mathbf{e}_i$).

4.2.3.2. Collaboration Balance Loss

To address the observed gap and potential interference between the reconstruction loss and the BPR loss during training, a collaboration balance loss ($\mathcal{L}_c$) is introduced. This loss acts as a regularizer on the generative output to ensure stability and better alignment between generation and preference learning. It is defined as an L2-norm of the generated item embeddings: $$\mathcal{L}_c = \frac{1}{\mathbf{N}} \|\hat{\mathbf{e}}^{(\mathbf{m})}\|_2 = \frac{1}{\mathbf{N}} \left( \sum_i |\hat{\mathbf{e}}_i^{(\mathbf{m})}|^2 \right)^{1/2}$$ where:

• $\mathcal{L}_c$ is the collaboration balance loss.
• $\hat{\mathbf{e}}^{(\mathbf{m})}$ is the matrix of generated item embeddings (where each row is $\hat{\mathbf{e}}_i^{(\mathbf{m})}$).
• $\mathbf{N}$ is the total number of items.
• $\|\cdot\|_2$ denotes the L2-norm. This loss encourages the generated embeddings to have controlled magnitudes, preventing them from becoming too large or too small, which helps stabilize training and coordinate objectives.

4.2.3.3. Total Optimization

The overall training objective is a weighted sum of all these components: $$\mathcal{L}_{total} = (1 - \lambda_b) \mathcal{L}_r + \lambda_b \mathcal{L}_{bpr} + \lambda_c \mathcal{L}_{con} + \lambda_l \mathcal{L}_c + \lambda_g \mathcal{L}_{reg}$$ where:

• $\mathcal{L}_{total}$ is the total loss function.

• $\mathcal{L}_r$ is the reconstruction loss.

• $\mathcal{L}_{bpr}$ is the BPR loss.

• $\mathcal{L}_{con}$ is the contrastive loss.

• $\mathcal{L}_c$ is the collaboration balance loss.

• $\mathcal{L}_{reg}$ is a regularization term.

• $\lambda_b, \lambda_c, \lambda_l, \lambda_g$ are non-negative coefficients that balance the influence of each loss term, tuned on a validation set.

  The regularizer $\mathcal{L}_{reg}$ is applied to the user and item embeddings to prevent overfitting and promote smoothness: $$\mathcal{L}_{reg} = \sum_{i=1}^{\mathrm{N}} (\|\mathbf{e}_i\|_2^2 + \|\mathbf{u}_i\|_2^2)$$ where:

• $\mathbf{e}_i$ are the item embeddings.

• $\mathbf{u}_i$ are the user embeddings.

  Multiple Types of Optimization: The framework is flexible. If multiple auxiliary metadata channels are used, the respective reconstruction, contrastive, and balance losses generated by each channel are summed and jointly optimized as part of the overall objective.

4.2.4. Inference Stage

During inference, InfoDCL avoids additional diffusion sampling. Instead, it directly uses the user and item representations learned during training and further refines them by incorporating higher-order co-occurrence information. This is achieved using the LightGCN architecture.

First, the observed interaction graph is converted into a symmetrically normalized adjacency matrix $\mathcal{\bar{H}}_{\mathbf{u}, \mathbf{i}}$: $$\mathcal{\bar{H}}_{\mathbf{u}, \mathbf{i}} = \frac{\mathcal{A}_{\mathbf{u}, \mathbf{i}}}{\sqrt{|\mathcal{N}_{\mathbf{u}}| |\mathcal{N}_{\mathbf{i}}|}}$$ where:

• $\mathcal{A}_{\mathbf{u}, \mathbf{i}}$ is the adjacency matrix representing user-item interactions (or user-item-user/item-user-item propagation paths).

• $|\mathcal{N}_{\mathbf{u}}|$ denotes the degree of user $\mathbf{u}$ (number of items interacted with).

• $|\mathcal{N}_{\mathbf{i}}|$ denotes the degree of item $\mathbf{i}$ (number of users interacted with).

  Then, $K$ layers of graph convolution are performed to obtain the final user and item representations: $$\hat{\mathbf{E}} = \mathcal{\bar{H}}_{\mathbf{u},*} \mathbf{E}, \qquad \hat{\mathbf{U}} = \mathcal{\bar{H}}_{*,\mathrm{i}} \mathbf{U}$$ where:

• $\mathbf{E}$ is the matrix of initial item embeddings (learned during training).

• $\mathbf{U}$ is the matrix of initial user embeddings (learned during training).

• $\hat{\mathbf{E}} \in \mathbb{R}^{\mathbf{N} \times \mathbf{d}}$ is the matrix of refined item embeddings.

• $\hat{\mathbf{U}} \in \mathbb{R}^{\mathbf{M} \times \mathbf{d}}$ is the matrix of refined user embeddings.

• The notation $\mathcal{\bar{H}}_{\mathbf{u},*}$ and $\mathcal{\bar{H}}_{*,\mathrm{i}}$ implies aggregation from neighbors, effectively performing graph convolution. Finally, the relevance score for a user-item pair is estimated by taking the inner product of their refined embeddings ($\hat{\mathbf{u}}^\top \hat{\mathbf{e}}$), and items are ranked accordingly.

4.2.5. Theoretical Analysis

The paper includes theoretical analyses in Appendix A and B to validate the effectiveness of the proposed informative noise.

4.2.5.1. Theorem A.1: Informative Noise Generation

Theorem A.1 provides a closed-form expression for the informative noise generated by one round of re-denoising in the latent space of the model. It demonstrates that the noise produced by PsNet conforms to the diffusion framework and effectively incorporates semantic information from auxiliary metadata.

Let $\mathbf{v}_0 \in \mathbb{R}^{\mathbf{d}}$ be the clean embedding of an item, and the forward diffusion process at any time $t$ follow the standard DDPM formulation: $$\mathbf{v}_t = \alpha_t \mathbf{v}_0 + \sigma_t \epsilon, \quad \epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_d)$$ where $\alpha_t, \sigma_t$ are parameters controlling the signal and noise levels. Let $\mathbf{v}_T \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_d)$ be a standard Gaussian sample at the final time step $T$. Structural semantic information $\mathbf{s} \in \mathbb{R}^{\mathbf{d}_s}$ is injected into the noise.

Theorem A.1 Statement: Let $\epsilon_{\theta}(\mathbf{v}, t \mid \star)$ be a trained noise prediction network that is $L$-Lipschitz with respect to its first argument. Assume the trajectory is smooth over a step size $k \ll T$, i.e., $\|\mathbf{v}_T - \mathbf{v}_{T-k}\| \leq Lk$. Let the classifier-free guidance (CFG) scales at step $T$ and T-k be $\omega_{\ell}$ and $\omega_w$, respectively. Then the output of one round of re-denoising, denoted $\mathbf{v}_T'$, satisfies: $$\mathbf{v}_T' = \mathbf{v}_T + \kappa \cdot \mathbf{g}_s, \quad \kappa = (\omega_{\ell} - \omega_w) \cdot \frac{\alpha_T \sigma_{T-k} - \alpha_{T-k} \sigma_T}{\alpha_{T-k}}$$ where: $$\mathbf{g}_s = \epsilon_{\theta}\left(\mathbf{v}_{T - \frac{k}{2}}, T - \frac{k}{2} \mid \mathbf{s}\right) - \epsilon_{\theta}\left(\mathbf{v}_{T - \frac{k}{2}}, T - \frac{k}{2} \mid \emptyset\right)$$ is referred to as the semantic gradient.

Explanation: This theorem shows that the generated informative noise $\mathbf{v}_T'$ is effectively the original Gaussian noise $\mathbf{v}_T$ translated by a scaled semantic gradient $\mathbf{g}_s$.

• Semantic Gradient ($\mathbf{g}_s$): This gradient captures the semantic information. It's the difference between the noise predicted by the model when conditioned on the semantic information $\mathbf{s}$ and when conditioned on an empty (unconditional) context $\emptyset$. This means $\mathbf{g}_s$ points in the direction of the semantic information.
• Scaling Factor ($\kappa$): This factor controls the magnitude of the semantic information injection. It depends on the classifier-free guidance scales ($\omega_{\ell}, \omega_w$) and the diffusion schedule parameters ($\alpha_t, \sigma_t$). By adjusting these, the strength of semantic injection can be controlled. The theorem formally proves that the informative noise is not just random noise but is explicitly steered by the auxiliary semantic information through the diffusion process.

4.2.5.2. Theorem B.1: Preference Alignment via Informative Noise

Theorem B.1 analyzes the impact of informative noise on user preference learning, demonstrating that it guides the diffusion process to yield representations with higher user preference scores.

Let $\mathbf{v}_T \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_d)$ be an initial Gaussian noise vector in the latent space. Let $\mathbf{s} \in \mathbb{R}^{\mathbf{d}_s}$ denote structural semantics, and let $\mathbf{u} \in \mathbb{R}^{\mathbf{d}}$ be a user embedding vector. Let $\mathbf{G} : \mathbb{R}^{\mathbf{d}} \to \mathbb{R}^{\mathbf{d}}$ be the deterministic generative mapping (e.g., a DDIM sampler over $K$ steps) such that $\mathbf{v}_0 = \mathbf{G}(\mathbf{v}_T)$ yields a generated item embedding.

Assumptions:

1. The noise predictor $\epsilon_{\theta}(\cdot, t \mid \cdot)$ is $L$-Lipschitz.
2. Informative noise is constructed via one re-denoise step: $\mathbf{v}_T^{\mathrm{info}} = \mathbf{v}_T + \kappa \cdot \mathbf{g}_s$, where $\mathbf{g}_s$ is the semantic gradient (as defined in Theorem A.1) and $\kappa > 0$ controls injection strength.
3. There exists $\delta > 0$ such that $\langle \mathbf{u}, \mathbf{g}_s \rangle \geq \delta$, meaning the semantic direction aligns positively with user preference.

Theorem B.1 Statement: Then the informative-noise-generated embedding $\mathbf{v}_0^{\mathrm{info}} = \mathbf{G}(\mathbf{v}_T^{\mathrm{info}})$ satisfies the following expected preference bound: $$\mathbb{E}_{\mathbf{v}_T} \left[ \langle \mathbf{u}, \mathbf{v}_0^{\mathrm{info}} \rangle \right] \geq \mathbb{E}_{\mathbf{v}_T} \left[ \langle \mathbf{u}, \mathbf{v}_0^{\mathrm{std}} \rangle \right] + \kappa \delta - \gamma \kappa^2 \|\mathbf{u}\| \cdot \|\mathbf{g}_s\|$$ where $\mathbf{v}_0^{\mathrm{std}} = \mathbf{G}(\mathbf{v}_T)$ is the item embedding generated from standard noise, and $\gamma$ is a constant related to the curvature of $\mathbf{G}$.

Explanation: This theorem mathematically proves that using informative noise leads to a higher expected preference score (measured by the inner product $\langle \mathbf{u}, \mathbf{v}_0 \rangle$) for a given user $\mathbf{u}$ compared to using standard Gaussian noise.

• The term $\kappa \delta$ represents the direct positive boost from the informative noise, as $\delta$ signifies a positive alignment between user preference and the semantic gradient.
• The term $-\gamma \kappa^2 \|\mathbf{u}\| \cdot \|\mathbf{g}_s\|$ is a penalty term related to the non-linearity of the generator $\mathbf{G}$ and the square of the injection strength $\kappa$. The theorem suggests an optimal injection strength $\kappa^* = \frac{\delta}{2 \gamma \|\mathbf{u}\| \cdot \|\mathbf{g}_s\|}$ that maximizes this improvement. This provides theoretical backing for why InfoDCL's approach of steering the diffusion with semantics yields better recommendations.

4.3. Full Content of the Research Paper

The full content of the research paper has been provided as context for this analysis.

5. Experimental Setup

5.1. Datasets

The experimental evaluations are conducted on five commonly used public recommendation datasets. These datasets are:

• ML-1M (MovieLens 1M): A classic dataset for movie recommendations.

• Amazon-Office: User-item interactions for office products from Amazon.

• Amazon-Electronics: User-item interactions for electronic products from Amazon.

• Amazon-Baby: User-item interactions for baby products from Amazon.

• Taobao: A dataset from the Taobao e-commerce platform.

  The choice of these datasets allows for evaluation across different domains (movies, various product categories) and varying levels of sparsity and scale, ensuring a robust assessment of InfoDCL's performance.

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

Analysis of Data Characteristics:

• Scale: Datasets range from medium (Office, Taobao) to large (ML-1M, Baby, Electronics) in terms of users, items, and interactions.
• Sparsity: All datasets exhibit very high sparsity, with the lowest being ML-1M at 95.53% and the highest reaching 99.92% for Taobao. This extreme sparsity is a key challenge that InfoDCL aims to address.

5.2. Evaluation Metrics

The effectiveness of the recommender system is assessed using two widely adopted ranking metrics: NDCG@K and Recall@K.

1. Recall@K

   • Conceptual Definition: Recall@K measures the proportion of relevant items that are successfully retrieved among the top-K recommended items. It indicates the model's ability to find as many relevant items as possible within the recommended list of size K. A higher Recall@K means the model is better at identifying relevant items.
   • Mathematical Formula: $ \mathrm{Recall@K} = \frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} \frac{|\mathrm{R}(u) \cap \mathrm{T}(u)|}{|\mathrm{T}(u)|} $
   • Symbol Explanation:
     • $|\mathcal{U}|$: The total number of users in the evaluation set.
     • $\mathrm{R}(u)$: The set of top-K items recommended to user $u$.
     • $\mathrm{T}(u)$: The set of relevant items for user $u$ in the test set.
     • $|\cdot|$: Denotes the cardinality (number of elements) of a set.

2. NDCG@K (Normalized Discounted Cumulative Gain at K)

   • Conceptual Definition: NDCG@K is a measure of ranking quality that takes into account the position of relevant items in the ranked list. It assigns higher scores to relevant items that appear at higher ranks (closer to the top of the list). The "discounted" part means that relevant items found lower in the list are penalized. The "normalized" part ensures that scores across different queries are comparable by dividing by the ideal DCG. A higher NDCG@K indicates a better-ranked list where highly relevant items are prominently featured.
   • Mathematical Formula: $ \mathrm{NDCG@K} = \frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} \frac{\mathrm{DCG@K}_u}{\mathrm{IDCG@K}_u} $ where \mathrm{DCG@K}_u = \sum_{j=1}^{K} \frac{2^{\mathrm{rel}_j} - 1}{\log_2(j+1)} and \mathrm{IDCG@K}_u = \sum_{j=1}^{|\mathrm{T}(u)|} \frac{2^{\mathrm{rel}_j^*} - 1}{\log_2(j+1)}.
   • Symbol Explanation:

     • $|\mathcal{U}|$: The total number of users in the evaluation set.

     • $\mathrm{DCG@K}_u$: Discounted Cumulative Gain for user $u$ at cutoff K.

     • $\mathrm{IDCG@K}_u$: Ideal Discounted Cumulative Gain for user $u$ at cutoff K (the DCG of the perfectly sorted list of relevant items).

     • $K$: The cutoff position in the ranked list.

     • $\mathrm{rel}_j$: The relevance score of the item at position $j$ in the recommended list. For implicit feedback, this is typically 1 if the item is relevant and 0 otherwise.

     • $\mathrm{rel}_j^*$: The relevance score of the item at position $j$ in the ideal (perfectly sorted) ranked list.

     • $\log_2(j+1)$: A logarithmic discount factor that penalizes relevant items appearing at lower ranks.

       The authors use an "all-ranking evaluation protocol" and report the average score over all test users.

5.3. Baselines

A comprehensive set of baseline methods is used for comparison, categorized as follows:

1. Classical Collaborative Filtering Models:

   • MF [12]: Matrix Factorization, a fundamental collaborative filtering technique that decomposes the user-item interaction matrix into user and item latent factors.
   • ENMF [2]: Efficient Neural Matrix Factorization, a neural matrix factorization method designed for efficiency without sampling.

2. Representative Graph Neural Network (GNN)-based Approaches:

   • NGCF [7]: Neural Graph Collaborative Filtering, which explicitly encodes the collaborative signal in the form of high-order connectivity by propagating embeddings on the user-item interaction graph.
   • LightGCN [6]: A simplified GCN model for recommendation that removes non-linear activations and feature transformations, focusing on linear aggregation for efficiency and effectiveness.

3. State-of-the-Art Generative Models based on Diffusion Processes:

   • DiffRec [30]: Diffusion Recommender Model, an early method applying diffusion models to recommendation, treating it as a denoising task.
   • DDRM [50]: Denoising Diffusion Recommender Model, which implements mutual conditioning between users and items for co-evolution during reverse diffusion.
   • GiffCF [53]: Graph signal diffusion model for collaborative filtering, simulating heat diffusion on the interaction graph.

4. Recent Contrastive Learning (CL)-based Techniques with Strong Accuracy:

   • SGL [34]: Self-supervised Graph Learning, which uses graph augmentations and node-level contrastive objectives.
   • NCL [16]: Neighborhood-enriched Contrastive Learning, leveraging structural and semantic neighbors for contrastive views.
   • SimGCL [41]: Simple Graph Contrastive Learning, which injects noise into embeddings instead of graph augmentations.
   • RecDCL [47]: Dual Contrastive Learning for Recommendation.
   • SGCL [49]: Symmetric Graph Contrastive Learning, designed to counter noisy views.
   • CoGCL [51]: Enhancing Graph Contrastive Learning with Reliable and Informative Augmentation.

5. Multimodal Baselines (for specific comparisons in Section 4.4):

   • MMSSL [33]: Multi-Modal Self-Supervised Learning for Recommendation.

   • LATTICE [48]: Mining Latent Structures for Multimedia Recommendation.

   • BM3 [52]: Bootstrap Latent Representations for MultiModal Recommendation.

   • LGMRec [5]: Local and Global Graph Learning for Multimodal Recommendation.

   • MGCN [43]: Multi-View GCN.

   • DiffMM [11]: Multi-Modal Diffusion Model for Recommendation (also listed above).

     These baselines represent a comprehensive spectrum of recommendation methodologies, from classic models to cutting-edge GNNs, diffusion models, and contrastive learning approaches, providing a strong basis for comparison.

5.4. Implementation Details

• Embedding Dimensionality: All models consistently use an embedding dimensionality of 64.
• Initialization: Embeddings are initialized using the Xavier scheme.
• Hyperparameter Optimization:
  • Learning Rate: Logarithmically sampled within $[1 \times 10^{-6}, 5 \times 10^{-1}]$.
  • Batch Sizes: Discretely selected based on dataset interaction density (e.g., 1024 for ML-1M, 2000 for Amazon-Office).
  • Loss Coefficients:
    • Diffusion reconstruction term ($\lambda_r$): Tuned within [0, 1.0].
    • Pairwise ranking term ($\lambda_b$): Tuned within [0, 1.0].
    • Regularization coefficient ($\lambda_g$): Searched in [0.001, 0.01].
    • Contrastive loss weight ($\lambda_{con}$): Varied between $5 \times 10^{-5}$ and $5 \times 10^{-6}$.
  • GCN Layers (Inference): Explored from 0 to 3 layers.
  • Diffusion Timesteps: Tested across a range of 100 to 500.
• Optimizers: Impact of Adam and AdamW optimizers was compared.

6. Results & Analysis

6.1. Core Results Analysis

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

Observations from Performance Comparison:

• Overall Superiority of InfoDCL: InfoDCL consistently outperforms all other state-of-the-art models across all five datasets and all evaluation metrics (Recall@20/50, NDCG@20/50). This demonstrates the robust effectiveness of its proposed framework. The improvement percentages are significant, especially on ML-1M (e.g., 42.70% Recall@20) and Taobao (e.g., 24.20% Recall@20).
• Performance of Diffusion-based Models: Diffusion-based models like DiffRec and GiffCF generally perform better than traditional CF (MF, ENMF) and GCN-based models (NGCF, LightGCN), highlighting the potential of generative approaches. However, they are still significantly outmatched by InfoDCL. DDRM performs notably poorly across all datasets, which might indicate issues with its specific implementation or sensitivity to dataset characteristics.
• Performance of Contrastive Learning Models: CL methods (SGL, NCL, SimGCL, RecDCL, SGCL, CoGCL) show strong performance, often surpassing GCN-based and basic diffusion models. This confirms the benefit of contrastive objectives in learning robust representations. However, their reliance on random perturbations limits their ability to capture authentic user preferences in sparse data, which InfoDCL addresses.
• InfoDCL-S vs. InfoDCL-D: The table presents InfoDCL-S and InfoDCL-D. The context indicates that these might refer to single-channel vs. dual-channel contrastive learning within the InfoDCL framework, or perhaps different types of auxiliary information. Without explicit definition, it's difficult to pinpoint the exact difference, but both versions perform exceptionally well. InfoDCL-D generally achieves the best performance (bolded values), suggesting benefits from incorporating multiple sources of auxiliary information or a more complex configuration.

Key Takeaways: The results strongly validate InfoDCL's core innovations:

1. Informative Noise: The use of auxiliary metadata and SVD to produce informative noise is crucial, enabling the diffusion process to generate embeddings that capture more semantic information and user preferences than random Gaussian noise.
2. Collaborative Training: The collaborative training objective strategy effectively integrates generation, contrastive learning, and preference learning, transforming potential conflicts into synergistic collaboration, leading to significant performance gains.

6.2. Comparison with Multimodal Baselines

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

Observations from Multimodal Comparison:

• InfoDCL's Dominance: InfoDCL consistently achieves superior performance across all five datasets and metrics when compared against state-of-the-art multimodal recommender systems (MMSSL, LATTICE, BM3, LGMRec, MGCN, DiffMM). This suggests its method of leveraging modality information is highly effective.
• Effectiveness on Single Modality Data (Taobao): The Taobao dataset, which only contains visual data (a single modality), shows InfoDCL significantly outperforming all competitors. This is a strong indicator that the proposed PsNet is highly effective at generating informative noise with semantic information, even from a single auxiliary modality. This enriched noise enables the diffusion process to capture genuine user preferences better.
• Semantic Injection into Embeddings: The results reinforce the idea that the generated item embeddings, enriched with semantic information from the PsNet and integrated through the collaborative training objective, lead to substantial improvements in recommendation performance.

6.3. Ablation Analysis

The following figure (Figure 2 from the original paper) shows the ablation analysis across five datasets:

Analysis of Ablation Study:

• w/o SR (Without Spectral Rectification): This variant removes the spectral rectification module, leaving only contextual re-encoding for integrating Gaussian noise with auxiliary metadata. This means SVD-based simulation of one-step diffusion is discarded, resulting in a simpler linear fusion of noise and semantics.
  • Result: Eliminating spectral rectification leads to a significant performance drop across all datasets and metrics. This highlights the crucial role of SVD-based decomposition and transformation in capturing and encoding auxiliary semantic information effectively to create informative noise.
• w/o CR (Without Contextual Re-encoding): This variant removes the contextual re-encoding module, relying solely on SVD (spectral rectification) to generate informative noise.
  • Result: Although the performance degradation is less severe than w/o SR, a decline is still observed. This indicates that while spectral rectification is powerful, it has limitations in fully approximating the one-step diffusion. The contextual re-encoding module is necessary to bridge this discrepancy and further enhance the semantic richness of the informative noise.
• w/o PsNet (Without Preference Signal Network): In this variant, the informative noise is replaced with randomly sampled Gaussian noise, effectively reverting to a standard diffusion process for generating item embeddings.
  • Result: Across almost all datasets and evaluation metrics, removing PsNet results in severe performance degradation. This is a critical finding, strongly indicating that injecting auxiliary semantic information into the noise (i.e., generating informative noise) is essential. Random Gaussian noise corrupts the sparse user preference signals, whereas informative noise leverages semantics to produce embeddings that better capture authentic user preferences.
• w/o CBL (Without Collaboration Balance Loss): This variant removes the collaboration balance loss ($\mathcal{L}_c$) from the overall optimization strategy.

  • Result: Omitting this loss term leads to performance degradation across all datasets and metrics. On some datasets (e.g., Amazon-Electronics), its impact is particularly pronounced. This underscores the importance of $\mathcal{L}_c$ in coordinating the multiple learning objectives (reconstruction, contrastive, BPR) and ensuring stable training, preventing the generative and preference learning objectives from interfering with each other.

    Conclusion from Ablation Study: Each component of InfoDCL—spectral rectification, contextual re-encoding, the entire PsNet for informative noise generation, and the collaboration balance loss—plays a vital and non-redundant role in achieving the model's superior performance. The informative noise generation (via PsNet) and the collaboration balance loss are particularly critical.

6.4. Training Efficiency

The following figure (Figure 3 from the original paper) shows the performance versus efficiency analysis on Amazon-Electronics:

Analysis of Training Efficiency:

• The plot compares Recall@20 (performance) against average training time per epoch (efficiency) on the large Amazon-Electronics dataset.
• LightGCN: Shows good efficiency (low training time) due to its simplified architecture but has lower performance compared to more complex models.
• Diffusion Models (DiffRec): Tends to have significantly longer training times due to their inherent complexity, with only marginal performance gains over simpler models.
• Contrastive Learning Models (NCL): Offers strong performance (second-highest Recall@20 among baselines) but with a slightly increased training time compared to LightGCN.
• InfoDCL: Achieves the highest Recall@20 while maintaining a relatively low training time per epoch. It strikes an optimal balance between performance and efficiency, outperforming all baselines in performance without incurring substantial computational overhead. This is particularly valuable as it allows the model to be deployed more practically on large datasets. The design choice of integrating GCN layers only during inference stage contributes to this efficiency.

6.5. In-depth Analysis

6.5.1. SNR Comparison Between Gaussian Noise and Informative Noise

The following figures (Figure 4 and 8 from the original paper) display the SNR comparison on Amazon-Baby, Amazon-Electronics, Taobao, Amazon-Office, and ML-1M datasets:

该图像是一个对比图表，展示了在三个数据集（Amazon-Baby、Amazon Electronics 和 Taobao）上，使用高斯噪声和信息噪声的信噪比（SNR）随时间步变化的情况。图中蓝线表示高斯噪声，橙线表示信息噪声，展示了不同噪声类型对信号质量的影响。

Analysis of SNR Comparison:

• SNR (Signal-to-Noise Ratio) is calculated as $\mathrm{SNR}(\mathbf{X}) = \frac{(\mathbb{E}[\mathbf{X}])^2}{\mathrm{Var}(\mathbf{X})}$, where $\mathbb{E}[\mathbf{X}]$ is the expectation and $\mathrm{Var}(\mathbf{X})$ is the variance of a random variable $\mathbf{X}$. A higher SNR indicates a stronger signal relative to background noise, implying more meaningful information.
• The plots show the SNR of the noisy embeddings $\mathbf{z}_T^{(\mathbf{m})}$ over time steps for both informative noise (orange/red lines) and randomly sampled Gaussian noise (blue lines).
• Consistent Advantage of Informative Noise: Across all five datasets (Amazon-Baby, Amazon-Electronics, Taobao, Amazon-Office, and ML-1M), the SNR of embeddings generated with informative noise is consistently higher than that of embeddings generated with Gaussian noise.
• Increasing Advantage with Timesteps: The advantage of informative noise becomes more pronounced as the number of noise-injection time steps increases. This suggests that as more noise is introduced, the guidance from the integrated semantic information becomes even more critical in maintaining signal integrity and informativeness.
• Implication: This analysis empirically confirms that PsNet successfully embeds rich semantic information into the noise. This semantically structured noise provides better guidance during the diffusion process, leading to generated item embeddings that more accurately capture user preferences and are more informative.

6.5.2. Visualization Comparison Between Gaussian Noise and Informative Noise

The following figure (Figure 5 from the original paper) shows the visualization of the item embeddings on the Amazon-Baby dataset using T-SNE:

Analysis of Visualization:

• T-SNE (t-Distributed Stochastic Neighbor Embedding) is a dimensionality reduction technique used for visualizing high-dimensional data, typically by mapping it to two or three dimensions. It's particularly effective at revealing clusters and relationships within data.
• Gaussian Noise Variant: When random Gaussian noise is used (left panel), the item embeddings cluster into limited, discrete regions. This crowding suggests that the random noise disrupts the inherent structure and relationships in the sparse recommendation data, making different items indistinguishable in the embedding space. This indicates poor representation learning.
• InfoDCL with Informative Noise: In contrast, after informative noise injection in InfoDCL (right panel), the generated embeddings exhibit a more balanced spatial arrangement and distinct clusters. This implies that the informative noise helps preserve and enrich the semantic content, allowing for better differentiation and clearer relationships between items. The distinct clusters are indicative of richer, more discriminative representations.
• Implication: This visualization provides intuitive evidence that informative noise mitigates the corruption caused by random noise in sparse data. By enriching the embeddings with semantic content, InfoDCL facilitates the learning of better user preferences and item representations, which is crucial for effective recommendation.

6.5.3. Analysis of Collaborative Training Objective Strategy

The following figure (Figure 6 from the original paper) compares the training behavior with and without the collaboration balance loss:

Analysis of Collaborative Training Objective Strategy:

• The figure illustrates the training curves for reconstruction loss (dashed green line) and BPR loss (solid blue line) on the Amazon-Electronics dataset.
• Without Collaboration Balance Loss (Left Panel): A clear divergence is observed between the reconstruction loss and the BPR loss. The reconstruction loss quickly diminishes to a very low value, suggesting that the generative model might be overfitting or becoming too dominant in the overall objective. This imbalance hinders further refinement of the denoising generative model, as its contribution to the total loss becomes negligible, leading to insufficient optimization.
• With Collaboration Balance Loss (Right Panel): The introduction of the collaboration balance loss ($\mathcal{L}_c$) significantly improves the situation. The two loss terms (reconstruction and BPR) align more closely in magnitude, enabling consistent and stable training. This balanced optimization ensures that both the generative process and the preference learning objectives receive adequate attention throughout training, preventing one from overwhelming the other.
• Implication: This analysis empirically validates the importance of the collaboration balance loss. It successfully transforms potential interference between the generative and preference learning objectives into mutual collaboration, leading to more stable training and ultimately, enhanced model effectiveness, as evidenced by the overall performance improvements.

6.5.4. Visualization of Similarity Between Eigenvectors

The following figure (Figure 7 from the original paper) shows the cosine similarity between the singular vectors of input and output in the diffusion process:

Analysis of Eigenvector Similarity:

• This figure shows a histogram of the absolute value of cosine similarity between the singular vectors obtained from the SVD decomposition of the diffusion model's inputs and outputs.
• High Similarity: The histogram clearly shows that most vector pairs achieve similarity scores very close to 1. This indicates a strong alignment in their feature representations.
• Implication: This finding provides strong theoretical justification and empirical support for the design of the PsNet, specifically its spectral rectification module. The high similarity confirms the theoretical underpinning (likely related to the Davis-Kahan theorem mentioned in the paper), demonstrating the feasibility of effectively transforming and learning characteristics between the input and output spaces using SVD. It validates that the SVD-based approximation of a single-step diffusion is structurally sound and preserves important feature information.

6.6. Hyperparameter Analysis

The following figure (Figure 9 from the original paper) shows the hyperparameter analysis across five datasets:

Analysis of Hyperparameter Sensitivity:

• The figure illustrates the impact of four key hyperparameters ($\lambda_b$, $\lambda_{con}$, $\lambda_l$, $\lambda_g$) on performance across the five datasets.
• BPR Loss Coefficient ($\lambda_b$):
  • Generally, increasing $\lambda_b$ improves performance across all datasets, especially for Taobao and Office. This suggests that a strong emphasis on the pairwise ranking objective (learning from observed interactions) is beneficial for InfoDCL.
  • Optimal values tend to be in the mid-to-high range.
• Collaboration Balance Loss Coefficient ($\lambda_l$):
  • The influence of $\lambda_l$ is dataset-dependent. ML-1M and Office datasets show optimal performance at mid-range values (around $1 \times 10^{-3}$ or $1 \times 10^{-4}$).
  • This highlights the need for careful tuning to balance the generative and preference learning components, which varies with data characteristics (e.g., sparsity levels). Too low or too high values can be detrimental.
• Regularization Coefficient ($\lambda_g$):
  • Lower values (e.g., $3 \times 10^{-3}$ to $5 \times 10^{-3}$) generally yield better performance, particularly on denser datasets like ML-1M.
  • This indicates that over-regularization can suppress the learning effect and degrade performance, suggesting that the model benefits from allowing the embeddings more freedom to learn from the rich signals provided by the informative noise and contrastive objectives.
• Contrastive Learning Coefficient ($\lambda_{con}$):

  • Smaller values (such as $5 \times 10^{-3}$) consistently lead to improved metrics across datasets.

  • This implies that while contrastive signals are beneficial, an overly strong contrastive loss might dominate the learning and hurt generalization, especially on sparse datasets like Baby and Electronics. A gentle nudge from contrastive learning is more effective than a strong push.

    Overall Hyperparameter Impact: The model's performance is sensitive to these hyperparameters, necessitating careful tuning. The collaboration balance loss and contrastive loss coefficients require particularly fine-grained adjustments to achieve optimal performance, reflecting the delicate balance required for the multi-objective optimization.

7. Conclusion & Reflections

7.1. Conclusion Summary

This work introduces InfoDCL, a novel diffusion-based contrastive learning framework designed to enhance recommender systems, particularly in the challenging context of sparse interaction data. The core innovation lies in replacing traditional randomly sampled Gaussian noise with informative noise. This informative noise is generated by a Preference Signal Network (PsNet) that integrates auxiliary semantic information (e.g., from user/item similarity graphs) through a single-step diffusion process, including spectral rectification and contextual re-encoding. This approach enables the diffusion model to generate preference-aware contrastive views that are richer in semantic content and more authentically reflect user preferences than views derived from simple perturbations.

To manage the interplay between the generative process and preference learning, InfoDCL proposes a collaborative training objective strategy. This strategy jointly optimizes reconstruction loss, contrastive loss, and Bayesian Personalized Ranking (BPR) loss, crucially incorporating a collaboration balance loss to prevent divergent optimization and foster mutual collaboration between these objectives. Furthermore, InfoDCL maintains training efficiency by integrating higher-order co-occurrence information via multiple GCN layers exclusively during the inference stage.

Extensive experiments on five real-world datasets (ML-1M, Amazon-Office, Amazon-Baby, Taobao, Electronics) demonstrate that InfoDCL significantly outperforms state-of-the-art methods across various metrics (Recall@K, NDCG@K). Ablation studies confirm the critical contribution of each proposed component. Theoretical analyses and empirical studies (e.g., SNR comparison, t-SNE visualization) provide strong interpretability, showing that informative noise indeed yields higher signal-to-noise ratios and more discriminative item embeddings.

7.2. Limitations & Future Work

The paper's conclusion section is concise regarding explicit limitations and future work, but the preceding analysis implies:

Potential Limitations:

• Dependence on Auxiliary Information Quality: The effectiveness of InfoDCL heavily relies on the quality and availability of auxiliary semantic information. If this information is noisy, scarce, or irrelevant, the PsNet might struggle to generate truly informative noise, potentially limiting performance. The paper uses similarity graphs based on interaction data, but for broader applicability, external rich auxiliary data is needed.
• Complexity of PsNet and Diffusion: While InfoDCL is more efficient than some diffusion models, the PsNet (with SVD, MLPs, concatenations) and the diffusion process itself still introduce complexity compared to simpler contrastive learning methods. This might lead to higher computational costs or memory requirements during training, especially for extremely large datasets or very high-dimensional embeddings, despite the efficiency gains in inference.
• Hyperparameter Sensitivity: The hyperparameter analysis shows that InfoDCL is sensitive to the weighting coefficients for its multiple loss terms. Tuning these parameters effectively across diverse datasets can be challenging and time-consuming.
• Generalizability of Theoretical Analysis: While the theoretical analyses (Theorem A.1 and B.1) provide strong justifications, they rely on certain assumptions (e.g., Lipschitz continuity, approximate linearity of the generative function). The extent to which these assumptions hold perfectly in highly complex, real-world recommendation scenarios could be a subject of further investigation.

Suggested Future Research Directions:

• Applying Diffusion to Sparse Recommendation: The paper suggests a "novel paradigm for applying diffusion method in contrastive learning frameworks" for "inherently sparse recommendation tasks." This implies further exploration into how diffusion models can be tailored to handle sparsity, perhaps by directly modeling sparse interaction tensors or incorporating sparse-aware noise injection mechanisms.
• Exploring Different Types of Auxiliary Information: The framework is flexible and extensible to various types of auxiliary information. Future work could investigate the impact of different external data sources (e.g., richer multimodal content, knowledge graphs, temporal context) on informative noise generation and overall recommendation performance.
• Adaptive Informative Noise Generation: Developing more adaptive or dynamic PsNet architectures that can automatically learn the optimal way to integrate and scale semantic information based on the current state of training or data characteristics could be beneficial.
• Further Optimizing Training Efficiency: While GCN is moved to inference, exploring more lightweight or sparse-aware diffusion model architectures tailored for recommendation could further improve training efficiency without sacrificing performance.
• Beyond Item Embeddings: The current framework focuses on generating informative item embeddings. Future work could extend this to user embeddings or joint user-item representations to capture preferences from multiple perspectives.

7.3. Personal Insights & Critique

InfoDCL presents a highly innovative and well-justified approach to improving contrastive learning in recommender systems. The paper successfully identifies a crucial limitation of existing CL methods in sparse data environments – the inability to generate truly semantic-rich views – and offers a powerful solution by integrating diffusion models with "informative noise."

Strengths and Inspirations:

• Synergistic Combination: The combination of diffusion models (for rich generation) and contrastive learning (for discriminative representation) is particularly elegant. It leverages the strengths of both paradigms to address the challenges of data sparsity.
• "Informative Noise" is a Key Concept: The idea of actively designing noise rather than randomly sampling it is a profound insight. By embedding semantic information directly into the noise, the model learns to generate outputs that are inherently more meaningful. This concept could be applied to other generative tasks where input noise often lacks specific guidance.
• Collaborative Training Objective: The explicit collaboration balance loss is a practical and effective solution to a common problem in multi-objective learning where different loss components might interfere or diverge. This strategy could inspire similar balancing mechanisms in other complex deep learning models.
• Efficiency Considerations: The strategic decision to move heavy GCN computations to the inference stage demonstrates a practical awareness of system design for real-world deployment, balancing model complexity with operational efficiency.
• Rigorous Validation: The extensive experimental results, supported by both ablation studies and theoretical analyses (SNR, t-SNE, eigenvector similarity), make a compelling case for the model's effectiveness and interpretability.

Potential Issues or Areas for Improvement:

• Transparency of Auxiliary Metadata: While the paper mentions constructing user and item similarity graphs, more detail on the specific features or methods used to derive these auxiliary metadata (especially beyond simple interaction counts) would be beneficial. The quality of this metadata is paramount.

• Computational Cost of PsNet: Although InfoDCL is efficient overall, the PsNet itself involves SVD, multiple MLPs, and concatenations. For extremely high-dimensional embeddings or very large item/user sets, the cost of generating informative noise for every diffusion step might still be substantial, even if it's "single-step" in its approximation. An analysis of the computational overhead of PsNet versus basic noise sampling would be informative.

• Scalability for Extreme Sparsity: While the paper addresses sparsity, some datasets in recommendation are orders of magnitude sparser than those tested. It would be interesting to see how InfoDCL performs in such extreme cold-start scenarios, where auxiliary information might also be very limited.

• Interpretability of $k$ and $\gamma$ in Theoretical Bounds: The theoretical bounds for preference alignment include parameters like $\kappa$ (injection strength) and $\gamma$ (related to generator curvature). While the paper provides guidance on how to interpret and potentially optimize $\kappa$, further practical methods or insights into estimating/controlling these parameters in real-world training could enhance the utility of the theoretical analysis.

  In conclusion, InfoDCL is a significant advancement in recommender systems, particularly for addressing data sparsity in contrastive learning. Its innovative use of informative noise within a diffusion framework, coupled with a well-designed collaborative training strategy, offers a powerful new direction for generative self-supervised recommendation. The principles and insights presented could have broader implications for other domains where generative models are used to enrich sparse or limited data for downstream discriminative tasks.