Exploring the Individuality and Collectivity of Intents behind Interactions for Graph Collaborative Filtering

Many studies have demonstrated that the GCN-based embedding encoding process (Kipf and Welling, 2017) can fully utilize the rich collaborative relationships on the user-item interaction graph $\mathcal{G}$ (He et al., 2020; Wu et al., 2021). Considering that BIGCF is a model-agnostic recommendation framework, we empirically employ lightweight GCN (He et al., 2020) to encode high-order collaborative relationships:

where $\mathbf{e}_{u,\boldsymbol{\mu}}^{(l)}$ and $\mathbf{e}_{i,\boldsymbol{\mu}}^{(l)}$ are $l$-th layer ($l\in[1,L]$) user and item structural embeddings, respectively (we define $\mathbf{e}_{u,\boldsymbol{\mu}}^{(0)}=\mathbf{e}_{u}^{(0)}$ and $\mathbf{e}_{i,\boldsymbol{\mu}}^{(l)}=\mathbf{e}_{i}^{(0)}$), and $\mathcal{S}_{u}$ and $\mathcal{S}_{i}$ are the one-order receptive field of user $u$ and item $i$, respectively. From a macroscopic view, the process is a cumulative multiplication of the graph Laplace matrix: $\mathbf{E}_{\boldsymbol{\mu}}^{(l)}=\left(\mathbf{D}^{-0.5}\mathbf{A}\mathbf{D}^{-0.5}\right)^{l}\mathbf{E}^{(0)}$, where $\mathbf{D}$ is the diagonal degree matrix of $\mathbf{A}$, and $\mathbf{A}\in\mathbb{R}^{(M+N)\times(M+N)}$ is the adjacent matrix of $\mathcal{G}$. After $L$ layers of propagation, we empirically construct the final structural embeddings by sum pooling for speed: $\mathbf{e}_{u,\boldsymbol{\mu}}={\textstyle\sum_{l=0}^{L}\mathbf{e}_{u,\boldsymbol{\mu}}^{(l)}}$ and $\mathbf{e}_{i,\boldsymbol{\mu}}={\textstyle\sum_{l=0}^{L}\mathbf{e}_{i,\boldsymbol{\mu}}^{(l)}}$. Considering the variability of neighbor contributions to the central node at each order on the interaction graph $\mathcal{G}$, the computation of $\mathbf{E}_{\boldsymbol{\mu}}=\{\mathbf{E}_{\mathcal{U},\boldsymbol{\mu}},\mathbf{E}_{\mathcal{I},\boldsymbol{\mu}}\}$ is included in each layer to capture user preferences and item characteristics with different structural semantics.

We now switch perspectives to investigate the correspondence between intents and interactions. Given the unique preferences of the user $u$ and the unique characteristics of the item $i$, we need to additionally consider the adaptability between intents and interactions. Specifically, we first measure the correlation scores between the structural embeddings $\{\mathbf{e}_{u,\boldsymbol{\mu}},\mathbf{e}_{i,\boldsymbol{\mu}}\}$ and the collective intents $\{\mathbf{C}_{\mathcal{U}},\mathbf{C}_{\mathcal{I}}\}$:

where $\kappa\in[0,1]$ is a temperature coefficient used to adjust the predicted distribution. In practice, the collective intent $\{\mathbf{C}_{\mathcal{U}},\mathbf{C}_{\mathcal{I}}\}$ is utilized to compute the affinity between the interaction and each intent $k\in\mathcal{K}$, reflecting the individuality and enabling deeper integration of disentangled intents with the graph structure:

where $\mathbf{e}_{u,\boldsymbol{\sigma}}$ and $\mathbf{e}_{i,\boldsymbol{\sigma}}$ are the individual intent embeddings of user $u$ and item $i$, respectively. In order to obtain the approximate posterior that can fit the user’s preferences, we regard the encoded structural embeddings and the individual intent embeddings as the variational parameters for distributions defined in Eq. 3. Considering the additivity of the Gaussian distributions (Ho et al., 2020), varying intents are seen as a combination of approximate Gaussians to adequately fit the individualized profile of user $u$ (or item $i$):

The item side has a similar definition. The mean $\mathbf{e}_{u,\boldsymbol{\mu}}$ represents the individual preferences for user $u$, which are independent of intents, while the variance $\mathbf{e}_{u,\boldsymbol{\sigma}}$ is a linear combination of distributions generated by multiple intents, which provides the variance with a variety of implications. For example, the user $u_{1}$ in Fig. 1(b) will contain more correlation scores from intents $c_{1}$ and $c_{2}$. And the intent $c_{1}$ should have the highest score because the primary premise for choosing this movie is the horror theme. Thus we can assume that $\mathbf{e}_{u_{1},\boldsymbol{\sigma}}=0.6c_{1}+0.3c_{2}+0.1c_{3}$.

Given a user $u$ and an item $i$, the mean and variance embeddings of the approximate posterior can be used to sample $\mathbf{e}_{u}\sim\mathcal{N}(\boldsymbol{\mu}_{u},\boldsymbol{\sigma}_{u}^{2})$ and $\mathbf{e}_{i}\sim\mathcal{N}(\boldsymbol{\mu}_{i},\boldsymbol{\sigma}_{i}^{2})$. Considering the non-differentiable nature of the sampling process, reparameterization trick (Kingma and Welling, 2013) is used here to cleverly circumvent the back-propagation problem:

where $\boldsymbol{\epsilon}_{u}$ and $\boldsymbol{\epsilon}_{i}$ are auxiliary noise variables sampled from $\mathcal{N}(\mathbf{0},\mathbf{I})$, and $\odot$ denotes an element-wise product. We emphasize the advantages of introducing reparameterization in the following ways: (1) $\boldsymbol{\epsilon}_{u}$ and $\boldsymbol{\epsilon}_{i}$ achieve fusion in an adaptive manner, thus avoiding tedious manual adjustments; (2) Eq. 8 does not depend on any additional process, such as attention mechanisms. Combining Eqs. 4, 5, 6, 7, and 8, we can obtain:

Eq. 9 further highlights the fusion process of structural information and intents based on the reparameterization process. It is important to note that the intent $\mathbf{C}_{\mathcal{U}}$ ($\mathbf{C}_{\mathcal{I}}$) is common to all users (items) and represents the collectivity of users (items), however the variance vector $\mathbf{e}_{u,\boldsymbol{\sigma}}$ ($\mathbf{e}_{i,\boldsymbol{\sigma}}$) is unique and represents the individuality of user $u$ (item $i$) on the interaction graph $\mathcal{G}$.

Based on the fused embeddings, we compute the probability scores of the existence of interaction edges $R_{ui}$ via inner product:

There is evidence from a number of recent research that self-supervised learning (SSL) provides extra supervised signals to assist ease the data sparsity problem, and is positively motivating for recommendation tasks (Wu et al., 2021). Contrastive learning (Chen et al., 2020), which builds a pair of views of anchor nodes via data augmentation and maximizes the mutual information between them, is generally the most popular strategy for self-supervised learning. Given arbitrary node $a$, the widely used contrastive loss infoNCE (Chen et al., 2020) is defined as follows:

where $\mathbf{a}^{\prime}$ and $\mathbf{a}^{\prime\prime}$ are two augmented embeddings of $a$, $\text{cos}(\cdot,\cdot)$ means the cosine similarity, and $\tau$ is a predefined temperature coefficient. Note that the other nodes $\{b\in\mathcal{B}\}$ in a mini-batch are considered as negative samples of node $a$ in Eq. 11 for simplicity. Most of existing recommendation paradigms adopt various types of data augmentation to perform SSL task, such as structural (Wu et al., 2021; Yang et al., 2022), feature (Lin et al., 2022), or semantic augmentation (Ren et al., 2023). Nevertheless, these augmentation techniques not only add significantly to the training cost but also have a high propensity to distort the semantic information of input data. For example, removing key node structures may destroy the original interaction graph $\mathcal{G}$.

Noting that the core of infoNCE lies in uniformizing the feature space (Wang and Isola, 2020; Wang and Liu, 2021), we regard the contrastive process $\mathcal{I}(\mathbf{a},\mathbf{b})$ as a regularization of all nodes $\mathcal{V}$ on the interaction graph $\mathcal{G}$. Specifically, regularization in the interaction space consists of two parts:

• $\bullet$

  Interaction Regularization (IR) $\mathcal{I}(\mathbf{e}_{u},\mathbf{e}_{i})$: Considering that the essence of recommendation lies in quantifying the user-item similarity, it is necessary to ensure a closer distance between the user $u$ and the positive interaction $i$, and it is also needed to keep the user $u$ as far away as possible from the negatives $\{j\in\mathcal{B}/i\}$.

• $\bullet$

  Homogeneous Node Regularization (HNR) $\mathcal{I}(\mathbf{e}_{u},\mathbf{e}_{u})$ and $\mathcal{I}(\mathbf{e}_{i},\mathbf{e}_{i})$: The Matthew effect caused by the popularity bias (Wei et al., 2021) also needs to be taken into account to prevent the learned embeddings from concentrating close to one direction, which would result in dimensional collapse (Jing et al., 2021). We keep the nodes themselves (e.g., $u$) as positive samples while keeping different nodes (e.g., $\{v\in\mathcal{B}/u\}$) far away from each other.

Based on the above analysis, we propose the graph contrastive regularization process in the interaction space:

where $\mathbf{e}_{u}$ and $\mathbf{e}_{i}$ are the user and item embeddings obtained via Eq. 9. We emphasise the superiority of Eq. 12 in two respects. Firstly, the loss $\mathcal{L}_{\text{inter}}$ does not depend on any form of data augmentation, which not only significantly reduces the time complexity but also ensures the integrity of the structural information. Secondly, Eq. 12 guarantees alignment between positive interaction pairs while keeping the entire interaction feature space uniform, whereas traditional CL strategies do not take into account the alignment relationships between users and items.

• $\bullet$

  Bilateral Intent Regularization (BIR) $\mathcal{I}(\mathbf{e}_{u,\boldsymbol{\sigma}},\mathbf{e}_{u,\boldsymbol{\sigma}})$ and $\mathcal{I}$ $(\mathbf{e}_{i,\boldsymbol{\sigma}},\mathbf{e}_{i,\boldsymbol{\sigma}})$: In the intent space, we obtain the individual intent embeddings $\{\mathbf{e}_{u,\boldsymbol{\sigma}},\mathbf{e}_{i,\boldsymbol{\sigma}}\}$ of user $u$ and item $i$ through the collective intents $\{\mathbf{C}_{\mathcal{U}},\mathbf{C}_{\mathcal{I}}\}$. Considering that intent embeddings $\{\mathbf{e}_{u,\boldsymbol{\sigma}},\mathbf{e}_{i,\boldsymbol{\sigma}}\}$ indicate different motivations (reasons) for user $u$ (item $i$) to select (be selected) items (by users), if an intent $k$ can be represented by other intents $\{k^{\prime}\in\mathcal{K}/k\}$, then such an intent $k$ may be redundant.

Based on this, we further model disentangled intent representations via graph contrastive regularization process in the intent space:

Similar to Eq. 12, $\mathcal{L}_{\text{intent}}$ also does not rely on any form of data augmentation. The process of graph contrastive regularization in both interaction and intent spaces is shown in Fig. 3. Notice that the computational chains of final interaction embeddings $\{\mathbf{e}_{u},\mathbf{e}_{i}\}$ and individual intent embeddings $\{\mathbf{e}_{u,\boldsymbol{\sigma}},\mathbf{e}_{i,\boldsymbol{\sigma}}\}$ contain the collective intent $\{\mathbf{C}_{\mathcal{U}},\mathbf{C}_{\mathcal{I}}\}$ and the structural embeddings $\{\mathbf{e}_{u,\boldsymbol{\mu}},\mathbf{e}_{i,\boldsymbol{\mu}}\}$, which allows the proposed graph contrastive regularization process to achieve cross-domain collaboration in dual spaces. We will further analyze in Subsection 2.4.2.

The time complexity of BIGCF mainly comes from the graph structure encoding and the intent modeling. Specifically, since BIGCF uses LightGCN (He et al., 2020) as the basic graph encoder, the time complexity of the process is $O(2|\mathcal{E}|Ld)$, where $|\mathcal{E}|$ is the number of edges on the interaction graph $\mathcal{G}$, and the time complexity of the intent modeling part is $O(|\mathcal{K}||\mathcal{V}|d)$, where $|\mathcal{V}|=M+N$ is the number of nodes of the interaction graph $\mathcal{G}$. Therefore, the total time complexity of BIGCF during training is $O(2|\mathcal{E}|Ld+|\mathcal{K}|\mathcal{|}V|d)$. Table 1 presents the horizontal comparisons of the time complexity of BIGCF with other same-type methods. $\rho$ is the the edge keep probability of SGL-ED (Wu et al., 2021), $H$ is the number of hyperedges in HCCF (Xia et al., 2022), and $q$ is the required rank for SVD in LightGCL (Cai et al., 2023). We have the following findings:

• $\bullet$

  The time complexity of BIGCF is slightly higher than that of LightGCN due to the modeling process of user and item intents.

• $\bullet$

  The time complexity of BIGCF has a significant advantage over SSL-based methods because the intent modeling is independent of the interaction graph $\mathcal{G}$ and the number of layers $L$.

• $\bullet$

  BIGCF introduces the idea of GCL to achieve contrastive regularization in both interaction and intent spaces. The process does not require any additional operations for graph or feature augmentation to construct multi-views, thus significantly reducing the time complexity of model training (e.g., additional graph augmentation is required for SGL-ED (Wu et al., 2021) and DCCF (Ren et al., 2023), and SVD-based preprocessing is required for LightGCL (Cai et al., 2023)).

In this section, we present the positive promotion of the graph contrastive regularization paradigm for BIGCF. Specifically, the graph contrastive regularization process achieves alignment and uniformity in dual spaces with the help of measuring the mutual information $\mathcal{I}(\mathbf{a},\mathbf{b})$ between vectors $\mathbf{a}$ and $\mathbf{b}$. In the interaction space, for the regularization process of user $u$ and item $i$ in Eq. 12, the gradient of user $u$ is as follows (Wu et al., 2021):

where $c(u)$ and $c(j)$ denote the contribution of user $u$ and negative sample ${j}$ to the gradients, respectively. For a large number of negative samples, the $L_{2}$ norm of $c(j)$ is defined as follows:

The above equation shows that the gradient optimization of $c(j)$ ultimately depends on the cosine similarity. For hard-negative samples $\{j\in\mathcal{B}/<u,i>\}$, the closer the similarity to $1.0$ will result in a significant increase for the $L_{2}$ norm $\left\|c(j)\right\|_{2}$. The above effects will produce a surprisingly exponential increase when moderated by smaller temperature coefficient $\tau$, resulting in a significant improvement in training efficiency (Wang and Liu, 2021).

Noting that in the second and third terms of both Eqs. 12 and 13, the two input parameters are identical (e.g., $\mathcal{I}(\underline{\mathbf{e}_{u},\mathbf{e}_{u}})$), we offer the following derivation:

This indicates that $\mathcal{I}(\mathbf{e}_{u},\mathbf{e}_{u})$ actually measures the uniformity among different nodes in a batch. The above corollary also holds for the graph contrastive regularization of item embedding $\mathbf{e}_{i}$ and intent embeddings $\{\mathbf{e}_{u,\boldsymbol{\sigma}},\mathbf{e}_{i,\boldsymbol{\sigma}}\}$. With gradient optimization, nodes with greater similarity will be penalized more, thus forcing them to move away from each other and making the feature space uniform:

• $\bullet$

  For interaction embeddings, a more uniform feature space ensures that cold-start items have more chances to be recommended.

• $\bullet$

  For intent embeddings, a more uniform feature space will ensure that the redundancy among the intents is reduced and better reflects the user’s potential preferences.

Furthermore, given final interaction embeddings ($\mathbf{e}_{u}$ and $\mathbf{e}_{v}$) and individual intent embeddings ($\mathbf{e}_{u,\boldsymbol{\sigma}}$ and $\mathbf{e}_{v,\boldsymbol{\sigma}}$) in a mini-batch, we have the following derivations:

The above equation demonstrates the GCR process across the dual feature spaces, i.e., the process of calculating $\text{cos}(\mathbf{e}_{u},\mathbf{e}_{v})$ will rely on the intent embeddings $\mathbf{e}_{u,\boldsymbol{\sigma}}$ and $\mathbf{e}_{,\boldsymbol{\sigma}}$, while the process of calculating $\text{cos}(\mathbf{e}_{u,\boldsymbol{\sigma}},\mathbf{e}_{,\boldsymbol{\sigma}})$ will take into account the interaction embeddings $\mathbf{e}_{u,\boldsymbol{\mu}}$ and $\mathbf{e}_{u,\boldsymbol{\mu}}$. The direct (solid line) or non-direct (dashed line) participation process of various types of embeddings executing the graph contrastive regularization is also presented in Fig. 3.

To validate the effectiveness of BIGCF, we adopt three widely used large-scale recommendation datasets: Gowalla, Amazon-Book, and Tmall (Ren et al., 2023), which are varied in scale, field, and sparsity. Table 2 provides statistical information for three datasets. To ensure fairness and consistency, we adopt the same processing method with existing efforts (He et al., 2020; Ren et al., 2023). Specifically, all explicit feedback is forced to be converted to implicit feedback (i.e., ratings are only 0 and 1). Items that a user has interacted with are considered positive samples, and other items apart from that are seen as negative samples for that user. We measure the performance of all recommendation models via Recall@K and NDCG@K (Wang et al., 2019; Zhang et al., 2023).

To validate the effectiveness of BIGCF, we choose the following state-of-the-art recommendation methods for comparison experiments:

• $\bullet$

  Only factorization-based method: MF (Rendle et al., 2009).

• $\bullet$

  Generative-based methods: Mult-VAE (Liang et al., 2018) and CVGA (Zhang et al., 2023).

• $\bullet$

  GCN-based methods: NGCF (Wang et al., 2019) and LightGCN (He et al., 2020).

• $\bullet$

  Intent modeling-based methods: DisenGCN (Ma et al., 2019a), DisenHAN (Wang et al., 2020b), MacridVAE (Ma et al., 2019b), DGCF (Wang et al., 2020a), DICE (Zheng et al., 2021), and DGCL (Li et al., 2021).

• $\bullet$

  SSL-based methods: SGL-ED (Wu et al., 2021), HCCF (Xia et al., 2022), LightGCL (Cai et al., 2023), and DCCF (Ren et al., 2023).

We implement BIGCF in PyTorch¹¹1https://github.com/BlueGhostYi/BIGCF. For a fair comparison, we use an experimental setup consistent with previous works (Ren et al., 2023). Specifically, the embedding size and batch size of all models are set to 32 and 10240, respectively. the default optimizer is Adam (Kingma and Ba, 2014), and initialization is done via the Xavier method (Glorot and Bengio, 2010). For all comparison methods, we determine the optimal hyperparameters based on the optimal settings given in the paper as well as the grid search. For BIGCF, the number of GCN layers is set in the range of {1, 2, 3}, the number of intent $|\mathcal{K}|$ is {16, 32, 64, 128}, and we empirically set the temperature coefficients $\kappa$ and $\tau$ to be 1.0 and 0.2, respectively. The weight of graph contrastive regularization $\lambda_{1}$ is set in the range of {0.1, 0.2, 0.3, 0.4, 0.5}, and the weight of $L_{2}$ regularization $\lambda_{2}$ is set in the range of {$1\text{e}^{-3}$, $1\text{e}^{-4}$, $\textbf{1e}^{\textbf{-5}}$, $1\text{e}^{-6}$}. Bolded positions indicate the default optimal settings.

Table 3 shows the performance of BIGCF and all baselines, and we have the following observations:

• $\bullet$

  BIGCF achieves the best performance across all metrics on three datasets. Quantitatively, BIGCF improves by 11.19%, 11.25% and 13.02% over the strongest baseline w.r.t. Recall@20 on Gowalla, Amazon-Book, and Tmall datasets, respectively. The above experimental results demonstrate the rationality and generalizability of the proposed BIGCF.

• $\bullet$

  BIGCF achieves a significant improvement over disentanglement-based approaches (e.g., DGCF (Wang et al., 2020a) and DCCF (Ren et al., 2023)), demonstrating the necessity of modeling intents at a finer granularity.

• $\bullet$

  BIGCF can achieve better performance than the same type of generative methods (e.g., Mult-VAE (Liang et al., 2018)) and SSL-based methods (e.g., SGL-ED (Wu et al., 2021) and LightGCL (Cai et al., 2023)), which demonstrates that the proposed BIGCF can mine user preferences more effectively.

• $\bullet$

  Fig. 4 presents comparisons of the training time of BIGCF with other disentanglement-based methods, and it can be seen that BIGCF has a significant advantage in training efficiency.

To verify whether proposed BIGCF can effectively deal with the data sparsity issue, we divide the testing set into three subsets based on the interaction scale, denoting sparse users, common users, and popular users, respectively. The experimental results are shown in Fig. 5. It is clearly evident that BIGCF achieves improvements on all testing subsets of all datasets. Despite the performance degradation of all methods on sparse groups, BIGCF still shows advantages and gradually increases the gap with the performance of DCCF, which indicates that BIGCF has better sparsity resistance. We attribute this to the proposed bilateral intent modeling as well as to the graph contrastive regularization in dual spaces. On the one hand, the finer-grained intents help to deeply understand users’ real preferences and prevent the recommender system from over-recommending popular items; on the other hand, the graph contrastive regularization further constrains the distribution of the node representations within the feature space as a way of preventing nodes from over-aggregating.

We construct a series of variants to verify the validity of each module in BIGCF:

• $\bullet$

  $\text{BIGCF}_{\text{w/o GCR}}$: remove the Graph Contrastive Regularization;

• $\bullet$

  $\text{BIGCF}_{\text{w/o IR}}$: remove the Interaction Regularization;

• $\bullet$

  $\text{BIGCF}_{\text{w/o HNR}}$: remove the Homogeneous Node Regularization;

• $\bullet$

  $\text{BIGCF}_{\text{w/o BIR}}$: remove the Bilateral Intent Regularization;

• $\bullet$

  $\text{BIGCF}_{\text{w/o BIGR}}$: remove the Bilateral Intent-guided Graph Reconstruction, and use $\mathbf{e}_{u,\boldsymbol{\mu}}$ and $\mathbf{e}_{i,\boldsymbol{\mu}}$ directly for downstream tasks;

• $\bullet$

  $\text{BIGCF}_{\text{w/o PGR}}$: remove the probability-based graph reconstruction (Eq. 9) and construct $\mathbf{e}_{u}$ and $\mathbf{e}_{i}$ by average pooling.

The experimental results for all variants with BIGCF on three datasets are shown in Table 4 and we have the following findings:

The performance of BIGCF is significantly degraded after removing the GCR module, and it is also degraded after removing the IR, HNR, and BIR modules, respectively. The experimental results demonstrate the necessity and effectiveness of graph contrastive regularization, which can regulate the uniformity of the feature spaces in an augmentation-free and self-supervised manner to prevent the node representations from converging to consistency.

Focusing on intent modeling: (1) firstly, removing the GCR module, $\text{BIGCF}_{\text{w/o GCR}}$ still outperforms LightGCN; (2) secondly, removing the BIR module decreases the performance of BIGCF; (3) lastly, removing the bilateral intent modeling, the performance of $\text{BIGCF}_{\text{w/o BIGR}}$ likewise suffers a certain degree of degradation. The ablation experiments with the above three variants validate the necessity of the proposed intent modeling. Finally, the removal of probabilistic modeling and reparameterization process results in a significant performance degradation of $\text{BIGCF}_{\text{w/o PGR}}$, which demonstrates the effectiveness of generative training strategy.

In this section, we present the effect of various hyperparameters on the performance of BIGCF, as shown in Fig. 6 and Table 5.

The experimental results in Table 5 show that BIGCF can effectively fuse graph structure information into the modeling process of user preferences with two GCN layers. However, stacking more GCN layers may introduce unnecessary noise information, which will negatively affect the performance of BIGCF.

As shown in Fig. 6, the performance of BIGCF further improves as the number of intents increases, which proves the effectiveness of intent modeling. When the number increases further, the performance improvement trend of BIGCF starts to slow down, which we argue could be because the excessive number of intents makes the boundaries among intents start to blur, which interferes with the modeling of user preferences. Increasing the strength of the GCR can further improve the performance, which demonstrates the necessity of constraining the uniformity of the feature space. It should be noted that there is a trade-off between alignment and uniformity, and that excessive graph contrastive regularization may cause troubles with the main recommendation task.

Finally, we evaluate the validity of individual and collective intents through case studies. Specifically, we randomly select a user $u_{2550}$ on Amazon-Book dataset and visualize his correlation scores (computed by Eq. 5) for collective intent (Fig. 7(a)). When not yet trained, all intents provide the same contribution, which cannot distinguish the user’s true preferences. And as the training proceeds, some aspects that the user does not care about are below the initial value (blue line), while the aspects that he cares about provide larger correlation scores (e.g., $\mathbf{C}_{\mathcal{U}}^{14}$, $\mathbf{C}_{\mathcal{U}}^{34}$, and $\mathbf{C}_{\mathcal{U}}^{45}$). It is worth noting that the user behavior is not entirely governed by the collective intent $\mathbf{C}_{\mathcal{U}}^{14}$, but is also more or less influenced by other intents (e.g., $\mathbf{C}_{\mathcal{U}}^{34}$ and $\mathbf{C}_{\mathcal{U}}^{45}$). And focusing on overall behavior, we randomly select 5,000 users and show their means and variances of their correlation scores for collective intent before, during, and after training (Fig. 7(b)). It can be seen that after training, most users demonstrate greater variances for collective intent, which indicates that each user focuses on different aspects of multiple intents (e.g., user $u_{2550}$ in Fig. 7(a)), and BIGCF can effectively utilize implicit feedback data to capture the variability among users and reflect user preferences more accurately.