1. Bibliographic Information

1.1. Title

Rethinking Popularity Bias in Collaborative Filtering via Analytical Vector Decomposition

1.2. Authors

• Lingfeng Liu (School of Artificial Intelligence and Data Science, University of Science and Technology of China, Hefei, China)
• Yixin Song (School of Artificial Intelligence and Data Science, University of Science and Technology of China, Hefei, China)
• Dazhong Shen (College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China)
• Bing Yin (iFLYTEK Research, iFLYTEK, Hefei, China)
• Hao Li (iFLYTEK Research, iFLYTEK, Hefei, China)
• Yanyong Zhang (School of Artificial Intelligence and Data Science, University of Science and Technology of China, Hefei, China)
• Chao Wang* (School of Artificial Intelligence and Data Science, University of Science and Technology of China, Hefei, China)

1.3. Journal/Conference

This paper is published at the 32nd ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD '26), August 09-13, 2026, Jeju Island, Republic of Korea. KDD is one of the premier international conferences for data mining, data science, and knowledge discovery, widely recognized for publishing high-quality, impactful research in the field. Its reputation signifies the paper's contribution to cutting-edge research in recommender systems.

1.4. Publication Year

2026 (based on the ACM Reference Format and the provided publication date 2025-12-11T14:35:13.000Z, suggesting an early online release for a 2026 conference).

1.5. Abstract

The abstract states that popularity bias in collaborative filtering (CF) models hinders personalization by over-recommending popular items and neglecting niche content. While existing methods view this as an external factor, this paper identifies it as an intrinsic geometric artifact of Bayesian Pairwise Ranking (BPR) optimization. Through mathematical analysis, it proves that BPR organizes item embeddings along a dominant "popularity direction" where embedding magnitudes correlate with interaction frequency. This distortion forces user embeddings to balance genuine preference with global popularity, leading to suboptimal configurations. The paper proposes Directional Decomposition and Correction (DDC), a universally applicable framework that corrects this embedding geometry using asymmetric directional updates. DDC guides positive interactions along personalized preference directions and steers negative interactions away from the global popularity direction, thereby disentangling preference from popularity. Extensive experiments on multiple BPR-based architectures show DDC significantly outperforms state-of-the-art debiasing methods, reducing training loss to less than 5% of baselines and achieving superior recommendation quality and fairness.

1.6. Original Source Link

https://arxiv.org/abs/2512.10688v1 (Preprint, publication status confirmed by ACM reference). PDF Link: https://arxiv.org/pdf/2512.10688v1.pdf

2. Executive Summary

2.1. Background & Motivation

• Core Problem: The paper addresses the pervasive issue of popularity bias in collaborative filtering (CF) recommender systems. This bias causes models to disproportionately recommend items that are already popular, thereby neglecting niche content and undermining the core goal of personalization.
• Why is this problem important? In modern recommender systems, CF models, particularly embedding-based ones like LightGCN, are foundational. They are almost universally trained using the Bayesian Pairwise Ranking (BPR) loss for implicit feedback datasets. However, popularity bias not only reduces the personalization and diversity of recommendations but also perpetuates a Matthew effect, where popular items gain more visibility while less popular but potentially relevant niche content remains undiscovered. This leads to user dissatisfaction, reduced engagement, and a less fair recommendation ecosystem.
• Challenges/Gaps in Prior Research: Traditional approaches typically treat popularity bias as an external confounding factor. They employ re-weighting methods (like Inverse Propensity Scoring - IPS), regularization, or causal inference to address the effects of bias. However, these methods often act at a macroscopic level, addressing symptoms rather than the root cause. They fail to explain how BPR optimization geometrically distorts the latent representation space to systematically favor popular items. This fundamental gap in understanding the intrinsic mechanism of bias generation is what the paper aims to fill.
• Paper's Entry Point/Innovative Idea: The paper's innovative idea is to reveal that popularity bias is not an external factor but an intrinsic geometric artifact of BPR optimization. It posits that BPR systematically organizes item embeddings along a dominant "popularity direction" where embedding magnitudes directly correlate with interaction frequency. This geometric distortion forces user embeddings to perform two conflicting tasks: expressing genuine preference and calibrating against global popularity. The paper then proposes to correct this geometric distortion directly.

2.2. Main Contributions / Findings

The paper makes the following primary contributions:

• Geometric Root of Popularity Bias: It identifies and theoretically characterizes a dominant popularity direction in BPR embedding spaces. This reveals the geometric root of popularity bias through rigorous mathematical analysis, showing that BPR inherently conflates an item's intrinsic qualities with its global popularity.
• Directional Decomposition and Correction (DDC) Framework: It proposes DDC, a novel and universally applicable framework. DDC transforms existing BPR-based CF models by surgically correcting the embedding geometry through asymmetric directional updates. It guides positive interactions along personalized preference directions and steers negative interactions away from the global popularity direction, effectively disentangling preference from popularity. A key advantage is its plug-and-play nature, working as a fine-tuning stage without architectural modifications.
• Comprehensive Experimental Validation and Superior Performance: The paper provides extensive experimental validation across multiple BPR-based architectures and benchmark datasets. It demonstrates that DDC significantly outperforms state-of-the-art debiasing methods. Notably, DDC reduces the final BPR loss on training data to less than 5% of heavily-tuned baselines, while achieving superior recommendation quality and fairness (e.g., dramatically reducing AvgPop@10 while boosting MRR@10 and NDCG@10). This indicates that DDC guides user embeddings toward more accurate representations of true preference.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

To understand this paper, a novice reader should grasp the following foundational concepts:

• Collaborative Filtering (CF): Collaborative Filtering is a fundamental technique in recommender systems that makes recommendations by collecting preferences or taste information from many users. It operates on the principle that if two users share similar tastes on some items, they are likely to have similar tastes on other items as well. The two main types are user-based CF (finding similar users) and item-based CF (finding similar items). This paper focuses on embedding-based CF.
• Implicit Feedback: In recommender systems, feedback can be explicit (e.g., star ratings) or implicit (e.g., clicks, purchases, views, watch time). Implicit feedback is much more abundant but doesn't directly convey preference strength (a click doesn't necessarily mean a strong like). The paper deals with implicit feedback datasets.
• Embedding-based Models / Latent Space: These models represent users and items as low-dimensional dense vectors, called embeddings, in a continuous latent space (or embedding space). The idea is that similar users or items will have similar embedding vectors (i.e., close to each other in the latent space). The similarity between a user and an item (representing a prediction of preference) is typically computed as the dot product of their respective embedding vectors.
  • Matrix Factorization (MF): A foundational embedding-based CF model. It directly learns user and item embeddings by factorizing the user-item interaction matrix. If $U$ is the user matrix and $V$ is the item matrix, then the predicted interaction $\hat{y}_{ui}$ for user $u$ and item $i$ is given by the dot product of their embeddings: $\hat{y}_{ui} = \mathbf{e}_u^T \mathbf{e}_i$.
  • LightGCN: A state-of-the-art Graph Neural Network (GNN)-based model for recommendation. It simplifies traditional Graph Convolutional Networks (GCNs) by only including the most essential component: neighborhood aggregation. It learns user and item embeddings by propagating information across the user-item interaction graph. Despite its sophisticated learning process, the final preference score is still calculated as an inner product between the user and item embeddings, similar to MF.
• Bayesian Pairwise Ranking (BPR) Loss: This is a de facto standard optimization objective for implicit feedback datasets in collaborative filtering. Instead of predicting a direct rating, BPR aims to learn a personalized ranking. It assumes that for a given user, an interacted item (a positive item) should be ranked higher than an un-interacted item (a negative item). The BPR loss maximizes the score difference between positive and negative items for a user. It is a pairwise learning approach.
  • Sigmoid Function ($\sigma(x)$): A common activation function in machine learning, defined as $\sigma(x) = 1 / (1 + e^{-x})$. It maps any real-valued number to a value between 0 and 1, often interpreted as a probability. In BPR, it's used to model the probability that a positive item is preferred over a negative item.
• Popularity Bias: This refers to the tendency of recommender systems to recommend items that are already popular, irrespective of a user's individual preferences for niche content. This can lead to a self-fulfilling prophecy or Matthew effect, where popular items become even more popular, and less popular items get overlooked.

3.2. Previous Works

The paper categorizes related work into three groups, highlighting how DDC differs.

• Macroscopic Debiasing Strategies:

  • Approach: These methods treat the recommender model as a black box and intervene at the data or objective function level. They primarily address the symptoms of popularity bias.
  • Examples:
    • Inverse Propensity Scoring (IPS): A common re-weighting method. It attempts to balance training data by down-weighting interactions with popular items and up-weighting interactions with niche items. The idea is to make the learning process less sensitive to the over-representation of popular items.
      • Limitation: Can suffer from high variance, especially when propensity scores are small (for very unpopular items), and may oversimplify the problem by assuming a simple inverse relationship with popularity.
    • Regularization approaches: These add penalty terms to the objective function to discourage correlations between recommendation scores and item popularity.
      • Limitation: Often achieve bias reduction at the cost of overall recommendation accuracy, as they impose a general constraint rather than targeting the source of the bias.
  • DDC's Differentiation: DDC argues these methods address symptoms without understanding the structural impact on learned representations.

• Causal and Disentangled Approaches:

  • Approach: These are more principled approaches that leverage causal inference to model and remove popularity's confounding influence, or aim to learn separate latent factors for genuine interest versus conformity to popular trends.
  • Examples:
    • Causal methods: Use causal graphs to represent relationships and counterfactual inference to estimate unbiased effects.
    • Disentanglement methods: Aim to learn distinct embedding components, one representing a user's true preference and another representing factors like popularity or conformity. The goal is to isolate a 'pure' preference representation for prediction.
  • Limitation: These methods often rely on strong causal assumptions or complex training schemes. Crucially, while they hypothesize a popularity component, they fail to explain *how* it is structurally and geometrically encoded by standard optimization like BPR.
  • DDC's Differentiation: DDC directly addresses this foundational gap by identifying the geometric mechanism of popularity bias in BPR.

• Representation Space Optimization (e.g., Contrastive Learning - CL):

  • Approach: Contrastive Learning has gained popularity for improving embedding quality in GNN-based models. It generally aims to pull similar samples closer together and push dissimilar samples further apart in the latent space.
  • Examples: SGL, SimGCL.
  • Limitation: These methods primarily pursue uniformity (spreading embeddings more evenly) as a general objective for representation quality. While this can indirectly alleviate popularity bias by preventing popular items from completely dominating the latent space, they do not directly address the specific, systematic geometric distortion that popularity imprints on the embedding space. Their solution to popularity bias is indirect and incomplete.
  • DDC's Differentiation: DDC identifies a principal "popularity direction" and applies a targeted, asymmetric correction, which is a more direct and fundamental approach to realigning the space to disentangle preference from popularity.

3.3. Technological Evolution

The field of recommender systems has evolved from early methods like basic collaborative filtering and matrix factorization to sophisticated deep learning models, particularly Graph Neural Networks (GNNs) like LightGCN. Throughout this evolution, the core objective of personalization has remained, but new challenges have emerged or become more apparent. The BPR loss has become a dominant training objective for implicit feedback due to its effectiveness in ranking tasks.

However, as models became more complex and data more abundant, the subtle ways bias can infiltrate latent representations became critical. Initially, popularity bias was often tackled retrospectively or externally, trying to adjust outputs or re-weight inputs. The evolution has moved towards understanding the internal mechanisms of bias generation. This paper represents a significant step in this direction, moving beyond macroscopic interventions to a deep analysis of how the fundamental optimization objective (BPR) itself sculpts the embedding space in a biased way. DDC fits into this lineage by offering a principled, geometric correction that can be applied as a fine-tuning step to existing, powerful BPR-trained CF models.

3.4. Differentiation Analysis

Compared to the main methods in related work, DDC offers several core differences and innovations:

• Intrinsic vs. External Perspective: The most significant differentiation is DDC's fundamental perspective on popularity bias. Unlike most prior methods that treat bias as an external confounding factor (e.g., IPS for data re-weighting, regularization for score modification, causal methods for external modeling), DDC proves that popularity bias is an intrinsic geometric artifact of BPR optimization. This re-framing shifts the focus from managing symptoms to addressing the root cause within the model's latent representation learning.
• Geometric Source Correction: DDC directly identifies and corrects the geometric distortion in the embedding space. It pinpoints a dominant "popularity direction" that BPR inherently creates. Previous methods, even disentanglement or contrastive learning, might indirectly or partially mitigate this, but they don't explicitly target and mathematically derive this specific geometric artifact and its systematic correction.
• Asymmetric Directional Updates: DDC's core mechanism is asymmetric directional updates. It surgically disentangles preference from popularity by guiding positive interactions along personalized preference directions and simultaneously steering negative interactions away from the global popularity direction. This is a more refined and targeted approach than general debiasing strategies or uniform embedding space regularization (like in contrastive learning).
• Model-Agnostic and Plug-and-Play: DDC is designed as a universally applicable framework that can be applied as a fine-tuning stage to any existing BPR-based CF model without requiring architectural modifications. This makes it highly practical and broadly usable, in contrast to methods that might require significant changes to the model architecture or complex training schemes.
• Efficiency and Effectiveness: The paper demonstrates that DDC not only achieves superior recommendation quality and fairness but also dramatically reduces BPR training loss and accelerates convergence. This suggests a more efficient optimization dynamic by moving user embeddings out of suboptimal local minima, which is a tangible benefit beyond just debiasing.

4. Methodology

4.1. Principles

The core idea of the Directional Decomposition and Correction (DDC) method stems from a rigorous mathematical analysis revealing that popularity bias is an intrinsic geometric artifact of Bayesian Pairwise Ranking (BPR) optimization in collaborative filtering (CF) models. The theoretical basis is that BPR systematically organizes item embeddings along a dominant "popularity direction" where embedding magnitudes directly correlate with interaction frequency. This geometric distortion forces user embeddings to handle two conflicting tasks: expressing genuine preference and calibrating against global popularity. The principle of DDC is to resolve this conflict by decoupling these two tasks within the BPR optimization process through asymmetric directional updates.

Specifically, DDC's principles are:

1. Identify the Popularity Direction: Mathematically derive and empirically confirm the existence of a dominant "popularity direction" in the latent space where popular items' embeddings are aligned.
2. Recognize Gradient Sub-optimality: Show that the standard BPR gradient for user embeddings is a confounded mixture of genuine preference and global popularity signals, leading to inefficient optimization and suboptimal local minima.
3. Decouple and Correct: Introduce an asymmetric update mechanism where updates related to positive interactions (items a user likes) are guided along a personalized preference direction, while updates related to negative interactions (items a user doesn't like or hasn't interacted with) are used to calibrate against the global popularity direction. This disentangles the conflicting signals at their geometric source.
4. Low-dimensional Correction: Achieve this decoupling by learning only two scalar coefficients per user, representing movements along pre-defined preference and popularity directions, thus avoiding increased embedding dimensionality or model complexity.

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

The DDC framework operates as a fine-tuning stage on top of an existing BPR-based CF model.

4.2.1. Problem Formulation

The paper considers a standard collaborative filtering problem with implicit feedback.

• Let $\mathcal{U}$ be the set of all users and $\mathcal{I}$ be the set of all items.
• $\mathcal{R} \subseteq \mathcal{U} \times \mathcal{I}$ represents the set of historical user-item interactions (implicit feedback). If $(u, i) \in \mathcal{R}$, user $u$ has interacted with item $i$.
• The goal is to predict a personalized ranked list of items for each user from the unobserved set.
• Embedding-based models learn low-dimensional embedding vectors $\mathbf{e}_u \in \mathbb{R}^d$ for each user $u$ and $\mathbf{e}_i \in \mathbb{R}^d$ for each item $i$, where $d$ is the latent dimension.
• The preference score $\hat{y}_{ui}$ (similarity between user $u$ and item $i$) is computed from their embeddings.

4.2.2. CF Models

The paper discusses two representative CF models that compute preference scores via inner product:

• Matrix Factorization (MF): The preference score $\hat{y}_{ui}$ that user $u$ is predicted to have for item $i$ is calculated as the inner product of their respective embeddings: $$\hat{y}_{ui} = \mathbf{e}_u^T \mathbf{e}_i$$ Here, $\mathbf{e}_u$ is the embedding vector for user $u$, and $\mathbf{e}_i$ is the embedding vector for item $i$. The superscript $T$ denotes the transpose, meaning it's a dot product.

• LightGCN: LightGCN learns user and item embeddings through a graph propagation mechanism on the user-item interaction graph. It starts with initial learnable embeddings $\mathbf{E}^{(0)}$ and refines them across $K$ layers. At each layer $k$, embeddings are aggregated from neighbors: $$\mathbf{E}^{(k)} = (\mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}) \mathbf{E}^{(k-1)}$$ Where $\mathbf{A}$ is the adjacency matrix of the user-item graph (representing interactions), $\mathbf{D}$ is the diagonal degree matrix (containing the number of connections for each node), $\mathbf{E}^{(k-1)}$ are the embeddings from the previous layer, and $\mathbf{E}^{(k)}$ are the updated embeddings. The final representation for user $u$ ($\mathbf{e}_u$) and item $i$ ($\mathbf{e}_i$) is a weighted sum of embeddings from all layers: $$\mathbf{e}_u = \sum_{k=0}^K \alpha_k \mathbf{e}_u^{(k)} \quad \text{and} \quad \mathbf{e}_i = \sum_{k=0}^K \alpha_k \mathbf{e}_i^{(k)}$$ Where $K$ is the number of layers, $\alpha_k$ are layer-wise combination weights, and $\mathbf{e}_u^{(k)}$ and $\mathbf{e}_i^{(k)}$ are the embeddings from layer $k$. Crucially, LightGCN also computes the final preference score using the inner product: $$\hat{y}_{ui} = \mathbf{e}_u^T \mathbf{e}_i$$ This shared final step means the geometric bias investigated by the paper is relevant across various architectures using this scoring mechanism.

4.2.3. Bayesian Pairwise Ranking (BPR) Loss

The BPR loss is the standard objective for implicit feedback. It samples triplets (u, i, j), where $u$ is a user, $i$ is a positive item (interacted with by $u$), and $j$ is a negative item (not interacted with by $u$). The objective is to maximize the score difference $\hat{y}_{ui} - \hat{y}_{uj}$. The BPR loss function is formulated as: $$\mathcal{L}_{BPR} = \sum_{(u,i,j) \in \mathcal{D}} -\ln \sigma(\mathbf{e}_u^T \mathbf{e}_i - \mathbf{e}_u^T \mathbf{e}_j) + \lambda ||\Theta||_2^2$$

• $\mathcal{D}$: The set of all training triplets (u, i, j).

• $\sigma(\cdot)$: The sigmoid function.

• $\mathbf{e}_u^T \mathbf{e}_i$: The predicted preference score for user $u$ and positive item $i$.

• $\mathbf{e}_u^T \mathbf{e}_j$: The predicted preference score for user $u$ and negative item $j$.

• $\lambda$: The L2 regularization hyperparameter.

• $||\Theta||_2^2$: The L2 norm of all learnable model parameters (embeddings).

  This objective encourages $\mathbf{e}_u^T \mathbf{e}_i$ to be greater than $\mathbf{e}_u^T \mathbf{e}_j$, thus ranking positive items higher than negative items.

4.2.4. Analysis of Popularity Bias (Section 3)

The paper's core theoretical contribution is to deconstruct how BPR optimization intrinsically creates popularity bias.

4.2.4.1. The Geometric Imprint of Popularity

First, the paper defines item popularity and identifies a popularity direction in the embedding space.

Definition 3.1 (Item Popularity): The popularity of an item $i$, denoted Pop(i), is its interaction frequency in the training data, i.e., the size of the set of users who have interacted with it. $Pop(i) := |\mathcal{U}_i^+|$ Where $\mathcal{U}_i^+$ is the set of users who have interacted with item $i$.

The popularity direction $\mathbf{e}_{pop}$ is defined empirically. Let $\mathcal{I}_{head}$ and $\mathcal{I}_{tail}$ be the sets of items with the highest and lowest interaction frequencies (e.g., top and bottom $\rho=0.05$ fraction). The popularity direction is the normalized difference between their centroids: $$\mathbf{e}_{pop} := \frac{\mathbf{v}_{diff}}{\left\| \mathbf{v}_{diff} \right\|}, \quad \mathrm{where~} \mathbf{v}_{diff} = \frac{1}{\left| \mathcal{I}_{head} \right|} \sum_{i \in \mathcal{I}_{head}} \mathbf{e}_i - \frac{1}{\left| \mathcal{I}_{tail} \right|} \sum_{i \in \mathcal{I}_{tail}} \mathbf{e}_i$$

• $\mathbf{e}_{pop}$: The normalized vector representing the dominant popularity direction.

• $\mathbf{v}_{diff}$: The difference vector between the centroid (average embedding) of high-popularity items and low-popularity items.

• $\mathcal{I}_{head}$: Set of high-popularity items.

• $\mathcal{I}_{tail}$: Set of low-popularity items.

• $\mathbf{e}_i$: Embedding of item $i$.

• $\| \cdot \|$: L2 norm (magnitude of the vector).

  The paper then empirically confirms a strong positive correlation between an item's popularity Pop(i) and the magnitude of its projection onto $\mathbf{e}_{pop}$: $Pop(i) \propto \mathbf{e}_i^T \mathbf{e}_{pop}$. This is visually demonstrated in Figure 1.

The following figure (Figure 1 from the original paper) geometrically visualizes and quantifies popularity bias in BPR-based CF models.

该图像是示意图，展示了基于BPR的协同过滤模型中的流行度偏差（图1）。(a) 图中展示了项目嵌入在二维空间中的投影，嵌入按主流行度方向 $( \mathbf { e } _ { p o p } )$ 结构性组织。(b) 项目嵌入在该方向上的投影幅度与实际流行度呈近乎完美的线性相关性（Pearson相关系数 $r = 0.99$），定量确认了几何偏差的存在。

The mathematical reason for this geometric imprint is derived from the BPR gradient for item embeddings. For each positive interaction (u, i), the BPR loss gradient pulls $\mathbf{e}_i$ towards the user embedding $\mathbf{e}_u$. The expected update for item $i$ is proportional to the sum of user embeddings who interacted with it: $$\mathbb{E}[\Delta \mathbf{e}_i] \propto \sum_{u \in \mathcal{U}_i^+} \mathbf{e}_u = Pop(i) \cdot \mathbb{E}_{u \in \mathcal{U}_i^+} [\mathbf{e}_u]$$

• $\mathbb{E}[\Delta \mathbf{e}_i]$: Expected change in item $i$'s embedding.

• $\mathcal{U}_i^+$: Set of users who interacted with item $i$.

• Pop(i): Popularity of item $i$.

• $\mathbb{E}_{u \in \mathcal{U}_i^+} [\mathbf{e}_u]$: Expected (average) embedding of users who interacted with item $i$.

  The key insight is:

• Popular Items: For popular items, which are interacted with by a large, diverse set of users, the Law of Large Numbers dictates that the average embedding of these users, $\mathbb{E}_{u \in \mathcal{U}_i^+} [\mathbf{e}_u]$, will approximate the global average user embedding (denoted as $\mathbf{v}$ in the appendix). This means popular items are consistently pulled in a stable, common direction (the mean-user direction).

• Niche Items: For niche items, liked by a small, specific group, their average user embedding is idiosyncratic and not aligned with the global average.

  Consequently, popular items receive powerful and consistently directed updates, forcing them to align along a common axis ($\mathbf{e}_{pop}$). The magnitude of this alignment is directly proportional to their popularity. This process embeds an item's global popularity as the principal axis of variation in the latent space.

4.2.4.2. Theoretical Analysis of BPR Gradient Sub-optimality

The existence of a strong popularity direction $\mathbf{e}_{pop}$ distorts the optimization for user embeddings. For a training triplet (u, i, j), the BPR gradient with respect to the user embedding $\mathbf{e}_u$ is: $$\nabla_{\mathbf{e}_u} \mathcal{L}_{uij} = - \sigma \big( - \mathbf{e}_u^T (\mathbf{e}_i - \mathbf{e}_j) \big) \big( \mathbf{e}_i - \mathbf{e}_j \big)$$

• $\nabla_{\mathbf{e}_u} \mathcal{L}_{uij}$: Gradient of the BPR loss with respect to user $u$'s embedding for a specific triplet (u, i, j).

• $\sigma(\cdot)$: Sigmoid function.

• $\mathbf{e}_u^T (\mathbf{e}_i - \mathbf{e}_j)$: Score margin between positive item $i$ and negative item $j$ for user $u$.

• $(\mathbf{e}_i - \mathbf{e}_j)$: Difference vector between the positive item and negative item embeddings.

  The expected total gradient for user $u$ (over all their positive items $i \in \mathcal{I}_u^+$ and negative items $j \notin \mathcal{I}_u^+$) can be decomposed. Let $w_{uij} = \sigma( - \mathbf{e}_u^T (\mathbf{e}_i - \mathbf{e}_j) )$ be the scalar coefficient from the loss derivative. The expected gradient is: $$\nabla_{\mathbf{e}_u} \mathcal{L}_u = \mathbb{E}_{i,j} \left[ w_{uij} \mathbf{e}_j \right] - \mathbb{E}_{i,j} \left[ w_{uij} \mathbf{e}_i \right]$$

• $\mathbb{E}_{i,j} \left[ w_{uij} \mathbf{e}_j \right]$ (Negative Sample Contribution): This term is an expectation over the vast set of unobserved items $j$. Due to data sparsity, this set approximates the global item population. The average embedding $\mathbb{E}_j [\mathbf{e}_j]$ is dominated by the popularity component and thus aligns strongly with $\mathbf{e}_{pop}$. This term provides a consistent push, moving $\mathbf{e}_u$ away from the popularity direction, acting as a popularity calibration signal.

• $\mathbb{E}_{i,j} \left[ w_{uij} \mathbf{e}_i \right]$ (Positive Sample Contribution): This term is an expectation over the user's interaction history $\mathcal{I}_u^+$. Ideally, it should represent the user's unique taste (a preference signal). However, if the user has interacted with even a few popular items, their large-magnitude embeddings (aligned with $\mathbf{e}_{pop}$) will disproportionately influence this sum. This "contaminates" the preference signal, pulling the gradient towards the global popularity direction $\mathbf{e}_{pop}$, even if the user's true taste is for niche items.

  The ideal update direction for user $u$, denoted $\mathbf{d}_u^*$, should be proportional to $\left( \mathbb{E}_{i \in \mathcal{I}_u^+} [\mathbf{e}_i] - \mathbb{E}_{j \notin \mathcal{I}_u^+} [\mathbf{e}_j] \right)$. However, due to popularity contamination in the positive term, the actual BPR gradient is misaligned with $\mathbf{d}_u^*$, leading to sub-optimality.

4.2.4.3. The Nature of the Conflict

The standard BPR framework forces a single user embedding $\mathbf{e}_u$ to perform two distinct and often contradictory tasks simultaneously:

1. Preference Expression: To rank liked items highly, $\mathbf{e}_u$ must have a high dot product with their embeddings $\mathbf{e}_i$. This requires $\mathbf{e}_u$ to align with the barycenter (average) of the user's true positive items.

2. Popularity Calibration: To rank un-interacted items lowly, $\mathbf{e}_u$ must minimize dot products with their embeddings $\mathbf{e}_j$. Since the centroid of un-interacted items aligns with the popularity direction, this requires $\mathbf{e}_u$ to be orthogonal to or pushed away from $\mathbf{e}_{pop}$.

   When a user's true preference is for niche items, their ideal preference direction is not aligned with $\mathbf{e}_{pop}$. The confounded BPR gradient forces the update for $\mathbf{e}_u$ into a compromised direction, leading to inefficient updates and suboptimal local minima.

4.2.4.4. Theoretical Basis for Decoupling

The paper proposes that the solution lies in decoupling these two tasks. The standard BPR loss uses the same vector $\mathbf{e}_u$ for both positive and negative terms: $$\mathcal{L}_{uij} = -\ln \sigma (\mathbf{e}_u^T \mathbf{e}_i - \mathbf{e}_u^T \mathbf{e}_j)$$ The paper suggests reframing this by using two separate (but initially identical) user vectors: $$\mathcal{L}_{uij} = -\ln \sigma ((\mathbf{e}_u^{pos-term})^T \mathbf{e}_i - (\mathbf{e}_u^{neg-term})^T \mathbf{e}_j)$$ Where initially $\mathbf{e}_u^{pos-term} = \mathbf{e}_u^{neg-term} = \mathbf{e}_u$. The core flaw of BPR is forcing these two to be identical throughout the learning process. The proposed solution is to allow each term to evolve independently for its specialized task. This provides the theoretical justification for the asymmetric update rule in DDC.

4.2.5. Directional Decomposition and Correction (DDC) (Section 4)

DDC is a fine-tuning framework that rectifies the distorted embedding geometry. It decouples the correction of the monolithic user embedding into two targeted, one-dimensional updates along a personalized preference axis and the global popularity axis.

The following figure (Figure 2 from the original paper) conceptually illustrates the proposed Directional Decomposition and Correction (DDC) framework.

该图像是示意图，展示了所提出的方向分解和校正（DDC）框架的概念。图(a)构建了全局流行方向 $\mathbf{e}_{pop}$，表示高流行项目和低流行项目的均值嵌入之间的差向量。图(b)为每个用户构建个性化偏好方向 $\mathbf{e}_{pref}$，基于用户的交互历史。图(c)展示了原始BPR更新方向为偏好和流行信号的混合。图(d)说明了DDC如何沿 $\mathbf{e}_{pref}$ 和 $\mathbf{e}_{pop}$ 轴修改更新方向，以更好地符合用户真实偏好，同时进行流行校正。

4.2.5.1. Decoupling the BPR Update

In the fine-tuning stage, the original model's user embeddings $\mathbf{e}_{u,orig}$ and item embeddings $\mathbf{e}_i^*$ are frozen. For each user $u$, DDC learns two scalar coefficients, $\alpha_u$ and $\beta_u$, which control corrections along two pre-defined directions.

• Positive Interaction: Preference Alignment: For a positive pair (u, i), the goal is to reinforce user $u$'s specific taste. The update should move the user embedding towards items they genuinely like. A personalized preference direction $\mathbf{e}_{pref_u}$ for user $u$ is constructed by leveraging the user's groundtruth interaction history. Specifically, the items in user $u$'s history $\mathcal{I}_u^+$ are evaluated using the pre-trained model scores $\hat{y}_{ui} = \mathbf{e}_{u,orig}^T \mathbf{e}_i^*$. Let $S_u^{top} \subset \mathcal{I}_u^+$ be the top $k$ fraction of these items based on these scores. The preference direction is then constructed by averaging the embeddings of these reliable items: $$\mathbf{e}_{pref_u} := \frac{\sum_{i \in S_u^{top}} \mathbf{e}_i^*}{\left\| \sum_{i \in S_u^{top}} \mathbf{e}_i^* \right\|}$$

  • $\mathbf{e}_{pref_u}$: Normalized personalized preference direction for user $u$.

  • $S_u^{top}$: Set of top $k$ fraction of items from user $u$'s interaction history $\mathcal{I}_u^+$, ranked by the pre-trained model scores.

  • $\mathbf{e}_i^*$: Frozen item embedding from the pre-trained model.

  • $\| \cdot \|$: L2 norm for normalization.

    For the positive term in the BPR loss, a modified user embedding is used, which can only be adjusted along this personalized preference direction: $$\mathbf{e}_u^{pos-term} = \mathbf{e}_{u,orig} + \beta_u \mathbf{e}_{pref_u}$$

  • $\mathbf{e}_u^{pos-term}$: The effective user embedding used for the positive term in the DDC loss.

  • $\mathbf{e}_{u,orig}$: The original, frozen user embedding from the pre-trained model.

  • $\beta_u$: A learnable, user-specific scalar that controls the magnitude of movement along the preference direction.

• Negative Interaction: Popularity Calibration: For a negative pair (u, j), the objective is to correctly rank an un-interacted item. This primarily requires calibrating against global popularity. Therefore, the global popularity direction $\mathbf{e}_{pop}$ (derived in Section 4.2.4.1) is used for this task. The effective user embedding for the negative term is: $$\mathbf{e}_u^{neg-term} = \mathbf{e}_{u,orig} + \alpha_u \mathbf{e}_{pop}$$

  • $\mathbf{e}_u^{neg-term}$: The effective user embedding used for the negative term in the DDC loss.
  • $\mathbf{e}_{u,orig}$: The original, frozen user embedding from the pre-trained model.
  • $\alpha_u$: A learnable, user-specific scalar for popularity calibration. Through optimization, $\alpha_u$ is expected to become negative, effectively calibrating the user's score profile away from popularity-biased scoring patterns.
  • $\mathbf{e}_{pop}$: The global popularity direction.

4.2.5.2. DDC Loss Function

By substituting these two asymmetric user representations into the BPR loss formulation (the decoupled version), the DDC fine-tuning objective is formulated. For each user $u$, the optimal scalar coefficients $(\alpha_u, \beta_u)$ are learned by minimizing: $$\mathcal{L}_{DDC} = \sum_{(u,i,j) \in \mathcal{D}} -\ln \sigma \left( (\mathbf{e}_u^{pos-term})^T \mathbf{e}_i^* - (\mathbf{e}_u^{neg-term})^T \mathbf{e}_j^* \right)$$ Substituting the definitions of $\mathbf{e}_u^{pos-term}$ and $\mathbf{e}_u^{neg-term}$: $$\mathcal{L}_{DDC} = \sum_{(u,i,j) \in \mathcal{D}} -\ln \sigma \left( (\mathbf{e}_{u,orig} + \beta_u \mathbf{e}_{pref_u})^T \mathbf{e}_i^* - (\mathbf{e}_{u,orig} + \alpha_u \mathbf{e}_{pop})^T \mathbf{e}_j^* \right)$$

• $\mathcal{L}_{DDC}$: The DDC loss function used for fine-tuning.

• $\mathcal{D}$: The set of training triplets.

• $\mathbf{e}_i^*, \mathbf{e}_j^*$: The frozen item embeddings from the pre-trained model.

  This objective disentangles the learning process. The gradient with respect to $\beta_u$ primarily depends on aligning with positive items (reinforcing preference), while the gradient with respect to $\alpha_u$ primarily depends on calibrating against negative items (correcting for popularity).

4.2.5.3. Final User Embedding

After fine-tuning and learning the optimal scalar coefficients $\alpha_u^*$ and $\beta_u^*$, the final, corrected user embedding for recommendation is constructed by applying both learned corrections to the original embedding: $$\mathbf{e}_u^{final} := \mathbf{e}_{u,orig} + \alpha_u^* \mathbf{e}_{pop} + \beta_u^* \mathbf{e}_{pref_u}$$

• $\mathbf{e}_u^{final}$: The final, corrected user embedding used for making recommendations.

• $\alpha_u^*$: The learned optimal scalar coefficient for popularity calibration.

• $\beta_u^*$: The learned optimal scalar coefficient for preference alignment.

  This framework does not increase the dimensionality of the base model's embeddings. Instead, it provides a principled, low-dimensional correction that guides the user embeddings out of suboptimal minima created by standard BPR, leading to improved recommendation performance.

5. Experimental Setup

5.1. Datasets

The experiments are conducted on three widely-used public benchmark datasets with varying characteristics and sparsity. To ensure data quality, the 10-core setting (meaning users and items with less than 10 interactions are removed) is applied.

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

• Amazon-Book: A dataset from Amazon product reviews, specifically for books. It represents user interactions with books.

• Yelp: A dataset from the Yelp platform, typically involving user reviews/ratings of businesses (e.g., restaurants, shops).

• Tmall: An e-commerce dataset from Tmall, a large online shopping platform, representing user purchase or interaction history with products.

  These datasets are chosen because they are standard benchmarks in recommender systems research, offering diverse domains (e-commerce, reviews) and characteristics (number of users, items, interactions, and high sparsity), making them effective for validating the generality and performance of the proposed method.

5.2. Evaluation Metrics

The paper evaluates top-N recommendation performance using three standard metrics and an additional metric for popularity bias mitigation. For all metrics, higher values generally indicate better performance, except for AvgPop@K where lower values indicate less popularity bias.

1. Recall@K:

   • Conceptual Definition: Recall@K measures the proportion of relevant items (items a user actually interacted with in the test set) that are successfully retrieved within the top-K recommendations. It focuses on how many of the truly relevant items the system managed to "recall" in its top list.
   • Mathematical Formula: $$\text{Recall@K} = \frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} \frac{|\text{Relevant}_u \cap \text{Recommended}_{u,K}|}{|\text{Relevant}_u|}$$
   • Symbol Explanation:
     • $|\mathcal{U}|$: The total number of users.
     • $\text{Relevant}_u$: The set of items user $u$ has interacted with in the test set (ground truth).
     • $\text{Recommended}_{u,K}$: The set of top-K items recommended to user $u$.
     • $|\cdot|$: Denotes the cardinality (number of elements) of a set.

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

   • Conceptual Definition: NDCG@K is a measure of ranking quality that accounts for the position of relevant items in the recommendation list. It assigns higher values to relevant items that appear higher in the list and gives a "discount" to relevant items that appear lower. It's often preferred over Recall when the order of recommendations matters.
   • Mathematical Formula: $$\text{NDCG@K} = \frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} \frac{\text{DCG@K}_u}{\text{IDCG@K}_u}$$ where \text{DCG@K}_u = \sum_{p=1}^K \frac{\text{rel}_p}{\log_2(p+1)} and \text{IDCG@K}_u = \sum_{p=1}^{|\text{Relevant}_u|} \frac{\text{rel}_p}{\log_2(p+1)}
   • Symbol Explanation:
     • $|\mathcal{U}|$: The total number of users.
     • $\text{rel}_p$: The relevance score of the item at position $p$ in the recommendation list (typically 1 if relevant, 0 if not).
     • $\text{DCG@K}_u$: Discounted Cumulative Gain for user $u$ at rank K.
     • $\text{IDCG@K}_u$: Ideal Discounted Cumulative Gain for user $u$ at rank K (DCG of the perfect ranking).
     • $\text{Relevant}_u$: The set of items user $u$ has interacted with in the test set.

3. Mean Reciprocal Rank (MRR@K):

   • Conceptual Definition: MRR@K measures the average of the reciprocal ranks of the first relevant item in a list of top-K recommendations. If the first relevant item is at rank $p$, its reciprocal rank is $1/p$. It's particularly useful when there's only one "correct" or highly important item for each query, or when you want to penalize systems for putting the first relevant item lower down.
   • Mathematical Formula: $$\text{MRR@K} = \frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} \frac{1}{\text{rank}_u}$$
   • Symbol Explanation:
     • $|\mathcal{U}|$: The total number of users.
     • $\text{rank}_u$: The rank of the first relevant item for user $u$ in the top-K recommendation list. If no relevant item is found within K, rank is often considered $\infty$ (contributing 0 to the sum).

4. Average Popularity (AvgPop@K):

   • Conceptual Definition: AvgPop@K measures the average popularity of items recommended in the top-K list across all test users. Popularity here is defined as the interaction frequency in the training data. A lower AvgPop@K indicates that the model is recommending less popular, more diverse items, and is thus more effective at mitigating popularity bias.
   • Mathematical Formula: $$\mathrm{AvgPop@K} = \frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} \frac{\sum_{i \in R_u} \phi(i)}{|R_u|}$$
   • Symbol Explanation:
     • $|\mathcal{U}|$: The total number of users.
     • $R_u$: The set of top-K recommended items for user $u$.
     • $\phi(i)$: The total number of interactions for item $i$ in the training data (i.e., its popularity).
     • $|R_u|$: The number of items in the recommendation list (which is K, or less if some are filtered).

5.3. Baselines

The paper compares DDC against two groups of baselines:

1. Backbone Models: To demonstrate the general applicability of DDC as a plug-and-play module, it is applied to five representative CF models which were trained until convergence using standard BPR loss:

   • MF (Matrix Factorization): A foundational collaborative filtering model.
   • LightGCN: A state-of-the-art Graph Neural Network (GNN)-based model for recommendation.
   • DGCF (Disentangled Graph Collaborative Filtering): A GNN-based model that aims to disentangle different factors of influence (e.g., user intent, item attributes) in the embedding space.
   • NCL (Neighborhood-enriched Contrastive Learning): A GNN-based model that incorporates contrastive learning to improve embedding quality by leveraging neighborhood information.
   • LightCCF (Lightweight Contrastive Collaborative Filtering): Another GNN-based model that uses contrastive learning for collaborative filtering, potentially with a focus on efficiency.

2. Debiasing Methods: To compare DDC's effectiveness in debiasing, it is benchmarked against seven state-of-the-art debiasing methods. For a fair comparison, these methods are applied to two of the backbone models, MF and LightGCN.

   • IPS (Inverse Propensity Scoring): A re-weighting method that adjusts the loss function by down-weighting interactions with popular items.

   • DICE (Disentangling User Interest and Conformity for Recommendation with Causal Embedding): A disentanglement method that attempts to separate user interest from conformity to popular trends.

   • MACR (Model-Agnostic Counterfactual Reasoning for Eliminating Popularity Bias in Recommender System): A causal inference-based method designed to remove popularity bias using counterfactual reasoning.

   • PC (Popularity Correlation): A regularization-based method that penalizes correlations between recommendation scores and item popularity.

   • PAAC (Popularity-Aware Alignment and Contrast for Mitigating Popularity Bias): A recent debiasing method that uses popularity-aware alignment and contrastive learning.

   • DCCL (Disentangled Causal Embedding With Contrastive Learning For Recommender System): Combines disentangled causal embeddings with contrastive learning.

   • TPAB (Temporal Popularity Awareness Bias): A debiasing method that considers the temporal dynamics of popularity distribution shifts.

     DDC is applied as a fine-tuning stage to pre-trained backbones (e.g., LightGCN-DDC).

5.4. Implementation Details

• Batch Size: The training batch size is set to 8192 for all methods.
• Framework: All methods are implemented under the RecBole framework, which is a unified, comprehensive, and efficient library for recommendation algorithms, ensuring a consistent experimental environment.
• Hyperparameter Tuning: A grid search is performed to find optimal hyperparameters for all models (backbones and debiasing baselines).
• Early Stopping: All models are trained until convergence. Early stopping is employed, terminating training if validation performance (MRR@10) does not improve for 50 consecutive epochs. The model achieving the best validation performance is selected for testing.
• Embedding Dimension: The embedding dimension $d$ is set to 64.
• DDC Specific Hyperparameter: For DDC, the primary hyperparameter is $k$, which represents the proportion of a user's interacted items used to construct their personalized preference direction $\mathbf{e}_{pref_u}$. For the main experiments (Tables 2 and 3), this value is uniformly set to $30\%$ ($k=0.3$) to demonstrate robust performance without extensive tuning. The paper later explores the sensitivity to $k$.

6. Results & Analysis

6.1. Core Results Analysis

The experimental results rigorously validate DDC's effectiveness in enhancing recommendation accuracy and mitigating popularity bias.

6.1.1. Effectiveness on Various Backbone Models (RQ1)

The following HTML table transcribes the results from Table 2 of the original paper, showing the performance of five backbone models before and after applying DDC fine-tuning. This table uses HTML to correctly render the merged header rows.

Analysis: The results clearly show that DDC consistently provides substantial improvements across all five backbone models (MF, LightGCN, DGCF, NCL, LightCCF) and all three datasets (Amazon-Book, Yelp, Tmall). The improvements are significant, with MRR@10 gains as high as 29.3% (MF-DDC on Yelp) and MAP@10 gains of 30.5% (MF-DDC on Yelp). This broad applicability, from the classic MF to advanced GNN and contrastive learning architectures, supports the paper's core claim that the geometric distortion caused by BPR is a fundamental and widespread issue. DDC's ability to rectify this distortion serves as a universal and effective solution, unlocking previously trapped performance potential.

6.1.2. Comparison with State-of-the-Art Debiasing Methods (RQ1)

The following HTML table transcribes the results from Table 3 of the original paper, comparing DDC with seven competitive debiasing baselines using MF and LightGCN as base models. This table uses HTML to correctly render the merged header rows.

Analysis: DDC decisively outperforms all other debiasing methods across MF and LightGCN backbones on all datasets.

• Many existing methods, such as IPS (MF-IPS, LightGCN-IPS) and MACR (MF-MACR, LightGCN-MACR), often degrade performance compared to their respective base models. This suggests that macroscopic approaches, like simple re-weighting or complex causal modeling, can be unstable or based on assumptions that don't always hold, leading to a poorer overall recommendation quality. PC also shows significant degradation.
• Other methods like PAAC, DCCL, and TPAB provide some gains over the baselines, but these gains are consistently surpassed by DDC. For instance, on Yelp with LightGCN, LightGCN-DDC achieves an MRR@10 of 0.0860, which is a 10.0% relative improvement over the strongest baseline (LightGCN-TPAB). This superior performance provides strong evidence that DDC's approach of directly identifying and correcting the geometric source of popularity bias is a more fundamental and effective solution than methods that treat its symptoms.

6.1.3. Analysis of Popularity Bias Mitigation (RQ1)

The paper claims that DDC improves recommendation accuracy by mitigating popularity bias at its geometric source. To validate this, the AvgPop@10 metric is used, which calculates the average interaction count of the top-10 recommended items. A lower AvgPop@10 indicates less popular and more diverse recommendations.

The following table transcribes the results from Table 4 of the original paper, showing the impact of DDC on recommendation accuracy and popularity on the Tmall dataset.

Analysis: The results in Table 4 are clear: DDC not only consistently improves recommendation accuracy (MRR@10, NDCG@10) across all backbones but also dramatically reduces the average popularity of recommended items.

• For example, LightGCN-DDC reduces AvgPop@10 by 39.1%, while simultaneously boosting MRR@10 and NDCG@10.
• LightCCF-DDC achieves an even more remarkable 47.2% reduction in AvgPop@10. This provides strong, direct evidence that DDC successfully mitigates popularity bias. The concurrent gains in accuracy and the reduction in popularity demonstrate that DDC is not making a simple trade-off between relevance and novelty. Instead, by correcting the underlying geometric flaw, it enables the model to escape popularity-driven local minima and discover more personalized items, leading to recommendations that are both more accurate and less biased towards popular content.

6.2. Ablation Studies / Parameter Analysis (RQ2, RQ3)

6.2.1. Effectiveness of the Asymmetric Update Rule (RQ2)

To validate the design choices of DDC, a detailed analysis on the Tmall dataset with MF as the backbone is conducted. The paper investigates different update strategies within the DDC loss by varying which corrections (popularity correction $\mathsf{a} = \alpha_u \mathbf{e}_{pop}$ or preference correction $\mathsf{b} = \beta_u \mathbf{e}_{pref_u}$) are applied to the positive term and negative term in the BPR loss. The notation 'pos-term_neg-term' is used. $DDC (b_a)$ represents the proposed method.

The following table transcribes the relevant section from Table 5 of the original paper, specifically focusing on the "Analysis of Asymmetric Update Rule".

Analysis: The results strongly support the design of the asymmetric update rule in DDC.

• Superiority of DDC (b_a): The proposed $DDC (b_a)$ (applying preference correction $\mathsf{b}$ to the positive term and popularity correction $\mathsf{a}$ to the negative term) significantly outperforms all other configurations. This validates its clear separation of concerns, which directly resolves the gradient conflict identified in standard BPR.
• Why DDC (b_a) Excels: This rule directs positive pair updates only along personal preference directions ($\mathbf{e}_{pref_u}$) to reinforce individual taste. Simultaneously, it restricts negative pair updates only to popularity directions ($\mathbf{e}_{pop}$) for global calibration. This disentangled gradient control prevents interference between preference learning and popularity calibration.
• Suboptimal Alternatives:

  • Configurations like $DDC (a_x)$ (e.g., $DDC (a_a)$, $DDC (a_b)$, $DDC (a_ab)$), where popularity correction is applied to positive items, perform poorly. This is because popularity correction suppresses the user's preference signal for items they like, harming personalization.

  • The symmetric rules $DDC (a_a)$ and $DDC (b_b)$ perform comparably to or slightly better than the baseline (MF (BPR Baseline)), but are limited because they use a single type of correction for both preference and popularity, failing to fully decouple the signals.

  • The worst performers, such as $DDC (a_b)$, directly fight against the learning objective, showing how misapplying the corrections can severely damage performance.

  • Variants like $DDC (ab_x)$ or $DDC (x_ab)$, which apply both corrections (a and b) to a single term, effectively re-introduce the confounding effect that DDC aims to eliminate, leading to confused gradients and inferior performance.

    This ablation study unequivocally confirms the necessity and effectiveness of the asymmetric design in DDC for properly disentangling preference and popularity signals.

6.2.2. Effectiveness of Final Embedding Composition (RQ2)

The paper also investigates the contribution of each directional component ($\alpha_u^* \mathbf{e}_{pop}$ and $\beta_u^* \mathbf{e}_{pref_u}$) to the final user embedding (Equation 14).

The following table transcribes the relevant section from Table 5 of the original paper, specifically focusing on the "Analysis of Final Embedding Composition".

Analysis: The results demonstrate that both the popularity calibration and preference alignment components are crucial for optimal performance.

• Removing Preference Alignment (DDC (w/o $\beta_{pref_u}$)): This variant (where the final embedding is $\mathbf{e}_{u,orig} + \alpha_u^* \mathbf{e}_{pop}$) performs significantly worse, nearly collapsing to the baseline MF level (0.0591 MRR@10 vs. 0.0599 for baseline). This confirms that enhancing the true preference signal (via $\beta_u^* \mathbf{e}_{pref_u}$) is the primary driver of improvement in DDC.

• Removing Popularity Correction (DDC (w/o $\alpha_{pop}$)): This variant (where the final embedding is $\mathbf{e}_{u,orig} + \beta_u^* \mathbf{e}_{pref_u}$) also shows a noticeable performance decrease compared to the full DDC (0.0672 MRR@10 vs. 0.0677 for full DDC). This indicates that explicit calibration against global popularity (via $\alpha_u^* \mathbf{e}_{pop}$) is vital for achieving the best results and fully mitigating bias.

  This analysis validates the design of combining both learned corrections to form the final, rectified user embedding, as both components play distinct and necessary roles in improving recommendation quality and debiasing.

6.2.3. Parameter Sensitivity (RQ3)

The paper investigates the sensitivity of DDC to its key hyperparameter, $k$, which is the proportion of a user's most relevant interacted items used to construct their personalized preference direction $\mathbf{e}_{pref_u}$. This analysis is conducted on the Tmall dataset for MF and LightGCN backbones, evaluating MRR@10 and AvgPop@10.

The following figure (Figure 3 from the original paper) shows the parameter sensitivity analysis of the proportion $k$ on the Tmall dataset, displaying its dual impact on recommendation accuracy.

Analysis: The analysis reveals a critical trade-off between accuracy and bias mitigation as a function of $k$.

• Recommendation Accuracy (MRR@10): For both MF-DDC and LightGCN-DDC, MRR@10 follows a concave trend, peaking at intermediate values of $k$ (approximately 0.5 for MF and 0.3 for LightGCN).

  • Small $k$ (e.g., 0.1): Using too few items makes the preference direction $\mathbf{e}_{pref_u}$ noisy and potentially not representative of the user's full interest profile, slightly hurting accuracy.
  • Large $k$ (e.g., 1.0): Including too many (potentially less relevant or popularity-contaminated) items makes the preference direction too generic and pulls it closer to the global popularity distribution, which also degrades personalized recommendation accuracy.

• Popularity Bias (AvgPop@10): For LightGCN-DDC, there is a strong positive correlation between $k$ and AvgPop@10. As $k$ increases, the model recommends significantly more popular items, confirming that a larger $k$ dilutes the personalized signal with global popularity. For MF-DDC, the trend is more subtle, but the lowest popularity bias is achieved around $k=0.5$, which aligns with its peak accuracy.

  Conclusion on $k$: DDC is robust across a reasonable range of $k$. The choice of $k=0.3$ (used in main experiments) is justified as it achieves near-optimal accuracy for both backbones while effectively keeping popularity bias in check, particularly for the more powerful LightGCN model. It represents a good balance in the accuracy-bias trade-off.

6.2.4. Convergence Analysis (RQ3)

The paper analyzes DDC's impact on convergence by tracking BPR loss and MRR@10 over epochs.

The following figure (Figure 4 from the original paper) shows the convergence curves of BPR loss and MRR @ 10 for MF and LightGCN on three datasets.

Analysis: The convergence curves reveal a dramatic improvement in optimization efficiency and outcome with DDC.

• BPR Loss Reduction: The left side of Figure 4 shows that standard BPR training (backbone model without DDC) leads to a slow decrease in loss, eventually plateauing at a relatively high value. However, once the backbone model converges and DDC fine-tuning begins, the BPR loss plummets dramatically.

  • The initial "jump" in loss at the start of the DDC phase is explained by the random initialization of DDC's coefficients ($\alpha_u, \beta_u$), which temporarily creates a deviation from the converged backbone state.
  • The loss shown during the DDC phase is not its specific optimization objective ($\mathcal{L}_{DDC}$) but the original BPR loss calculated using the final corrected embeddings ($\mathbf{e}_u^{final}$). The formula used for this evaluation loss is: $$\mathcal{L}_{eval} = \sum_{(u,i,j) \in \mathcal{D}} -\ln \sigma \left( (\mathbf{e}_u^{final})^T \mathbf{e}_i^* - (\mathbf{e}_u^{final})^T \mathbf{e}_j^* \right)$$ Where $\mathbf{e}_u^{final}$ is the final user embedding after DDC correction, and $\mathbf{e}_i^*, \mathbf{e}_j^*$ are the frozen item embeddings.
  • Quantitatively, for LightGCN on Yelp, the loss drops from 1.5055 (baseline) to 0.0267 (with DDC), which is about 1.8% of the original. For MF on Amazon-Book, the loss drops from 1.2922 to 0.0191 (about 1.5% of original). This represents an extremely efficient optimization trajectory.

• MRR@10 Improvement: The right side of Figure 4 demonstrates that this massive reduction in loss corresponds to a rapid and substantial increase in MRR@10. The DDC fine-tuning quickly surpasses the baseline's peak performance.

  Conclusion on Convergence: This analysis provides powerful evidence that DDC's principled geometric correction allows embeddings to escape the suboptimal local minima created by BPR's geometric bias. By guiding the embeddings to a fundamentally superior solution, DDC achieves a much better representation of true user preference, leading to both significantly lower loss and higher recommendation quality.

7. Conclusion & Reflections

7.1. Conclusion Summary

This work fundamentally re-thinks popularity bias in collaborative filtering (CF) by demonstrating it is an intrinsic geometric artifact of Bayesian Pairwise Ranking (BPR) optimization, rather than an external confounding factor. Through rigorous mathematical analysis, the paper proves that BPR systematically organizes item embeddings along a dominant "popularity direction", where embedding magnitudes are directly correlated with interaction frequency. This geometric distortion forces user embeddings to simultaneously handle preference expression and popularity calibration, leading to suboptimal configurations that favor popular items.

To address this, the paper proposes Directional Decomposition and Correction (DDC), a universally applicable framework. DDC surgically corrects this embedding geometry through asymmetric directional updates. It guides positive interactions along personalized preference directions and steers negative interactions away from the global popularity direction, effectively disentangling preference from popularity at the geometric source.

Extensive experiments across multiple BPR-based architectures and benchmark datasets show that DDC significantly outperforms state-of-the-art debiasing methods. It dramatically reduces BPR training loss (to less than 5% of heavily-tuned baselines) and achieves superior recommendation quality (e.g., higher MRR@10, NDCG@10) and fairness (significantly lower AvgPop@10). The plug-and-play nature and accelerated convergence of DDC further underscore its practical value.

7.2. Limitations & Future Work

The paper does not explicitly dedicate a section to limitations and future work. However, based on the scope and focus of the research, potential limitations and future directions can be inferred:

• Focus on BPR Loss: The core theoretical analysis and DDC framework are specifically designed around the Bayesian Pairwise Ranking (BPR) loss. While BPR is a de facto standard, other loss functions exist (e.g., point-wise losses, list-wise losses, or contrastive losses that are not strictly pairwise). The applicability of DDC's direct geometric correction to models trained with these other loss functions would require further investigation and potentially new theoretical derivations.
• Static Popularity Direction: The global popularity direction $\mathbf{e}_{pop}$ and personalized preference directions $\mathbf{e}_{pref_u}$ are derived once at the beginning of the fine-tuning stage from frozen embeddings. While this simplifies the process, popularity can be dynamic and user preferences can evolve over time. DDC in its current form might not fully capture temporal shifts in popularity or user taste. Future work could explore dynamic updates or adaptive computation of these directions.
• Hyperparameter $k$ for $\mathbf{e}_{pref_u}$: The parameter $k$ (proportion of top items for preference direction) requires tuning and presents a trade-off. While the paper shows robustness across a reasonable range, a more adaptive or learned approach to determine $k$ per user or per item type could be beneficial.
• Interpretability of $\alpha_u, \beta_u$: While $\alpha_u$ and $\beta_u$ are scalar coefficients, their exact interpretation and how they might vary across different user demographics or item categories could be further explored for deeper insights into individual user biases and preferences.
• Generalizability to Other Biases: DDC specifically targets popularity bias. Recommender systems are subject to other biases (e.g., exposure bias, selection bias, conformity bias). Future work could investigate if the directional decomposition principle can be extended or adapted to address multiple types of biases simultaneously or sequentially.
• Computational Cost for Very Large Systems: While DDC is lightweight as a fine-tuning stage, for extremely large user bases, the overhead of learning individual $\alpha_u$ and $\beta_u$ coefficients and computing $\mathbf{e}_{pref_u}$ for every user might still be a consideration, though it's likely minor compared to full model re-training.

7.3. Personal Insights & Critique

This paper offers a highly insightful and elegant solution to a pervasive problem in recommender systems. My personal insights and critique are as follows:

• Elegance of Geometric Interpretation: The core strength of this paper lies in its rigorous and intuitive geometric interpretation of popularity bias. By proving that BPR inherently creates a popularity direction, it moves beyond merely observing the bias to fundamentally understanding its origin within the latent space. This is a powerful shift from symptom-based mitigation to root-cause correction. The visualization in Figure 1 effectively conveys this core insight, making the theoretical claim much more tangible.

• Principled Decoupling: The idea of decoupling the preference signal and popularity calibration into asymmetric directional updates is conceptually sound and mathematically elegant. It addresses the inherent conflict forced upon user embeddings by BPR in a precise and targeted manner. This "surgical" approach is likely why it outperforms more macroscopic debiasing methods.

• Plug-and-Play Nature: The design of DDC as a fine-tuning stage is a significant practical advantage. It means existing, well-performing BPR-based CF models can be enhanced without undergoing architectural redesign or extensive retraining from scratch. This makes the method highly adoptable in real-world systems.

• Dramatic Performance Gains: The sheer magnitude of BPR loss reduction (to less than 5% of baselines) and the consistent, substantial improvements in accuracy and fairness are compelling. This suggests that DDC genuinely helps user embeddings find a "better home" in the latent space, moving beyond suboptimal minima.

• Potential for Broader Impact: The principle of directional decomposition and asymmetric updates could potentially inspire solutions for other types of biases or multi-objective optimization problems in embedding spaces. For instance, one could imagine disentangling other factors like recency, diversity, or specific item attributes from core preference.

• Minor Critique on $\mathbf{e}_{pref_u}$ Construction: While the construction of $\mathbf{e}_{pref_u}$ using top-$k$ pre-trained scores is simple and effective, it still relies on the potentially biased pre-trained model's ranking to define "relevant" items. This might introduce a subtle feedback loop, though the tuning of $k$ likely mitigates it. More robust, or causally-informed, ways to identify genuine preference signals could be an interesting avenue for future refinement. For instance, using explicit feedback where available, or employing techniques that inherently identify causal preference rather than observed interaction.

• Unverified Assumptions: The theoretical proofs rely on assumptions like "non-zero empirical mean" for user embeddings (Assumption A.1) and Law of Large Numbers approximations. While these are common in such analyses, their exact fidelity in all real-world scenarios might vary.

  Overall, this paper presents a significant theoretical and practical advancement in understanding and mitigating popularity bias in collaborative filtering. Its geometric perspective and elegant solution provide a robust foundation for future research in debiased recommendation.