SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Dual Problem Formation We construct the Lagrangian function by introducing the Lagrangian multipliers  $\pmb{\lambda} \in \mathbb{R}^n$  as  $\mathcal{L}(\pmb{w},\pmb{v},\pmb{\lambda}) = f(\pmb{w}) + g(\pmb{v}) - \langle \pmb{\lambda},\pmb{T}\pmb{w} - \pmb{v}\rangle$ . We establish the corresponding dual problem as Problem (8) by introducing the conjugate functions  $f^{*},g^{*}$  for simpler notation (Appendix E.3).
$$
\min  _ {\boldsymbol {\lambda}} F (\boldsymbol {\lambda}) + G (\boldsymbol {\lambda}), \tag {8}
$$
with  $F(\boldsymbol{\lambda}) = f^{*}(\boldsymbol{T}^{\intercal}\boldsymbol{\lambda}), \quad G(\boldsymbol{\lambda}) = g^{*}(-\boldsymbol{\lambda})$ . By Slater's condition, we know that strong duality holds as long as  $\boldsymbol{v}$  resides in the range of  $\boldsymbol{T}$ . Therefore, the optimal values for Problem (7) and (8) are identical, and the optimal solution for Problem (8) can be attained. Consequently, we can apply the FISTA algorithm to the dual problem to generate the dual sequence that converges to the optimal solution, and construct the corresponding optimal solution back for the primal problem.
Dual Problem with FISTA It is not hard to show that  $F(\lambda)$  is differentiable and  $\nabla F(\lambda) = T\pmb{x}^*$ , where  $\pmb{x}^{*} = \arg \max_{x}\{(T^{\intercal}\lambda)^{\intercal}\pmb{x} - f(\pmb{x})\}$  (Appendix E.4). Therefore, the dual problem minimizes the sum of a differentiable convex function  $F$  and a closed proper convex function  $G$ . This structure of the dual Problem (8) immediately paves the way for applying proximal gradient methods. Here, for the sake of a better convergence rate, we apply the FISTA algorithm with a fixed step size to the dual Problem (8), and the following iteration schemes are performed. We first choose any  $\lambda^0 = \lambda^{-1}$  and fix the step size as  $t = \frac{2\beta}{l + u}$ . Henceforth,  $k = 1,2,\dots$ , we repeat the following two steps as
$$
\boldsymbol {\mu} ^ {k} = \boldsymbol {\lambda} ^ {k - 1} + \frac {k - 2}{k + 1} \left(\boldsymbol {\lambda} ^ {k - 1} - \boldsymbol {\lambda} ^ {k - 2}\right), \tag {9}
$$
$$
\boldsymbol {\lambda} ^ {k} = \operatorname {p r o x} _ {t G} \left(\boldsymbol {\mu} ^ {k} - t \nabla F \left(\boldsymbol {\mu} ^ {k}\right)\right), \tag {10}
$$
Hence, based on the above-mentioned properties of  $\nabla F(\pmb{\mu}^k)$ , we have  $\nabla F(\pmb{\mu}^k) = \pmb{T}\bar{\pmb{w}}^k$  with  $\bar{\pmb{w}}^k$  set as  $\bar{\pmb{w}}^k = \arg \max_{\pmb{w}}\left\{(\pmb{T}^\top \pmb{\mu}^k)^\top \pmb{w} - f(\pmb{w})\right\} = \left[\frac{\pmb{T}^\top\pmb{\mu}^k - z}{2\beta}\right]_+$  (Appendix E.5).
Let  $\pmb{p}^k = \pmb{\mu}^k - t\pmb{T}\bar{\pmb{w}}^k$ . By the extended Moreau decomposition [43],  $\mathrm{prox}_{\gamma h}(z) + \gamma \mathrm{prox}_{\gamma^{-1}h^*}(z / \gamma) = z, \forall z$ . We have  $\mathrm{prox}_{tG}(\pmb{p}^k) = \pmb{p}^k - t\mathrm{prox}_{t^{-1}G^*}(t^{-1}\pmb{p}^k) = \pmb{\mu}^k - t(\pmb{T}\bar{\pmb{w}}^k - \bar{\pmb{u}}^k)$ . Here,  $\bar{\pmb{u}}^k = \mathrm{prox}_{t^{-1}g}(\pmb{T}\bar{\pmb{w}}^k - t^{-1}\pmb{\mu}^k)$ . Note that  $g(\pmb{v}) = -\alpha \mathbf{1}^\top \log(\pmb{v})$  and by the definition of proximal mapping, it is easy to prove that  $\mathrm{prox}_{t^{-1}g}(\pmb{v}) = \frac{1}{2} (\pmb{v} + \sqrt{\pmb{v} \odot \pmb{v} + 4\alpha t^{-1}\pmb{1}})$ . Therefore, we can simplify the updating step (10) as the following three steps (Appendix E.6).
$$
\bar {\boldsymbol {w}} ^ {k} = \left[ \frac {\boldsymbol {T} ^ {\intercal} \boldsymbol {\mu} ^ {k} - \boldsymbol {z}}{2 \beta} \right] _ {+}, \tag {11}
$$
$$
\bar {\boldsymbol {u}} ^ {k} = \frac {1}{2} \left(\boldsymbol {T} \bar {\boldsymbol {w}} ^ {k} - t ^ {- 1} \boldsymbol {\mu} ^ {k}\right) + \frac {1}{2} \sqrt {\left(\boldsymbol {T} \bar {\boldsymbol {w}} ^ {k} - t ^ {- 1} \boldsymbol {\mu} ^ {k}\right) \odot \left(\boldsymbol {T} \bar {\boldsymbol {w}} ^ {k} - t ^ {- 1} \boldsymbol {\mu} ^ {k}\right) + 4 \alpha t ^ {- 1} \mathbf {1}}, \tag {12}
$$
$$
\boldsymbol {\lambda} ^ {k} = \boldsymbol {\mu} ^ {k} - t (\boldsymbol {T} \bar {\boldsymbol {w}} ^ {k} - \bar {\boldsymbol {u}} ^ {k}), \tag {13}
$$
Finally, we arrive at Procedure GWBI, where the optimal graph weights are inferred for one block like  $W_{uu}$  in the optimal graph structure suggested in Proposition 1. Here, instead of directly optimizing the primal variable of the block graph weight vector  $w$ , we consider its corresponding dual variable  $\lambda$  for a better convergence rate. In each iteration step, we first find an extrapolated point  $\mu$  based on the points  $\lambda$  from two previous steps (line 3). We then perform the proximal gradient update on this extrapolated point (lines 4-6) to obtain  $\lambda$  for the next iteration. Note that lines 4-6 are the detailed instantiation of Eq. (10). Finally, we can convert the dual variable  $\lambda$  back to the desired primal variable  $w$  based on line 4 after its convergence. With this core procedure in hand, we plug it into the proposed Algorithm 1, Block-wise Affinity Graph Learning (BAGL) algorithm, in Appendix C.
We prove that our proposed optimization method for Problem (7) in Procedure GWBI enjoys the guarantee of the global convergence rate, where the generated primal sequence converges to the global optimal solution at a rate of  $O\left(\frac{1}{k}\right)$  with a fixed step size. To begin with, it is well known that the FISTA algorithm enjoys the global convergence rate of  $O\left(\frac{1}{k^2}\right)$  [2]. For simplicity, we focus on the results when optimizing Problem (5), which can be viewed as a general case of Problem (6) when  $l \neq u$ . Therefore,  $Q(\lambda) \equiv F(\lambda) + G(\lambda)$  converges to the dual optimal value  $Q(\lambda^{*})$  at a rate of  $O\left(\frac{1}{k^2}\right)$  due to this well-known fact [2], which yields Theorem 2.
Theorem 2. Let  $\{\lambda^k\}$  be the dual sequence generated by Procedure GWBI, then
$$
Q \left(\boldsymbol {\lambda} ^ {k}\right) - Q \left(\boldsymbol {\lambda} ^ {*}\right) \leq \frac {l + u}{\beta (k + 1) ^ {2}} \| \boldsymbol {\lambda} ^ {0} - \boldsymbol {\lambda} ^ {*} \| _ {2} ^ {2}. \tag {14}
$$
Input: Distance vector  $\mathbf{z}$ , linear mapping matrix  $T$ , balancing parameters  $\alpha$  and  $\beta$ , step size  $t$ , error tolerance parameter  $\epsilon$ .
Output: Block graph weight vector  $\hat{w}$ .
1 Initialize  $\lambda^0 = \lambda^{-1}$  at random and set  $k = 1$
2 do
Procedure GraphWeightBlockInference(z,T,α,β,t,ε)
$\pmb{\mu}^{k} = \pmb{\lambda}^{k - 1} + \frac{k - 2}{k + 1} (\pmb{\lambda}^{k - 1} - \pmb{\lambda}^{k - 2})$    
4  $\bar{w}^k = \left[\frac{T^\intercal\mu^k - z}{2\beta}\right]_+$    
5  $\bar{\pmb{u}}^k = \frac{1}{2} (T\bar{\pmb{w}}^k -t^{-1}\pmb {\mu}^k) + \frac{1}{2}\sqrt{(T\bar{\pmb{w}}^k - t^{-1}\pmb{\mu}^k)\odot(T\bar{\pmb{w}}^k - t^{-1}\pmb{\mu}^k) + 4\alpha t^{-1}\mathbf{1}};$    
6  $\lambda^k = \mu^k -t(T\bar{\pmb{w}}^k -\bar{\pmb{u}}^k);$    
7  $k\gets k + 1$    
8 while  $\| \lambda^k -\lambda^{k - 1}\|_{\infty} > \epsilon$    
9 return  $\hat{\boldsymbol{w}} = \bar{\boldsymbol{w}}^{k}$
We then consider the corresponding primal sequence  $\{\pmb{w}^k\}$  and its convergence rate. Since the strong duality holds and a dual optimal solution  $\lambda^{*}$  exists, any primal optimal point  $(\pmb{w}^{*},\pmb{v}^{*})$  is also a minimizer of  $\mathcal{L}(\pmb {w},\pmb {v},\pmb{\lambda}^{*})$ . This motivates us to construct the primal sequence  $\{\pmb{w}^k\}$  based on the dual sequence  $\{\lambda^k\}$  as  $\pmb{w}^{k} = \arg \min_{\pmb{w}}\mathcal{L}(\pmb {w},\pmb {v},\pmb{\lambda}^{k}) = \arg \max_{\pmb{w}}\{\langle T^{\top}\pmb{\lambda}^{k},\pmb {w}\rangle -f(\pmb {w})\}$ . Thanks to Theorem 3, this primal sequence  $\{\pmb{w}^{k}\}$  is guaranteed to converge to the optimal primal solution  $\pmb{w}^{*}$  of Problem (7) at the rate of  $O(\frac{1}{k})$ .
Theorem 3. Let  $\pmb{w}^k = \arg \max_{\pmb{w}} \{\langle \pmb{T}^\top \pmb{\lambda}^k, \pmb{w} \rangle - f(\pmb{w})\}$  with the dual sequence  $\{\pmb{\lambda}^k\}$  given by Procedure GWBI.  $\pmb{w}^*$  and  $\pmb{\lambda}^*$  are the optimal solution of Problem (7) and the optimal solution of Problem (8), respectively. We have,
$$
\left\| \boldsymbol {w} ^ {k} - \boldsymbol {w} ^ {*} \right\| _ {2} \leq \frac {\sqrt {l + u}}{\beta (k + 1)} \left\| \boldsymbol {\lambda} ^ {0} - \boldsymbol {\lambda} ^ {*} \right\| _ {2}. \tag {15}
$$
Motivated by [45], Theorem 3 establishes that the proposed method exhibits a sub-linear convergence rate, which represents a state-of-the-art result for optimization-based graph construction methods, accompanied by a global convergence guarantee. This improved convergence rate is primarily attributed to the utilization of the FISTA algorithm applied to the dual problem. Time complexity analysis is included in Appendix G.5. More results regarding the robustness of our method are discussed in Appendix D.
In this section, we conduct numerical experiments on both synthetic and real-world datasets to demonstrate the advantages of our proposed BAGL method in terms of efficacy and convergence. Robustness analysis (Appendix G.3) and more experimental results are included in Appendix G.
We choose the following graph construction methods in GSSL for comparison. Radial basis function kernel (RBF) [85] and kNN graph [17] are two classic methods. Smooth graph learning (SGL) [29] is a popular method in the graph signal processing domain. RGCLI [7] is another label-informed graph construction method. Anchor Graph Regularization (AGR) [40] deals with large-scale graph construction. GraphEBM [14] and BCAN [63] are two state-of-the-art methods. The former exploits the energy-based model while the latter constructs a bipartite graph.
All the hyper-parameters are fine-tuned with the grid search method. We repeat the experiment 20 times for each case and report the average result with optimal parameter setting in the efficacy analysis. Unless otherwise specified, the default label inference algorithm is LGC, and the label rate is ten labeled samples per class. More details on the experimental settings can be found in Appendix F.
(a) Synthetic dataset
(b) RBF
(c) BAGL (Ours)
Table 1: Description of datasets
Dataset	#Samples n	#Features d	#Classes c
ORHD	5,620	64	10
USPS	9,298	256	10
COIL100	7,200	1,024	100
TDT2	9,394	36,771	30
MNIST	70,000	784	10
EMNIST Letters	145,600	784	20
Figure 1: Visualization of classification results on the synthetic dataset.
Figure 2: Comparison of BAGL against baseline models with the Bonferroni-Dunn test. CD=2.154 at 0.05 significance level.
4.2 Synthetic Dataset
We generate a synthetic dataset as shown in Fig. 1 (a). The constructed dataset contains two clusters, a dense Gaussian cluster surrounded by a sparse ring-like cluster. With only one labeled sample given in each cluster, we compare the result of our proposed BAGL method (Fig. 1 (c)) with the result of the most popular method RBF (Fig. 1 (b)). We use the coordinates as the node feature and set the width in the RBF kernel as 0.7. For visualization purposes, we show the adjacency matrix with yellow line segments connecting the node pairs if the weight associated with the edge is greater than 0.5 after normalization. The direction of the edge is ignored in Fig. 1 (c). For a fair comparison, we perform label propagation [85] on both constructed affinity graphs. We can see that BAGL can recover two ground-truth clusters much better. Unlike RBF, BAGL can improve the connectivity by the logarithm penalty term and reduce the inter-cluster links by the block-wise design.
4.3 Real-world Datasets
Classification tasks are implemented to assess the performance of BAGL against all graph construction baseline methods on six real-world datasets, listed in Table 1. ORHD (Optical Recognition of Handwritten Digits Data Set), USPS, MNIST, and EMNIST Letters are four popular digits image datasets. COIL100 is an object image dataset. TDT2 is a text dataset. We fix the number of anchor nodes as 1000 in four datasets (COIL100, USPS, ORHD and TDT2), while for the rest two datasets (MNIST, EMNIST-Letters), the number of anchors is fixed as 2000. We perform tfidf and principal component analysis (PCA) as the pre-processing step on TDT2 dataset. The default label inference algorithm is LGC, with ten labels per class. Further details can be found in Appendix F.1.
4.3.1 Efficacy
We fix the number of labeled samples per class to ten and select three label inference methods for the second phase of GSSL. We report average results by performing 20 trials for each algorithm over all the settings in Table 2. Our algorithm outperforms all methods in the USPS, TDT2, and EMNIST-Letters datasets, indicating that BAGL can learn an optimal graph for label inference algorithms in the unified framework. Moreover, we perform the Friedman test with the Bonferroni-Dunn post hoc test for statistical significance analysis. Fig. 2 illustrates the critical difference (CD) diagram on the accuracy, where the average rank is marked along the axis with lower (better) ranks to the left. If the average rank difference between two models is greater than one CD, the relative performance is believed to be different. Accordingly, BAGL significantly outperforms all other baselines by a large margin. We also conduct experiments under low label rates with LGC fixed as the label inference method. Fig. 3 (a) and (b) demonstrate that BAGL performs relatively well with low label rates. This phenomenon can be attributed to the utilization of label information in BAGL. (Appendix G.1)
Table 2: Classification accuracy and standard deviation (%) on real-world datasets.
RBF	kNN	SGL	RGCLI	AGR	GraphEBM	BCAN	BAGL
ORHD	GRF	97.46±0.36	86.59±1.26	94.68±0.66	88.24±3.11	97.63±0.53	95.13±0.41	97.49±0.47	97.88±0.40
LGC	97.56±0.29	87.61±2.30	95.75±0.74	89.32±2.58	96.90±0.47	95.78±0.53	97.27±0.63	98.04±0.71
GCN	98.11±0.44	90.64±3.54	95.80±0.59	89.37±3.02	98.02±0.44	98.42±0.50	98.08±0.70	98.15±0.62
USPS	GRF	94.53±0.65	81.42±0.98	87.67±0.40	84.15±1.85	93.64±0.62	94.26±0.36	93.98±0.60	96.56±0.93
LGC	94.75±0.42	85.13±1.18	86.44±0.51	84.86±1.63	95.92±0.51	94.30±0.30	95.07±0.75	96.77±0.66
GCN	94.98±0.21	86.20±2.03	90.31±0.32	86.09±1.72	95.78±0.49	95.81±0.48	95.79±0.55	97.20±0.64
COIL100	GRF	94.40±0.19	81.24±1.64	92.65±0.82	87.48±2.30	86.54±0.40	85.22±0.57	84.51±0.59	94.78±0.53
LGC	95.13±0.37	83.66±1.35	93.27±1.03	87.79±2.47	87.81±0.57	88.50±0.47	87.06±0.63	94.93±0.45
GCN	94.31±0.25	87.64±1.72	93.52±0.91	89.30±1.65	94.63±0.51	90.15±0.39	90.02±0.48	94.99±0.88
TDT2	GRF	89.22±0.79	80.09±2.69	92.13±0.99	86.51±3.42	94.47±0.79	93.64±0.74	95.95±0.60	96.01±0.91
LGC	89.67±0.46	82.35±3.04	92.96±1.24	87.60±2.84	94.15±0.67	93.97±0.61	94.13±0.79	95.42±0.71
GCN	92.89±0.68	85.77±2.41	94.39±0.83	89.94±3.15	95.36±0.85	94.78±0.69	96.30±0.77	96.33±0.85
MNIST	GRF	83.60±0.24	64.20±1.82	95.03±0.77	87.65±2.07	91.02±0.31	95.39±0.31	92.41±0.47	95.40±0.62
LGC	84.12±0.17	68.86±1.63	94.40±0.52	88.22±2.36	94.79±0.37	95.43±0.47	93.55±0.58	95.42±0.51
GCN	87.03±0.32	74.93±1.77	95.18±0.47	90.47±2.11	95.30±0.21	95.51±0.40	94.84±0.37	95.47±0.43
EMNIST Letters	GRF	50.74±0.16	41.85±1.35	62.34±0.58	54.04±1.92	64.38±0.65	65.03±0.27	67.69±0.39	67.81±0.40
LGC	54.35±0.27	49.51±1.57	63.02±0.41	57.18±2.30	66.49±0.49	66.67±0.23	68.92±0.46	69.03±0.44
GCN	59.28±0.24	51.48±1.44	66.51±0.37	58.82±1.74	67.21±0.71	68.56±0.19	68.97±0.50	69.14±0.47
(a) ORHD
(b) USPS
(c) COIL100
Figure 3: (a) (b) Classification results and (c) (d) convergence results on real-world datasets.
(d) TDT2
4.3.2 Convergence
As an essential part of the overhead for BAGL, Procedure GWBI needs to be efficient for practice. Compared with two other optimization-based methods sharing a similar objective, SGL and its accelerated version by ADMM [61], BAGL enjoys the fastest convergence rate on both datasets. We sample  $1\%$  nodes in each dataset for the convenience of presentation in Fig. 3 (c) and (d). Its outstanding performance confirms the theoretical analysis in Theorem 3. (Appendix G.2).
4.3.3 Ablation Study
To obtain a better understanding of why the proposed BAGL works, we perform some ablation studies to empirically show how the key design of BAGL will potentially affect performance. We create three variants based on the original version of BAGL. First, to demonstrate the significance of the optimal asymmetric structure, we now do not differentiate labeled nodes and unlabeled nodes, and let any graph structure be the potential optimal structure without the constraints of  $W_{ul} = O$  and  $W_{uu} = O$ . We call this variant BAGL w/o optimal structure. Second, to reveal the importance of the connectivity regularization term, we set  $\alpha = 0$  in Procedure GWBI to remove the connectivity consideration. This variant is termed BAGL w/o connectivity. Third, to investigate the effects of sparsity control, we set  $\beta = 10e - 5 \approx 0$  in Procedure GWBI to allow the sparsity of the constructed graph to vary arbitrarily. The last variant of BAGL is abbreviated as BAGL w/o sparsity. We conduct the experiments on the ORHD dataset with the same setting as Table 2 and report the classification accuracy results in Table 3. The proposed optimal asymmetric graph structure contributes most to the success of BAGL. The connectivity regularization term and the sparsity control term also matter since they together encourage a more sparse graph (the latter) but without disconnected components (the former), which is a more favorable graph for the second label inference step.
Table 3: Ablation study of BAGL on the ORHD dataset.
BAGL w/o optimal structure	BAGL w/o connectivity	BAGL w/o sparsity	BAGL
ORHD	GRF	95.71±0.84	96.71±0.62	96.05±0.73	97.88±0.40
LGC	96.34±0.66	96.60±0.57	97.19±0.48	98.04±0.71
GCN	97.29±0.59	97.89±0.64	97.74±0.53	98.15±0.62
In this paper, we propose a novel approach to graph construction for graph-based semi-supervised learning. Building upon the optimal asymmetric graph structure derived from theoretical insights, we develop an efficient block-wise graph construction method that guarantees faster convergence. Our approach combines theoretical insights with practical considerations to provide a more effective and reliable framework for the graph construction step in graph-based semi-supervised learning.
BAGL is an optimization-based method for the graph construction step in graph-based semi-supervised learning. Graph Neural Networks (GNNs) excel in learning representations for graph-structured data [64, 54, 12, 49, 39, 78, 41, 75, 13, 53, 77, 51, 73, 52, 50, 38, 76, 11]. More recent graph structure learning methods aim to learn a clean graph structure from the given noisy graph so that the subsequent GNNs trained on this learned clean graph can obtain better performance. In GSSL, however, there is no given graph structure, and we need to learn the graph structure based on the node features only. Therefore, it is a more challenging task compared to graph structure learning. Therefore, we do not compare our method with other graph structure learning methods since their settings and goals are slightly different. We leave the investigation of graph structure learning for GSSL as future work since it is currently out of the scope of this work.
The other limitation of BAGL is it is only suitable for the transductive setting. If we have nodes or samples unseen in the training set, we have to construct the affinity graph again by executing BAGL again to infer their labels, which is often time-consuming and troublesome regarding efficiency. This is not desirable in real-world applications since we often come across new training samples after we build the affinity graph. We also leave the investigation of the inductive extension of BAGL as future work since this lack of inductive generalization is a well-known challenge in graph-based semi-supervised learning.
Even though BAGL is quite efficient in terms of convergence rate, it may still have computational issues when dealing with extremely large-scale datasets with billions of samples because the time spent on finishing one iteration during the optimization would increase dramatically when the number of training samples is extremely large. We leave the exploration of graph construction methods for extremely large-scale datasets as future work. Other potential applications of our method can be explored in the hyperbolic space [67, 70, 68, 71, 69, 66, 72] or in the natural language processing domain [32, 34, 33, 21, 20, 55, 42, 80, 81, 79].
The work described in this paper was partially supported by the National Key Research and Development Program of China (No. 2018AAA0100204) and RGC General Research Funding Scheme (GRF) 14222922 (CUHK 2151185).
[1] Ahmed El Alaoui, Xiang Cheng, Aaditya Ramdas, Martin J. Wainwright, and Michael I. Jordan. Asymptotic behavior of  $\ell_p$ -based laplacian regularization in semi-supervised learning. CoRR, abs/1603.00564, 2016.
[2] Amir Beck and Marc Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci., 2(1):183-202, 2009.
[76] Yifei Zhang, Hao Zhu, Ziqiao Meng, Piotr Koniusz, and Irwin King. Graph-adaptive rectified linear unit for graph neural networks. In Proceedings of the ACM Web Conference 2022, 2022.
[77] Yifei Zhang, Hao Zhu, Zixing Song, Piotr Koniusz, and Irwin King. COSTA: covariance-preserving feature augmentation for graph contrastive learning. In KDD, pages 2524-2534. ACM, 2022.
[78] Yifei Zhang, Hao Zhu, Zixing Song, Piotr Koniusz, and Irwin King. Spectral feature augmentation for graph contrastive learning and beyond. In AAAI, pages 11289-11297. AAAI Press, 2023.
[79] Yuji Zhang and Jing Li. Time will change things: An empirical study on dynamic language understanding in social media classification. arXiv e-prints, pages arXiv-2210, 2022.
[80] Yuji Zhang, Jing Li, and Wenjie Li. Vibe: Topic-driven temporal adaptation for twitter classification. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing (EMNLP), Singapore, December 2023. Association for Computational Linguistics.
[81] Yuji Zhang, Yubo Zhang, Chunpu Xu, Jing Li, Ziyan Jiang, and Baolin Peng. #HowYouTagTweets: Learning user hashtagging preferences via personalized topic attention. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 7811-7820, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics.
[82] Dengyong Zhou, Olivier Bousquet, Thomas Navin Lal, Jason Weston, and Bernhard Schölkopf. Learning with local and global consistency. In NIPS, pages 321-328. MIT Press, 2003.
[83] Dengyong Zhou, Jiayuan Huang, and Bernhard Scholkopf. Learning from labeled and unlabeled data on a directed graph. In Proceedings of the 22nd International Conference on Machine Learning, ICML '05, page 1036-1043, New York, NY, USA, 2005. Association for Computing Machinery.
[84] Dengyong Zhou, Bernhard Scholkopf, and Thomas Hofmann. Semi-supervised learning on directed graphs. In NIPS, pages 1633-1640, 2004.
[85] Xiaojin Zhu, Zoubin Ghahramani, and John D. Lafferty. Semi-supervised learning using gaussian fields and harmonic functions. In ICML, pages 912-919. AAAI Press, 2003.
[86] Liansheng Zhuang, Zihan Zhou, Shenghua Gao, Jingwen Yin, Zhouchen Lin, and Yi Ma. Label information guided graph construction for semi-supervised learning. IEEE Trans. Image Process., 26(9):4182-4192, 2017.