SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathcal {L} (\boldsymbol {d}, \alpha , \alpha_ {i}, \beta_ {j}, \lambda) = \alpha + \frac {1}{2} \| \boldsymbol {d} \| ^ {2} + \sum_ {i = 1} ^ {M} \alpha_ {i} \left(\nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {L} _ {i} \left(\boldsymbol {\theta} _ {t}\right) ^ {T} \boldsymbol {d} - \alpha\right) \tag {14} \\ + \sum_ {j \in I _ {\epsilon} (\boldsymbol {\theta} _ {t})} \beta_ {j} \left(\nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {G} _ {j} (\boldsymbol {\theta} _ {t}) ^ {T} \boldsymbol {d} - \alpha\right) + \lambda \cdot \mathbb {I} \left(\mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) \geq \xi\right) \left(\nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) ^ {T} \boldsymbol {d} - \alpha\right), \\ \end{array}
$$
where  $\alpha_{i}\geq 0,\beta_{j}\geq 0,\lambda \geq 0$  are the Lagrange multipliers. By requiring that the derivatives of  $\mathcal{L}(\pmb {d},\alpha ,\alpha_i,\beta_j,\lambda)$  with respect to both  $\pmb{d}$  and  $\alpha$  be zero, we have the following equations
$$
\boldsymbol {d} = - \sum_ {i = 1} ^ {M} \alpha_ {i} \nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {L} _ {i} (\boldsymbol {\theta} _ {t}) - \sum_ {j \in I _ {e} \left(\boldsymbol {\theta} _ {t}\right)} \beta_ {j} \nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {G} _ {j} \left(\boldsymbol {\theta} _ {t}\right) - \lambda \cdot \mathbb {I} \left(\mathcal {L} _ {\mathrm {B P R}} \left(\boldsymbol {\theta} _ {t}\right) \geq \xi\right) \nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {L} _ {\mathrm {B P R}} \left(\boldsymbol {\theta} _ {t}\right), \tag {15}
$$
$$
\sum_ {i = 1} ^ {M} \alpha_ {i} + \sum_ {j \in I _ {\epsilon} (\boldsymbol {\theta} _ {t})} \beta_ {j} + \lambda \cdot \mathbb {I} \left(\mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) \geq \xi\right) = 1.
$$
Next, we compute the minimal value of Eq. (14) by substituting the corresponding terms in Eq. (15), and obtain the dual problem of Eq. (13) as  $\max_{\alpha_i,\beta_j,\lambda}\min_{\boldsymbol{d},\boldsymbol{\alpha}}\mathcal{L}(\boldsymbol{d},\alpha,\alpha_i,\beta_j,\lambda)$ , equals to
$$
\begin{array}{l} \min  _ {\alpha_ {i}, \beta_ {j}, \lambda} \frac {1}{2} \left\| \sum_ {i = 1} ^ {M} \alpha_ {i} \nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {L} _ {i} (\boldsymbol {\theta} _ {t}) + \sum_ {j \in I _ {\epsilon} (\boldsymbol {\theta} _ {t})} \beta_ {j} \nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {G} _ {j} (\boldsymbol {\theta} _ {t}) + \lambda \cdot \mathbb {I} (\mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) \geq \xi) \nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) \right\| ^ {2}, \\ s. t. \sum_ {i = 1} ^ {M} \alpha_ {i} + \sum_ {j \in I _ {\epsilon} (\boldsymbol {\theta} _ {t})} \beta_ {j} + \lambda \cdot \mathbb {I} \left(\mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) \geq \xi\right) = 1, \tag {16} \\ \end{array}
$$
which can be efficiently solved by the gradient-based optimization methods such as MGDA [24].
Denote the solutions of Eq. (13) and Eq. (16) as  $(\pmb{d}^{*},\alpha^{*})$  and  $(\alpha_{i}^{*},\beta_{j}^{*},\lambda^{*})$ , respectively. To obtain the solution of the dual problem, according to the KKT conditions, we have
$$
\alpha_ {i} ^ {*} \left(\nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {L} _ {i} (\boldsymbol {\theta} _ {t}) ^ {T} \boldsymbol {d} ^ {*} - \alpha^ {*}\right) = 0, i = 1, \dots , M,
$$
$$
\beta_ {j} ^ {*} \left(\nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {G} _ {j} \left(\boldsymbol {\theta} _ {t}\right) ^ {T} \boldsymbol {d} ^ {*} - \alpha^ {*}\right) = 0, \forall j \in I _ {\epsilon} \left(\boldsymbol {\theta} _ {t}\right), \tag {17}
$$
$$
\lambda \cdot \mathbb {I} (\mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) \geq \xi) \left(\nabla_ {\boldsymbol {\theta} _ {t}} \mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) ^ {T} \boldsymbol {d} ^ {*} - \alpha^ {*}\right) = 0, \text {i f} \mathcal {L} _ {\mathrm {B P R}} (\boldsymbol {\theta} _ {t}) \geq \xi .
$$
By adding all equations in Eq. (17) together, we have  $\alpha^{*} = -\| d^{*}\|^{2}$ . In this way,
- If  $\theta_t$  is Pareto optimal, then no other solution in its neighborhood can achieve better objective values, and we obtain the solution  $d^* = 0$ , indicating that no direction can simultaneously improve all objective values.
- If  $\theta_{t}$  is not Pareto optimal, we have the following conclusions from Eq. (13)
$$
\nabla \mathcal {L} _ {i} \left(\boldsymbol {\theta} _ {t}\right) ^ {T} \boldsymbol {d} ^ {*} \leq \alpha^ {*} \leq - \| \boldsymbol {d} ^ {*} \| ^ {2} <   0, i = 1, \dots , M,
$$
$$
\nabla \mathcal {G} _ {j} \left(\boldsymbol {\theta} _ {t}\right) ^ {T} \boldsymbol {d} ^ {*} \leq \alpha^ {*} \leq - \left\| \boldsymbol {d} ^ {*} \right\| ^ {2} <   0, j \in I _ {\epsilon} \left(\boldsymbol {\theta} _ {t}\right), \tag {18}
$$
$$
\nabla \mathcal {L} _ {\mathrm {B P R}} \left(\boldsymbol {\theta} _ {t}\right) ^ {T} \boldsymbol {d} ^ {*} \leq \alpha^ {*} \leq - \left\| \boldsymbol {d} ^ {*} \right\| ^ {2} <   0, \text {i f} \mathcal {L} _ {\mathrm {B P R}} \left(\boldsymbol {\theta} _ {t}\right) \geq \xi ,
$$
therefore  $d^{*}$  will be a descent direction that at least decreases all the IGF losses simultaneously, as well as the recommendation accuracy loss when  $\mathcal{L}_{\mathrm{BPR}}(\pmb{\theta}_t) \geq \xi$ .
Finally, the model parameters are updated as  $\theta_{t + 1} = \theta_t + \eta d_t$ , where  $\eta$  denotes the step size. Remarkably, throughout the optimization procedure, the values of all IGF objectives and the distance between the obtained solution and the preference region decrease, indicating the Pareto optimality within the preference region. Moreover, to tackle the problem of high-dimensional parameter  $\theta$  in the multi-objective optimization, we propose to optimize its dual problem rather than directly solving the primal problem. As a result, the numbers of parameters  $(\alpha_i,\beta_j)$  in the dual problem is significantly reduced to be same as the numbers of IGF objectives and constraints, respectively, which greatly increases the scalability of the proposed algorithm to the large-scale recommendation datasets. We summarize the whole gradient-based fair recommendation training process in Algorithm 1.
Algorithm 1: Social-Aware Flexible Fair Recommendation with Controllable Accuracy (SoFA)  
Input: Boundary threshold  $\epsilon$ , accuracy threshold  $\xi$ , step size  $\eta_r$  and  $\eta$ .  
1 Set the fairness preference vectors  $\{s_1, s_2, \dots, s_N\}$ ;  
2 for  $n = 1$  to  $N$  do  
3 Randomly initialize the model parameters  $\pmb{\theta}_r^{(n)}$ ;  
4 Find the nearest feasible solution  $\pmb{\theta}_0^{(n)}$  from  $\pmb{\theta}_r^{(n)}$  for region  $\Omega_n$ ;  
5 for  $t = 0$  to  $T$  do  
6 if  $\mathcal{L}_{\mathrm{BPR}}(\pmb{\theta}_t^{(n)}) \leq \xi$  then  
7 Obtain  $\alpha_i, i = 1, \dots, M$ , and  $\beta_j, j \in I_\epsilon(\pmb{\theta}_t^{(n)})$  by solving the dual optimization;  
8 Compute the update direction  
9 else  
10 Obtain  $\alpha_i, i = 1, \dots, M, \beta_j, j \in I_\epsilon(\pmb{\theta}_t^{(n)})$ , and  $\lambda$  by solving the dual optimization;  
11 Compute the update direction  $\pmb{d}_t^{(n)} = -\sum_{i=1}^M \alpha_i \nabla_{\pmb{\theta}_t^{(n)}} \mathcal{L}_i(\pmb{\theta}_t^{(n)}) - \sum_{j \in I_\epsilon(\pmb{\theta}_t^{(n)})} \beta_j \nabla_{\pmb{\theta}_t^{(n)}} \mathcal{G}_j(\pmb{\theta}_t^{(n)})$ ;  
12 end  
13 Update the model parameters  $\pmb{\theta}_{t+1}^{(n)} = \pmb{\theta}_t^{(n)} + \eta d_t^{(n)}$ ;  
14 end
In this section we aim to answer the following research questions:
- RQ1: How does our proposed method perform compared with the existing approaches?
- RQ2: Do the preference regions in our method lead to more flexible Pareto solutions? Does the accuracy constraint result in more accurate recommendations while achieving IGF?
- RQ3: How does varying the accuracy thresholds affect the performance of our method?
Datasets and Pre-processing. The experiments are performed on two benchmark recommendation datasets that include user social attributes and item features:
- KuaiRec is a short-video recommendation dataset on the video-sharing platform Kuaishou<sup>3</sup>, which contains interactions between users and videos. In our experiments, we select user clicks spanning 10 consecutive days as the outcome of interest [29] and apply 10-core filtering to eliminate users and items with minimal interactions, which results in a pre-processed dataset consisting of 7,010 users, 1,405 items, and 863,819 interactions. For each user, the number of followers of that user is treated as the user social attribute, and we define the social utility derived from the user social attribute as the total fans count of that user. As to videos, we use tags to categorize them into six item groups.
- Epinions is a dataset derived from a trust network, comprising users' rating scores for products. Following previous work [10], we only keep the interactions with ratings greater than 3 and also adopt the 10-core filtering setting. Subsequently, we obtain a dataset consisting of 11,710 users, 13,682 items, and 215,262 interactions. For each user, the dataset includes their trust relations with other users, while for each product, it specifies its category. We let user's social utility equal to the number of trustors, and classify the products into six groups based on their categories.
Throughout our experiment, we randomly split the interactions of each dataset into training set  $(60\%)$ , validation set  $(20\%)$ , and testing set  $(20\%)$ . We use the validation set for tuning hyper-parameters and the testing data for evaluation the prediction and fairness performance of the trained models.
<sup>3</sup>https://www.kuaishou.com/.
Table 1: Performance comparison using SP and NSP as IGF notions, where SoFA is implemented with five preference regions. The best and second best results are bolded and underlined, respectively.
KuaiRec	Epinions
N@5↑	SP↓	NSP↓	F1SP ↓ deg	N@5↑	SP↓	NSP↓	F1SP ↓ deg
BPRMF	0.2426	0.0966	0.1119	0.1037 49.2°	0.0443	0.0252	0.0286	0.0268 48.6°
+ SP Reg	0.2389	0.0062	0.0168	0.0091 69.7°	0.0450	0.0140	0.0196	0.0163 54.5°
+ NSP Reg	0.2279	0.0366	0.0142	0.0205 21.2°	0.0378	0.0224	0.0188	0.0205 40.0°
+ SP&NSP Reg	0.2369	0.0090	0.0245	0.0132 69.8°	0.0448	0.0154	0.0205	0.0176 53.2°
+ SP Post	0.2412	0.0388	0.0545	0.0454 54.5°	0.0445	0.0141	0.0196	0.0164 54.2°
+ NSP Post	0.2348	0.0844	0.0311	0.0455 20.3°	0.0398	0.0212	0.0185	0.0197 41.2°
+ SP&NSP Post	0.2405	0.0817	0.0562	0.0666 34.5°	0.0443	0.0152	0.0207	0.0175 53.7°
MOOMTL	0.2229	0.0069	0.0238	0.0107 73.8°	0.0446	0.0138	0.0193	0.0161 54.4°
SoFA region 0	0.2349	0.0296	0.0096	0.0145 18.0°	0.0364	0.0909	0.0294	0.0445 17.9°
SoFA region 1	0.2376	0.0179	0.0105	0.0133 30.4°	0.0441	0.0326	0.0225	0.0266 34.6°
SoFA region 2	0.2329	0.0103	0.0146	0.0121 54.8°	0.0451	0.0153	0.0210	0.0177 53.9°
SoFA region 3	0.2413	0.0074	0.0194	0.0107 69.1°	0.0427	0.0118	0.0177	0.0142 56.3°
SoFA region 4	0.2402	0.0046	0.0227	0.0077 78.5°	0.0185	0.0095	0.0314	0.0146 73.1°
Table 2: Performance comparison using EO and NEO as IGF notions, where SoFA is implemented with five preference regions. The best and second best results are bolded and underlined, respectively.
KuaiRec	Epinions
N@5↑	EO↓	NEO↓	F1EO ↓ deg	N@5↑	EO↓	NEO↓	F1EO ↓ deg
BPRMF	0.2426	0.0189	0.1031	0.0319 79.6°	0.0443	0.1182	0.2059	0.1502 60.1°
+ EO Reg	0.2340	0.0044	0.1082	0.0085 87.7°	0.0425	0.1126	0.1956	0.1429 60.1°
+ NEO Reg	0.2329	0.0147	0.0992	0.0256 81.6°	0.0430	0.1289	0.1557	0.1410 50.4°
+ EO&NEO Reg	0.2346	0.0054	0.1035	0.0102 87.0°	0.0432	0.1136	0.1465	0.1280 52.2°
+ EO Post	0.2371	0.0061	0.1082	0.0116 86.8°	0.0437	0.1100	0.1850	0.1379 59.3°
+ NEO Post	0.2280	0.0143	0.0672	0.0236 78.0°	0.0427	0.1274	0.1565	0.1405 50.8°
+ EO&NEO Post	0.2413	0.0133	0.1039	0.0235 82.7°	0.0436	0.1131	0.1630	0.1335 55.3°
MOOMTL	0.2332	0.0031	0.1003	0.0061 88.2°	0.0444	0.1093	0.1449	0.1246 53.0°
SoFA region 0	0.1635	0.1454	0.0557	0.0806 21.0°	0.0224	0.3273	0.1494	0.2051 24.5°
SoFA region 1	0.2077	0.1135	0.0558	0.0749 26.2°	0.0353	0.1752	0.1394	0.1552 38.5°
SoFA region 2	0.2254	0.0610	0.0528	0.0566 40.9°	0.0448	0.1094	0.1473	0.1256 53.4°
SoFA region 3	0.2241	0.0284	0.0713	0.0406 68.3°	0.0443	0.1160	0.1604	0.1347 54.1°
SoFA region 4	0.2352	0.0023	0.0685	0.0045 88.1°	0.0447	0.1157	0.1574	0.1334 53.7°
Evaluation Protocols. We evaluate the recommendation accuracy with NDCG@5 [30]. For evaluating IGF, we report both the previous fairness metrics and social attribute-aware metrics, where lower value indicates fairer performance. We study two settings with (SP, NSP) and (EO, NEO) as the IGF objectives, respectively. Under each setting, the overall fairness is assessed using the harmonic-mean of the two IGF metrics denoted as F1SP. In addition, to quantify the trade-offs between the two IGF metrics, we calculate the angles formed by these metrics via  $\deg = \arctan(\mathrm{NSP} / \mathrm{SP})$ , where a lower angle degree signifies a greater emphasis on the social utility.
Hyper-Parameter Settings. For all compared methods, we use a pre-trained BPRMF [9] as the backbone with fixed optimal batch size and regularization coefficient, then fine-tune the model by using the baselines methods for achieving IGF notions. For the proposed SoFA, we set  $N = 5$  as the number of preference vectors that equally divide the first quadrant, and tune the learning rate for finding the initial solution within  $\{0.3\mathrm{lr},\mathrm{lr},3\mathrm{lr}\}$ , where  $\mathrm{lr}$  denotes the learning rate for the pre-trained BPRMF. This optimization step stops once a feasible solution is found. The learning rate for all methods is searched within  $\{0.03\mathrm{lr},0.1\mathrm{lr},0.3\mathrm{lr},\mathrm{lr},3\mathrm{lr},10\mathrm{lr}\}$ , and the coefficients of IGF terms for regularization-based and post-processing baselines are searched within  $\{0.1,0.2,0.5,1,2,5\}$ . The early stopping strategy is employed when the F1SP or F1EO value does not decrease over 5 epochs.
(a) SP and NSP trade-off
(b) F1SP and NDCG@5 trade-off
Figure 2: The trade-offs on KuaiRec between (a) IGF metrics, i.e., SP and NSP, and (b) overall fairness and accuracy, i.e., F1SP and NDCG, where yellow lines refer to the results when only optimizing one IGF metric, and scatters located on the upper right indicate better overall performance.
Table 3: Ablation studies of SoFA on the accuracy constraint, IGF objectives, and preference region.
KuaiRec	Epinions
N@5↑	SP↓	NSP↓	F1SP ↓ deg	N@5↑	SP↓	NSP↓	F1SP ↓ deg
SoFA region 0	0.2349	0.0296	0.0096	0.0145 18.0°	0.0364	0.0909	0.0294	0.0445 17.9°
w/o con	0.2181	0.0415	0.0124	0.0191 16.6°	0.0339	0.0814	0.0260	0.0395 17.7°
SoFA region 1	0.2376	0.0179	0.0105	0.0133 30.4°	0.0441	0.0326	0.0225	0.0266 34.6°
w/o con	0.2254	0.0211	0.0121	0.0154 29.8°	0.0429	0.0371	0.0267	0.0311 35.8°
SoFA region 2	0.2329	0.0103	0.0146	0.0121 54.8°	0.0451	0.0153	0.0210	0.0177 53.9°
w/o con	0.2117	0.0085	0.0131	0.0103 57.0°	0.0445	0.0163	0.0224	0.0189 53.9°
SoFA region 3	0.2413	0.0074	0.0194	0.0107 69.1°	0.0427	0.0118	0.0177	0.0142 56.3°
w/o con	0.2212	0.0077	0.0216	0.0113 70.4°	0.0417	0.0131	0.0222	0.0165 59.4°
SoFA region 4	0.2402	0.0046	0.0227	0.0077 78.5°	0.0185	0.0095	0.0314	0.0146 73.1°
w/o con	0.2193	0.0069	0.0256	0.0109 74.9°	0.0169	0.0093	0.0313	0.0144 73.4°
w/o SP	0.2220	0.0206	0.0064	0.0098 17.4°	0.0365	0.0216	0.0147	0.0175 34.3°
w/o NSP	0.2415	0.0050	0.0173	0.0077 74.0°	0.0447	0.0148	0.0200	0.0170 53.5°
w/o region	0.2412	0.0046	0.0193	0.0074 76.7°	0.0447	0.0141	0.0197	0.0164 54.4°
5.2 Performance Comparison (RQ1)
We compare our proposed SoFA with the vanilla BPRMF [9] and several baseline methods to achieve recommendation fairness. These baselines include: (1) Regularization-based method [31], which incorporates IGF metrics as regularization terms into the BPR loss. (2) Post-processing method [32], which reconstructs the recommendation results while imposing fairness objectives as constraints. (3) Multi-objective optimization-based method [28], which optimizes the formulated multi-objective problem without the preference regions and recommendation accuracy constraint. We adopt three versions of regularization-based methods and post-processing methods, which considers only direct utility (SP or EO), only social utility (NSP or NEO), and both utilities, respectively.
Tables 1 and 2 show the overall performance of compared methods under the (SP, NSP) and (EO, NEO) settings, respectively, from which we have the following observations: (1) In all cases, SoFA outperforms other baselines in terms of accuracy and fairness, which validates its effectiveness in fairly recommending with accuracy guarantee. (2) SoFA can effectively find solutions in different preference regions, showing the ability to accommodate customized trade-off between the direct utility and social utility as the IGF metrics. Figure 2a visualizes the trade-off between SP and NSP, showing that SoFA is able to find pareto optimal solutions between the IGF metrics. Figure 2b suggests that F1SP and NDCG do not always exhibit a conflicting trade-off, meaning that enhancing fairness does not necessarily sacrifice recommendation accuracy in certain scenarios.
(a) NDCG@5
(b) F1SP
Figure 3: Effects of varying accuracy thresholds in the optimization on KuaiRec with 100 repeats.
5.3 In-depth Analysis
Ablation Study (RQ2). We conduct the following ablations to evaluate the effectiveness of specific designs in SoFA: (1)  $w/o$  con, which trains the model without the recommendation accuracy constraint. (2)  $w/o$  region, which trains the model without specifying the preference region. (3)  $w/o$  SP, which trains the model without the SP fairness loss, thus does not require the preference region specification. (4)  $w/o$  NSP, similar to  $w/o$  SP but trains the model without the NSP fairness loss. Table 3 shows the performance of SoFA and its ablated versions, we find that training the model without the accuracy constraint encounters significant drop in recommendation accuracy compared to SoFA, indicating the effectiveness of the proposed accuracy constraint. In addition, removing the specification of the preference region may not hurt the performance in terms of accuracy and fairness, because the preference region do not directly contribute to the recommendation accuracy and overall fairness. However, it fails to obtain a recommendation model with flexibility to trade-off these IGF metrics.
Effects of Accuracy Requirement (RQ3). To further investigate the effect of recommendation accuracy constraint, we evaluate SoFA with relaxed constraints by increasing the threshold  $\xi$  in Eq. (9). In particular, we compare the default choice of the accuracy threshold with 1.1 times and 1.2 times of that value, as well as the optimization without the recommendation accuracy constraint. We conduct 100 runs for each optimization process on KuaiRec with varying preference regions. Figure 3 shows the accuracy and fairness performance, and we have observations as below: (1) NDCG has a clear decrease trend as the threshold increases, which further reveals that constraining the recommendation loss can control the accuracy of the obtained solution. (2) F1SP improves as the threshold decreases in most regions, which indicates that the accuracy constraint may also benefit the fairness. Therefore, we suggest to use the original BPRMF loss as the accuracy threshold in practice.
This study addresses a new problem in IGF motivated by the effect of user social influence on item utility. First, we propose two IGF notions, namely NSP and NEO, as extensions to the existing IGF notions that only considers the direct utility of item exposures. Next, we formulate a multi-objective optimization problem with pre-defined preference regions to ensure the flexibility of the trade-off between these IGF metrics. Then, we incorporate a recommendation accuracy constraint to control the accuracy sacrifice for satisfying the fairness requirements. We further propose an algorithm to solve the optimization problem and theoretically demonstrate its Pareto optimality. Extensive experiments on two real-world datasets validate the effectiveness of our method. One interesting future research direction is to generalize the user social influence studies in this paper to ranking-based IGF settings.
This work is supported by the National Key Research and Development Program of China (No. 2022YFB3104701) and the National Natural Science Foundation of China (No. 12301370).
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