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+# Incentivizing Dual Process Thinking for Efficient Large Language Model Reasoning
+
+Xiaoxue Cheng $^{1,2,3*}$ , Junyi Li $^{4*}$ , Zhenduo Zhang $^{5}$ , Xinyu Tang $^{1}$ , Wayne Xin Zhao $^{1,2,3\dagger}$ , Xinyu Kong $^{5}$ , Zhiqiang Zhang $^{5}$
+
+<sup>1</sup> Gaoling School of Artificial Intelligence, Renmin University of China
+
+$^{2}$  Beijing Key Laboratory of Research on Large Models and Intelligent Governance
+
+<sup>3</sup> Engineering Research Center of Next-Generation Intelligent Search and Recommendation, MOE
+
+<sup>4</sup> Department of Data Science, City University of Hong Kong <sup>5</sup> Ant Group chengxiaoxue@ruc.edu.cn, junyili@cityu.edu.hk, batmanfly@gmail.com
+
+# Abstract
+
+Large reasoning models (LRMs) have demonstrated strong performance on complex reasoning tasks, but often suffer from overthinking, generating redundant content regardless of task difficulty. Inspired by the dual process theory in cognitive science, we propose Adaptive Cognition Policy Optimization (ACPO), a reinforcement learning framework that enables LRMs to achieve efficient reasoning through adaptive cognitive allocation and dynamic system switch. ACPO incorporates two key components: (1) introducing system-aware reasoning tokens to explicitly represent the thinking modes thereby making the model's cognitive process transparent, and (2) integrating online difficulty estimation and token length budget to guide adaptive system switch and reasoning during reinforcement learning. To this end, we propose a two-stage training strategy. The first stage begins with supervised fine-tuning to cold start the model, enabling it to generate reasoning paths with explicit thinking modes. In the second stage, we apply ACPO to further enhance adaptive system switch for difficulty-aware reasoning. Experimental results demonstrate that ACPO effectively reduces redundant reasoning while adaptively adjusting cognitive allocation based on task complexity, achieving efficient hybrid reasoning.
+
+# 1 Introduction
+
+Recent advances in large reasoning models (LRMs) [1] have demonstrated remarkable success on complex tasks such as mathematical reasoning [2, 3, 4, 5], largely attributed to reinforcement learning that encourages the generation of detailed, step-by-step reasoning processes. LRMs improve answer accuracy through self-reflection and self-verification during long reasoning paths. As the reasoning length increases, the performance of model tends to improve accordingly [2, 6, 7]. Although the long chain-of-thought (CoT) [8] reasoning in LRMs is effective for solving complex problems [2, 9], it often leads to overthinking [10, 11], producing redundant reasoning paths. Most existing LRMs rely on fixed reasoning strategies, lacking the ability to dynamically switch between different thinking modes based on task complexity. This rigidity results in inefficient inference, particularly for simple problems that could be resolved more effectively with concise and direct reasoning.
+
+Several recent efforts have explored long CoT compression for efficient reasoning [12, 13]. One line of work fine-tunes LRMs using supervision from shorter chain-of-thought exemplars [14, 15], encouraging the model to arrive at correct answers with fewer intermediate steps. Another line
+
+introduces length penalties into reinforcement learning reward functions [3, 16, 17, 18], explicitly discouraging unnecessarily long reasoning trajectories during training. Since many prior approaches overlook task difficulty and treat all samples uniformly with the sole objective of shortening reasoning paths, some recent methods attempt to address this by estimating length budgets offline and training the model on sampled trajectories accordingly [19]. However, such offline strategies depend on precomputed budgets and fixed preference data, restricting their adaptability and scalability.
+
+Inspired by the dual process theory in cognitive science [20], which states that humans have two systems for thinking — fast, intuitive thinking (System 1) and slow, deliberate thinking (System 2), we pose the following research question: Can LRMs learn to dynamically switch between fast and slow thinking modes based on task complexity, enabling more efficient and adaptive reasoning?
+
+In this paper, we propose Adaptive Cognition Policy Optimization (ACPO), a reinforcement learning framework that enables LRMs to perform efficient and adaptive reasoning through dynamic system switch between fast and slow thinking modes. To achieve this, we first introduce system-aware reasoning tokens (e.g., <fast Think>, <slow Think>) to explicitly indicate the model engagement in fast or slow thinking modes during the reasoning process. Based on these tokens, we construct a dataset from which the reasoning trajectory is interleaved with fast thinking and slow thinking steps, followed by the final answer. Then, we propose a two-stage training strategy to enable dynamic system switch reasoning. In the first stage, we perform supervised fine-tuning with the constructed dataset as a cold start, establishing the model's foundational ability to generate explicit thinking process. In the second stage, we apply reinforcement learning with ACPO for further enhancement. Specifically, we introduce an online token length budget (TLB) mechanism that estimates task difficulty based on the model's sampling success rate, providing a real-time signal to adjust the reasoning budget according to task complexity. These estimates are further incorporated into a reward function that guides cognitive allocation through two components: a TLB reward that encourages difficulty-aware length control, and a system pattern reward that incentivizes appropriate system switch between fast and slow thinking modes.
+
+We evaluate ACPO on a range of complex reasoning benchmarks. Unlike traditional reinforcement learning methods that focus solely on accuracy or fixed-length compression, ACPO dynamically adjusts both reasoning length and cognitive effort based on task difficulty. For challenging problems, it effectively reduces redundant reasoning steps, while for simpler tasks, it avoids overcompression and maintains high accuracy. These results highlight the advantages of difficulty-aware reasoning, demonstrating the effectiveness of dynamic cognitive control through adaptive reward optimization. The main contributions of this work are as follows:
+
+- We introduce system-aware reasoning tokens to explicitly annotate fast and slow thinking steps in LRMs, enabling transparent reasoning paths and providing a foundation for dynamic control of the model's cognitive strategies.  
+- We propose ACPO, a reinforcement learning framework that integrates online difficulty estimation and token length budget to dynamically calibrate the reward function and steer difficulty-aware cognitive allocation.  
+- We demonstrate through extensive experiments that ACPO effectively compresses redundant reasoning content while maintaining accuracy, achieving a robust balance between efficiency and performance by avoiding overcompression and underexploration.
+
+# 2 Preliminary
+
+# 2.1 Group Relative Policy Optimization (GRPO)
+
+Group Relative Policy Optimization (GRPO) [21] is a reinforcement learning framework that eliminates the need for a value function by estimating advantages in a group-relative manner. Given a specific question-answer pair  $(q, a)$ , the behavior policy  $\pi_{\theta_{\mathrm{old}}}$  samples a group of  $G$  individual responses  $\{y_i\}_{i=1}^G$ . For each token  $y_{i,t}$  in response  $y_i$ , its normalized advantage is computed based on the group-level rewards  $\{R_i\}_{i=1}^G$  as follows:
+
+$$
+\hat {A} _ {i, t} = \frac {R _ {i} - \operatorname {m e a n} \left(\left\{R _ {i} \right\} _ {i = 1} ^ {G}\right)}{\operatorname {s t d} \left(\left\{R _ {i} \right\} _ {i = 1} ^ {G}\right)}. \tag {1}
+$$
+
+Similar to Proximal Policy Optimization (PPO) [22], GRPO adopts a clipped surrogate objective with an additional KL regularization term to stabilize optimization:
+
+$$
+\begin{array}{l} \mathcal {J} _ {\mathrm {G R P O}} (\theta) = \mathbb {E} _ {(q, a) \sim \mathcal {D}, \{y _ {i} \} _ {i = 1} ^ {G} \sim \pi_ {\theta_ {\text {o l d}}} (\cdot | q)} \\ \left[ \frac {1}{G} \sum_ {i = 1} ^ {G} \frac {1}{| y _ {i} |} \sum_ {t = 1} ^ {| y _ {i} |} \left(\min  \left(r _ {i, t} (\theta) \hat {A} _ {i, t}, \operatorname {c l i p} \left(r _ {i, t} (\theta), 1 - \varepsilon , 1 + \varepsilon\right) \hat {A} _ {i, t}\right) - \beta D _ {\mathrm {K L}} \left(\pi_ {\theta} \| \pi_ {\text {r e f}}\right)\right) \right], \tag {2} \\ \end{array}
+$$
+
+where  $r_{i,t}(\theta) = \frac{\pi_{\theta}(y_{i,t}|q,y_{i, < t})}{\pi_{\theta_{\mathrm{old}}}(y_{i,t}|q,y_{i, < t})}$  is the importance sampling ratio for the  $t$ -th token  $y_{i,t}$ .
+
+# 2.2 Token Length Budget
+
+The token length budget (TLB) [19] is a method introduced for estimating an appropriate token-level budget for reasoning trajectories. Instead of prescribing a fixed or hard constraint, TLB adaptively adjusts the expected output length based on the sampling accuracy of candidate responses, capturing the inherent difficulty of the task. Specifically, for a given question  $q$ ,  $N$  candidate responses are sampled from a language model. The TLB, denoted as  $L_{\mathrm{budget}}$ , is computed as:
+
+$$
+L _ {\text {b u d g e t}} = p \cdot L _ {r} + (1 - p) \cdot L _ {\max }, \tag {3}
+$$
+
+where  $p = \frac{c}{N}$  represents the sampling success rate, with  $c$  denoting the number of correct responses,  $L_{r}$  denotes the average token length among the correct responses,  $L_{\mathrm{max}}$  is the maximum token length among all responses. This formulation allows the budget to adapt to task complexity, providing flexible length guidance based on sampling accuracy without external supervision. In the DAST method [19], TLB is computed in an offline manner by sampling multiple responses for each question before training. These responses are then converted to preference pairs for further fine-tuning according to their estimated TLB scores.
+
+# 3 Method
+
+In this section, we propose ACPO, a reinforcement learning framework that enables dynamic and adaptive system switch of LRMs to improve reasoning efficiency and adaptability to task complexity. This framework incorporates system-aware reasoning tokens to explicitly represent fast and slow thinking steps of LRMs, and integrates an online length budget estimation in RL training to dynamically adjust reasoning length based on task difficulty, guiding the model to balance accuracy and efficiency through adaptive cognitive effort allocation. We first describe the use of system-aware reasoning tokens for system explicitization, followed by the construction of explicit reasoning paths for supervised fine-tuning. Next, we present the reinforcement learning process of ACPO, providing a detailed overview of the online token length budget estimation and reward design. The overall framework of the proposed ACPO is illustrated in Figure 1.
+
+# 3.1 Explicit Dual Process Reasoning
+
+The dual process theory [20] models human thinking as a combination of fast, intuitive processes (System 1) and slower, deliberate reasoning (System 2). Inspired by this theory, we introduce system-aware reasoning tokens as explicit indicators of different thinking modes within the model's reasoning process. Specifically, we define four special tokens, <fast_think>, </fast_think>, <slow_think> and </slow_think>, to explicitly wrap fast and slow reasoning steps, respectively. The introduction of reasoning tokens serves two purposes. First, it makes the model's internal thinking modes explicitly observable, revealing the process of system switch. Second, it provides a fine-grained mechanism for controlling and monitoring the model's cognitive dynamics. With this explicit thinking process, we can train the model for system switch, optimizing its cognitive allocation based on task difficulty.
+
+# 3.1.1 Data Construction
+
+To enable the model to learn explicit thinking patterns, we construct a training dataset for supervised fine-tuning. Each sample consists of a question paired with its corresponding answer consisting of interleaved fast thinking and slow thinking segments. To ensure the quality of the training data, we utilize the LIMO dataset [23], a carefully curated, high-quality dataset of complex mathematical
+
+![](images/028fc42cf31bbeceec7fdb6debb888dd907af2db8ba9fb0ccea14c8de199afbf.jpg)  
+Figure 1: The overall framework of ACPO. The upper section illustrates the system explicitization process and cold start training via SFT. The lower section presents the ACPO training phase.
+
+reasoning tasks with high challenge and diverse knowledge coverage. These tasks demand precise, multi-step reasoning to arrive at correct answers, making them well-suited for constructing explicit thinking processes. Specifically, our data construction involves two main steps: Candidate Response Sampling and Response Comparison and Annotation.
+
+- Candidate Response Sampling: Building on the approach in TOPS [24], we prompt the DeepSeek-R1-Distill-Qwen-32B [2] model to generate multiple candidate responses with varying lengths for each question. We adopt the prompting strategies in TOPS to encourage diversity of the reasoning lengths. After generating these candidates, we filter out incorrect responses and select the longest and shortest correct answers for further processing.  
+- Response Comparison and Annotation: We employ GPT-4 [25] as an evaluator to perform fine-grained comparison and annotation between the selected response pairs. For those reasoning steps that are present in both short and long responses, we consider them as essential and detailed components that should be marked as slow thinking, as they represent critical, non-omittable reasoning processes. For other steps that appear in the long response but are omitted or summarized in the short response, we view them as trivial steps that should be labeled as fast thinking. Based on this principle, we reframe the short response by enclosing essential reasoning steps with <slow Think></slowThink> tags and trivial reasoning steps with</fastThink> tags. An illustrative example is provided in Figure 4.
+
+# 3.1.2 Supervised Fine-Tuning (Cold Start)
+
+After completing the data construction process, we obtain a dataset containing 745 annotated samples with explicit reasoning tokens. This dataset is used for supervised fine-tuning, allowing the model to learn to generate outputs with clearly interleaved fast and slow thinking modes. This cold start phase equips the model with the foundational ability to explicitly switch between different thinking modes. We train the model with a standard cross-entropy loss to ensure its thinking process aligns with the intended system switch patterns.
+
+# 3.2 Reinforcement Learning with ACPO
+
+While the SFT process allows the model to align its outputs with the annotated reasoning tokens, explicitly capturing fast and slow thinking patterns, this approach may lead to overfitting and a tendency to memorize fixed reasoning paths. To overcome this limitation, we introduce a reinforcement learning phase, optimizing the fine-tuned model to achieve more flexible and adaptive cognition allocation through interaction and exploration. In the following sections, we present the proposed ACPO method, including the online token length budget estimation and reward design that are specifically tailored for dynamic system switch.
+
+# 3.2.1 Reward Design
+
+In this work, we adopt GRPO as the underlying RL algorithm, extending its framework with a customized reward structure to explicitly guide the model's reasoning behavior. Formally, during the ACPO optimization process, we sample a set of  $N$  candidate responses  $\{y_{1},y_{2},\ldots ,y_{N}\}$  from the current policy model for each training query  $q_{i}$ . For each response, we compute a composite reward that consists of three components: accuracy reward, online TLB reward, and system pattern reward to encourage accurate and efficient reasoning with adaptive cognition allocation, as described below.
+
+Accuracy Reward. For each sampled response  $y_{i}$ , we assess its correctness by comparing the generated answer with the ground-truth label in the training data. The accuracy reward is assigned as:
+
+$$
+R _ {\mathrm {a c c}, i} = \left\{ \begin{array}{l l} + 1, & \text {i f} y _ {i} \text {i s c o r r e c t ,} \\ - 1, & \text {o t h e r w i s e .} \end{array} \right. \tag {4}
+$$
+
+Online TLB Reward. To guide the model in generating reasoning paths of appropriate length that align with the task difficulty, we introduce an online token length budget (TLB) reward. Unlike the offline approach in DAST, which requires extensive pre-sampling and pairwise preference construction, our method estimates the length budget  $L_{\mathrm{budget}}$  online during training, without additional computational overhead. This is achieved by leveraging the natural group-based sampling process in GRPO, allowing the sampling success rate  $p$  and token length budget  $L_{\mathrm{budget}}$  to be directly calculated from the existing candidate responses, as defined in Eq. 3. Given a sampled response  $y_{i}$  with actual length  $L_{i}$ , we compute its TLB reward based on the deviation from the estimated  $L_{\mathrm{budget}}$ . The online TLB score  $R_{\mathrm{TLB},i}$  for each response is then defined as:
+
+$$
+R _ {\mathrm {T L B}, i} = \left\{ \begin{array}{l l} \tanh  (- \lambda_ {i}), & \text {i f y _ {i} i s c o r r e c t ,} \\ \tanh  (\lambda_ {i}), & \text {o t h e r w i s e ,} \end{array} \right. \tag {5}
+$$
+
+where  $\lambda_{i} = \frac{L_{i} - L_{\mathrm{budget}}}{L_{\mathrm{budget}}}$  and the tanh function is used to smoothly bound the reward within  $(-1, 1)$ . The TLB reward adapts the model's reasoning strategy to the difficulty of each task, promoting efficient fast thinking for easier problems while allowing longer and more deliberate reasoning for harder problems. Additionally, the online TLB estimation enables real-time adaptation and avoids reliance on static preference data, leading to better generalization across diverse tasks.
+
+System Pattern Reward. To further encourage difficulty-aware system switch, we use the sampling success rate  $p$  from the TLB estimation as a proxy for task difficulty. For easier queries with higher  $p$ , the model is encouraged to allocate a larger proportion of its reasoning path to fast thinking. In contrast, for harder queries with lower  $p$ , the model is incentivized to allocate more steps to slow thinking for more careful deliberation. Given a sampled response  $y_{i}$ , we compute the proportions of fast and slow thinking tokens within the total reasoning sequence, denoted as  $\rho_{\mathrm{fast},i}$  and  $\rho_{\mathrm{slow},i}$ , respectively. The system pattern reward is then defined as:
+
+$$
+R _ {\text {t h i n k}, i} = \left\{ \begin{array}{l l} \rho_ {\text {f a s t}, i}, & \text {i f} p > p _ {\text {t h r e s h}}, \\ \rho_ {\text {s l o w}, i}, & \text {o t h e r w i s e}, \end{array} \right. \tag {6}
+$$
+
+where  $p_{\mathrm{thresh}}$  is a predefined complexity threshold separating easy and hard questions.
+
+The final reward  $R_{i}$  for each sampled response  $y_{i}$  is computed as a weighted combination of the three components:
+
+$$
+R _ {i} = \left\{ \begin{array}{l l} \max  \left(w _ {\text {a c c}} \cdot R _ {\text {a c c}, i} + w _ {\text {l e n}} \cdot R _ {\text {T L B}, i} + w _ {\text {t h i n k}} \cdot R _ {\text {t h i n k}, i}, 0. 1\right), & \text {i f y} _ {i} \text {i s c o r r e c t ,} \\ \min  \left(w _ {\text {a c c}} \cdot R _ {\text {a c c}, i} + w _ {\text {l e n}} \cdot R _ {\text {T L B}, i} + w _ {\text {t h i n k}} \cdot R _ {\text {t h i n k}, i}, - 0. 1\right), & \text {i f y} _ {i} \text {i s i n c o r r e c t .} \end{array} \right. \tag {7}
+$$
+
+The weights  $w_{\mathrm{acc}}$ ,  $w_{\mathrm{len}}$ , and  $w_{\mathrm{think}}$  are hyperparameters that balance the importance of different reward signals during optimization. In order to ensure that the reward for correct responses remains strictly positive and the reward for incorrect responses remains strictly negative, we apply max and min operations to clip the final reward within the desired range. Based on the final reward, we leverage the GRPO objective in Eq. 2 to optimize the policy model.
+
+Table 1: Evaluation results of ACPO on three different reasoning models across the MATH 500 and AIME 2024 datasets. Bold fonts indicate the best performance for each reasoning model.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">MATH 500</td><td colspan="3">AIME 2024</td></tr><tr><td>Accuracy</td><td>#Token</td><td>ACU</td><td>Accuracy</td><td>#Token</td><td>ACU</td></tr><tr><td>DeepSeek-R1-Distill-Qwen-1.5B</td><td>83.9</td><td>5708</td><td>0.98</td><td>28.9</td><td>16894</td><td>0.11</td></tr><tr><td>THINKPRUNE</td><td>82.9</td><td>2356</td><td>2.35</td><td>27.0</td><td>7574</td><td>0.24</td></tr><tr><td>ACPO-1.5B</td><td>81.0</td><td>1679</td><td>3.22</td><td>30.0</td><td>6670</td><td>0.30</td></tr><tr><td>DeepScaleR-1.5B-Preview</td><td>87.8</td><td>3914</td><td>1.50</td><td>43.1</td><td>17206</td><td>0.17</td></tr><tr><td>L1-Exact</td><td>79.8</td><td>1044</td><td>5.09</td><td>16.7</td><td>1798</td><td>0.62</td></tr><tr><td>L1-Max</td><td>81.8</td><td>999</td><td>5.46</td><td>23.3</td><td>2230</td><td>0.69</td></tr><tr><td>DeepSeek-R1-Distill-Qwen-7B</td><td>92.8</td><td>3977</td><td>0.33</td><td>55.5</td><td>13254</td><td>0.06</td></tr><tr><td>SFT_Shortest</td><td>91.8</td><td>2954</td><td>0.44</td><td>50.0</td><td>10757</td><td>0.07</td></tr><tr><td>SimPO_Shortest</td><td>87.8</td><td>970</td><td>1.29</td><td>33.3</td><td>2737</td><td>0.17</td></tr><tr><td>SimPO_DAST</td><td>92.6</td><td>2802</td><td>0.47</td><td>53.3</td><td>6337</td><td>0.12</td></tr><tr><td>ACPO-7B</td><td>91.6</td><td>1405</td><td>0.93</td><td>52.8</td><td>4520</td><td>0.17</td></tr><tr><td>DeepSeek-R1-Distill-Llama-8B</td><td>89.1</td><td>5003</td><td>0.22</td><td>42.9</td><td>16374</td><td>0.04</td></tr><tr><td>ACPO-8B</td><td>87.4</td><td>2232</td><td>0.49</td><td>43.3</td><td>7405</td><td>0.07</td></tr></table>
+
+# 4 Experiment
+
+# 4.1 Experimental Setup
+
+Datasets and Evaluation Metrics. We conduct training on the DeepScaleR-Preview-Dataset [26], a mathematical dataset consisting of 40K question-answer pairs drawn from AIME, AMC, Omni-Math [27] and STILL [28]. For evaluation, we assess model performance on three mathematical datasets: GSM8K [29], AIME 2024, and MATH 500 [30]. For each test question, we generate 16 responses using a sampling temperature of 0.6 and a top- $p$  value of 0.95, and compute  $pass@1$  to measure accuracy. We report the average number of tokens generated per response in each dataset to assess reasoning efficiency. Moreover, we use Accuracy per Computation Unit (ACU) metric [31] to capture the balance between reasoning accuracy and efficiency, which is defined as:
+
+$$
+\mathrm {A C U} = \frac {\text {A c c u r a c y}}{\# \text {P a r a m s} \times \# \text {T o k e n s}} \tag {8}
+$$
+
+Baselines. We conduct experiments on DeepSeek-R1-Distill-Qwen-1.5B, DeepSeek-R1-Distill-Qwen-7B, and DeepSeek-R1-Distill-Llama-8B [2] and compare ACPO against the following methods.
+
+- DeepScaleR-1.5B-Preview [26] is a model trained on DeepSeek-R1-Distill-Qwen-1.5B with GRPO. We include it as an important baseline in our evaluations.  
+- THINKPRUNE [17] trains models with a fixed token limit, pruning unfinished responses beyond this limit with zero reward. We include THINKPRUNE with 2k token length constraint trained on DeepSeek-R1-Distill-Qwen-1.5B for comparison.  
+- L1 [18] trains models to follow specified response lengths by introducing exact and maximum length penalties in reinforcement learning, yielding two variants, L1-Exact and L1-Max, both trained on DeepScaleR-1.5B-Preview.  
+- DAST [19] introduces an offline approach to estimate token length budgets through sampling and construct length preference data for SimPO [32] training. We include the comparison methods SFT_Shmost, SimPO_Shmost, and SimPO_DAST from DAST in our experiments.
+
+Implementation Details. In the cold start phase, we fine-tune the models for 3 epochs using 745 annotated samples with explicit reasoning tokens. For ACPO training, we adopt the same hyperparameter settings as used in DeepScaleR-1.5B-Preview. Specifically, we use a learning rate of  $1 \times 10^{-6}$ , a batch size of 128, and a maximum context length of 8K tokens during training. The models are trained for one epoch, and both the SFT and RL stages are conducted using the VeRL
+
+Table 3: Performance comparison between three reasoning models trained with GRPO and ACPO on the MATH 500 and AIME 2024 datasets.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">MATH 500</td><td colspan="3">AIME 2024</td></tr><tr><td>Accuracy</td><td>#Token</td><td>ACU</td><td>Accuracy</td><td>#Token</td><td>ACU</td></tr><tr><td>DeepSeek-R1-Distill-Qwen-1.5B</td><td>83.9</td><td>5708</td><td>0.98</td><td>28.9</td><td>16894</td><td>0.11</td></tr><tr><td>+GRPO</td><td>84.5</td><td>3098</td><td>1.82</td><td>29.0</td><td>12990</td><td>0.15</td></tr><tr><td>+ACPO</td><td>81.0</td><td>1679</td><td>3.22</td><td>30.0</td><td>6670</td><td>0.30</td></tr><tr><td>DeepSeek-R1-Distill-Qwen-7B</td><td>92.8</td><td>3977</td><td>0.33</td><td>55.5</td><td>13254</td><td>0.06</td></tr><tr><td>+GRPO</td><td>92.5</td><td>3700</td><td>0.36</td><td>53.2</td><td>8577</td><td>0.08</td></tr><tr><td>+ACPO</td><td>91.6</td><td>1405</td><td>0.93</td><td>52.8</td><td>4520</td><td>0.17</td></tr><tr><td>DeepSeek-R1-Distill-Llama-8B</td><td>89.1</td><td>5003</td><td>0.22</td><td>42.9</td><td>16374</td><td>0.04</td></tr><tr><td>+GRPO</td><td>90.4</td><td>2172</td><td>0.52</td><td>43.3</td><td>8883</td><td>0.06</td></tr><tr><td>+ACPO</td><td>87.4</td><td>2232</td><td>0.49</td><td>43.3</td><td>7405</td><td>0.07</td></tr></table>
+
+framework [33]. We set  $p_{\mathrm{thresh}} = 0.5$  in Eq. 6 and set the reward weights as  $w_{\mathrm{acc}} = 0.6$ ,  $w_{\mathrm{len}} = 0.3$ , and  $w_{\mathrm{think}} = 0.1$  in Eq. 7.
+
+# 4.2 Main Results
+
+The evaluation results of our method and the baselines are presented in Table 1 and Table 2.
+
+For more challenging datasets, such as MATH 500 and AIME 2024 in Table 1, ACPO achieves significant token reduction while maintaining competitive accuracy. For instance, on the AIME 2024 dataset, ACPO-1.5B reaches  $30.0\%$  accuracy with an average token count of 6670, representing a  $60.5\%$  reduction in token usage compared to 16,894 tokens required by DeepSeek-R1-Distill-Qwen-1.5B.
+
+For MATH 500 dataset, ACPO reduces the token count for DeepSeek-R1-Distill-Qwen-7B from 3977 to 1405 with only a slight accuracy decrease. Although L1-Max and L1-Exact also achieve substantial token compression, they suffer noticeable accuracy drops compared to DeepScaleR-1.5B-Preview. This demonstrates that ACPO can effectively shorten reasoning lengths without sacrificing too much accuracy, particularly on complex reasoning tasks.
+
+For simpler GSM8K dataset in Table 2, where the reasoning paths of DeepSeek-R1-Distill-Qwen-1.5B and 7B tend to be less redundant,
+
+Table 2: Evaluation results of ACPO with the three reasoning models on the GSM8K dataset.  
+
+<table><tr><td>Methods</td><td>Accuracy</td><td>#Token</td><td>ACU</td></tr><tr><td>R1-Distill-Qwen-1.5B</td><td>79.9</td><td>643</td><td>8.28</td></tr><tr><td>ACPO-1.5B</td><td>81.3</td><td>572</td><td>9.48</td></tr><tr><td>R1-Distill-Qwen-7B</td><td>86.5</td><td>445</td><td>2.78</td></tr><tr><td>ACPO-7B</td><td>88.3</td><td>413</td><td>3.05</td></tr><tr><td>R1-Distill-Llama-8B</td><td>82.9</td><td>1026</td><td>1.01</td></tr><tr><td>ACPO-8B</td><td>86.7</td><td>732</td><td>1.48</td></tr></table>
+
+ACPO maintains the original length scale while achieving notable accuracy improvements. Specifically, ACPO-1.5B reduces average token usage from 643 to 572 while improving accuracy from  $79.9\%$  to  $81.3\%$ . Similarly, ACPO-7B achieves  $88.3\%$  accuracy with 413 tokens, outperforming the baseline in both accuracy and efficiency. However, for DeepSeek-R1-Distill-Llama-8B, which has relatively long reasoning paths with 1026 tokens, ACPO achieves effective compression by reducing the token count to 732 tokens with  $3.8\%$  accuracy improvement. These results highlight the adaptability of the online token length budget estimation, enabling selective compression that preserves necessary reasoning length while avoiding excessive compression.
+
+# 4.3 Further Analysis
+
+# 4.3.1 Ablation Analysis
+
+To assess the impact of the reward design in ACPO on both accuracy and reasoning length, we conduct ablation studies by training models with the GRPO reward setting, keeping all other parameters identical to those in ACPO. Specifically, we evaluate the three models on the MATH 500 and AIME 2024 datasets: DeepSeek-R1-Distill-Qwen-1.5B, DeepSeek-R1-Distill-Qwen-7B, and DeepSeek-R1-
+
+![](images/8adbbc9051a16cb3d9e91bf11f6d4a4653c8b08c39fc8b20b5ddc15c35b20b18.jpg)  
+Figure 2: Average response length and accuracy across different difficulty levels on MATH 500.
+
+![](images/30e16c76d2abe11be7be9f70c4cee5dd785a40cff384df69999e718fccc9428d.jpg)  
+Figure 3: Average fast and slow thinking ratios across different difficulty levels on MATH 500.
+
+Distill-Llama-8B, each trained for one epoch with GRPO and ACPO. We report accuracy, average token length, and ACU scores for comparison in Table 3.
+
+From the results, we observe that both GRPO and ACPO can compress the reasoning length of the original models. In contrast, ACPO consistently achieves more significant compression, often exceeding  $50\%$  reduction in token length. For instance, it reduces the average reasoning length of DeepSeek-R1-Distill-Qwen-1.5B on the MATH dataset to 1679 tokens, compared to 3098 tokens for GRPO. In terms of accuracy, GRPO shows a slight advantage, especially on the MATH dataset. However, ACPO demonstrates comparable performance on the more challenging AIME dataset. For DeepSeek-R1-Distill-Qwen-1.5B, ACPO reaches an accuracy of  $30.0\%$  compared to  $29.0\%$  for GRPO, effectively balancing accuracy and efficiency. While GRPO focuses solely on correctness, the reward design of ACPO jointly optimizes answer accuracy, reasoning length, and system switch, enabling a flexible coordination between correctness, efficiency, and cognitive allocation.
+
+# 4.3.2 Difficulty Adaptability Analysis
+
+To validate the difficulty adaptability of ACPO, we perform analyses on the MATH 500 dataset, which is divided into five difficulty levels. In Figure 2, we compare the accuracy and average response length of ACPO-1.5B, L1-Exact, and L1-Max. Unlike the L1 approach, which applies uniform token constraints regardless of the difficulty of questions, ACPO demonstrates a more adaptive length control strategy. It effectively shortens responses for simpler problems while preserving the necessary reasoning length for complex questions, resulting in minimal accuracy loss on more challenging levels (e.g., level 4, level 5). This adaptive capability highlights the advantage of our approach in difficulty-aware reasoning, aligning the token length budget more closely with problem complexity.
+
+In Figure 3, we further analyze the fast and slow thinking ratios for ACPO-7B on the MATH dataset. It is observed that, as the problem difficulty increases, the proportion of fast thinking decreases while the slow thinking component increases. Notably, the proportion of slow thinking increases rapidly from difficulty level 1 to level 3, and then grows more gradually beyond level 3. This trend suggests that the model is able to allocate more reasoning effort when solving more difficult problems, dynamically allocating cognitive resources based on task complexity. Such adaptive behavior aligns with the dual process theory of human thinking, demonstrating the effectiveness of our method in reasonable cognition allocation and dynamic system switch.
+
+# 4.3.3 Case Study
+
+In Figure 4, we present an example comparing the reasoning behavior of DeepSeek-R1-Distill-Qwen-1.5B trained with GRPO and ACPO on a number theory problem from the Math 500 dataset. The task requires finding the smallest positive integer multiple of 30 that can be written using only the digits 0 and 2, which requires applying divisibility rules: the number must end in 0 and the sum of its digits must be divisible by 3. The model trained with GRPO adopts a cautious and exhaustive strategy, enumerating multiple candidate numbers (e.g., 30, 60, 90) step by step, even after identifying a valid solution. This leads to a lengthy reasoning path with 1555 tokens, filled with redundant self-verification and repeated constraint checking. In contrast, the model trained with ACPO efficiently identifies the necessary constraints, rapidly filtering out invalid candidates and
+
+What is the least positive integer multiple of 30 that can be written with only the digits 0 and 2?
+
+# DeepSeek-R1-Distill-Qwen-1.5B with ACPO
+
+# DeepSeek-R1-Distill-Qwen-1.5B with GRPO
+
+<slow Think>Okay, so I need to find the smallest positive integer multiple of 30 that can be written using only the digits 0 and 2. Hmm, let's break this down. First, I know that 30 is 3 times 10, so any multiple of 30 must be divisible by both 3 and 10. That means the number has to end with a 0 because of the divisibility by 10. Next, the number must also be divisible by 3. The rule for divisibility by 3 is that the sum of the digits must be a multiple of 3. Since the number can only have digits 0 and 2, I need to figure out how many 2s I need so that their sum is divisible by 3. Let me try with the smallest number of digits. Starting with one digit: 2. But 2 isn't divisible by 3, so that's out. Next, two digits: 20. The sum of the digits is  $2 + 0 = 2$ , which isn't divisible by 3. So 20 doesn't work. How about three digits? Let's try 220. The sum is  $2 + 2 + 0 = 4$ , still not divisible by 3. Next, 2220. The sum is  $2 + 2 + 2 + 0 = 6$ , which is divisible by 3. So 2220 should work.</slow Think>  
+<fast Think>Wait, let me double-check. 2220 divided by 30 is 74, which is an integer. So yes, 2220 is a multiple of 30 and only uses the digits 2 and 0. I don't think there's a smaller number that fits these criteria because the previous attempts with fewer digits didn't work</fast Think>
+
+The least positive integer multiple of 30 that can be written with only the digits 0 and 2 is 2220. <answer> The least positive integer multiple of 30 that can be written with only the digits 0 and 2 is \boxed{2220}. </answer> (476 Tokens)
+
+<think>Okay, so I need to find the smallest positive integer multiple of 30 that can be written using only the digits 0 and 2. Hmm, let's break this down. First, I know that 30 is equal to 3 multiplied by 10, so any multiple of 30 must be a multiple of both 3 and 10. That means the number I'm looking for has to end with a 0 because of the 10 part, and the sum of its digits has to be a multiple of 3 because of the 3 part. Alright, so the number must end with 0, and the rest of the digits can only be 2s or 0s. Let me think about how to approach this. Maybe I can start by considering numbers made up of 2s and 0s that end with 0 and check if their digit sums are divisible by 3. Let's start with the smallest possible numbers. The smallest multiple of 30 is 30 itself. Is 30 made up only of 0s and 2s? Well, it has a 3 and a 0, so no. So 30 is out. Next, let's try 60, 60 has a 6 and a 0. Again, not just 2s and 0s. So 60 is also out. How about 90? That's 9 and 0. Still not good. Hmm, maybe I need to think differently. Instead of looking at multiples of 30, maybe I should construct numbers with only 2s and 0s that are divisible by 30. Since the number must end with 0, let's consider numbers like 20, 220, 2220, etc. Let's check their divisibility by 3. Starting with 20: The sum of the digits is  $2 + 0 = 2$ .2 is not divisible by 3, so 20 isn't a multiple of 30. Next, 220:
+
+Therefore, 2220 is the smallest such number. Let me double-check: 2220 divided by 30 is 74, which is an integer. So yes, 2220 is a multiple of 30, and it's made up only of 2s and 0s. I think that's the answer. So, the least positive integer multiple of 30 that can be written with only the digits 0 and 2 is 2220.<think> <answer>The least positive integer multiple of 30 that can be written with only the digits 0 and 2 is \boxed{2220}. </answer>(1555 Tokens)
+
+Figure 4: An case study comparing the reasoning process of DeepSeek-R1-Distill-Qwen-1.5B trained with GRPO and ACPO in MATH 500 Dataset.
+
+quickly converging to the correct answer in just 476 tokens. Notably, the model exhibits effective system switch, employing slow thinking during problem analysis and fast thinking for verification once a solution is found, achieving substantial token savings especially in the verification stage by eliminating repetitive constraint checks. The system-aware reasoning tokens offer a transparent view of the model's cognitive process, enabling interpretable analysis of how reasoning behavior aligns with fast and slow thinking modes.
+
+# 5 Related Work
+
+Efficient Reasoning in LRMs. Large reasoning models (LRMs) like DeepSeek-R1 [2] and OpenAI o1 [4] have demonstrated strong reasoning capabilities, but their long reasoning paths often suffer from overthinking, introducing redundant content and reducing inference efficiency [10, 13, 11]. To address this, one line of work focuses on supervised fine-tuning with concise, high-quality data to reduce the length of reasoning paths [14, 15]. For instance, TokenSkip [14] generates short reasoning chains through token importance sampling, while Self-Training [15] selects correct short reasoning paths through sampling. Another line of work seeks to improve efficiency through reinforcement learning by modifying reward functions to penalize excessively long reasoning paths [3, 17]. Among these methods, length budgeting is a direct way to explicitly control reasoning length. For example, L1 [18] introduces a fixed length budget to penalize responses that exceed a predefined length during RL training. DAST [19] estimates task difficulty and token length budget through sampling, and constructs preference data for SimPO [32] training. In this work, we integrate online difficulty-aware length budgeting into reinforcement learning process, enabling real-time reasoning budget estimation and reward allocation for efficient reasoning.
+
+Fast and Slow Thinking in LLMs. The dual process theory [20] describes two modes of human thinking: fast, intuitive thinking (System 1) and slow, deliberate thinking (System 2). Recent studies have investigated fast and slow thinking in LLMs, focusing on system switch based on task complexity or uncertainty. Specifically, System1.x [34] adapts a controller and System-1/2 planner to adjust reasoning systems based on task difficulty. FaST [35] develops a switching adapter that transitions between System 1 and System 2 for visual reasoning, depending on factors like uncertainty. HaluSearch [36] leverages model performance to generate supervised labels, enabling hierarchical dynamic switch between reasoning systems within MCTS. Dyna-Think[37] implements a training-free dynamic thinking mechanism, allowing the model to autonomously determine when to apply slow reasoning. In this work, we introduce system-aware reasoning tokens to explicitly
+
+represent fast and slow thinking modes, and leverage ACPO with online token length budget to enable adaptive cognitive allocation and dynamic system switch based on task difficulty.
+
+# 6 Conclusion
+
+In this paper, we proposed ACPO, a reinforcement learning framework designed to address the overthinking problem in LRMs through adaptive cognitive allocation and dynamic system switch. In our approach, we first introduce system-aware reasoning tokens to explicitly represent fast and slow thinking modes, making the model's cognitive process transparent and interpretable. Next, we proposed a two-stage training strategy, first fine-tuning the model to establish the ability to generate explicit thinking process, followed by reinforcement learning with ACPO to enhance adaptive cognition allocation. ACPO integrates online difficulty estimation and token length budget during training, guiding dynamic reasoning through a carefully designed reward function. Experimental results demonstrate that ACPO effectively reduces redundant reasoning without sacrificing too much accuracy. Our work provides a flexible and interpretable framework for adaptive hybrid reasoning, supporting efficient and difficulty-aware cognitive processes in LRMs.
+
+# 7 Limitation
+
+Although ACPO effectively reduces redundant reasoning and enables adaptive cognitive allocation, it has several limitations. First, its online difficulty estimation and token length budget mechanisms rely on verifiable data, which may limit generalization to open-domain tasks. Future work should explore more generalizable estimation methods for broader applicability. Second, ACPO can introduce accuracy trade-offs despite improving reasoning efficiency, highlighting the need for more refined reward designs to better balance efficiency and correctness.
+
+# Acknowledgments
+
+This work was partially supported by National Natural Science Foundation of China under Grant No. 92470205 and 62222215, Beijing Natural Science Foundation under Grant No. L233008 and Beijing Municipal Science and Technology Project under Grant No. Z231100010323009. Xin Zhao is the corresponding author.
+
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+
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+
+# A Prompts for Explicit Thinking Annotation
+
+In Section 3.1.1, we prompt DeepSeek-R1-Distill-Qwen-32B to generate responses of varying reasoning lengths for candidate response sampling, and leverage GPT-4 for fine-grained comparison to annotate different thinking modes. The system prompts for these two components are provided below, where the system prompts for candidate sampling are adapted from TOPS [24].
+
+# System Prompts for Candidate Response Sampling
+
+Low Reasoning Effort: You have extremely limited time to think and respond to the user's query. Every additional second of processing and reasoning incurs a significant resource cost, which could affect efficiency and effectiveness. Your task is to prioritize speed without sacrificing essential clarity or accuracy. Provide the most direct and concise answer possible. Avoid unnecessary steps, reflections, verification, or refinements UNLESS ABSOLUTELY NECESSARY. Your primary goal is to deliver a quick, clear and correct response.
+
+High Reasoning Effort: You have unlimited time to think and respond to the user's question. There is no need to worry about reasoning time or associated costs. Your only goal is to arrive at a reliable, correct final answer. Feel free to explore the problem from multiple angles, and try various methods in your reasoning. This includes reflecting on reasoning by trying different approaches, verifying steps from different aspects, and rethinking your conclusions as needed. You are encouraged to take the time to analyze the problem thoroughly, reflect on your reasoning promptly and test all possible solutions. Only after a deep, comprehensive thought process should you provide the final answer, ensuring it is correct and well-supported by your reasoning.
+
+# System Prompts for Response Comparison and Annotation
+
+Given a problem, a short answer, and a long answer, compare the short answer with the long answer and annotate the short answer based on the following rules. If the short answer omits certain reasoning or calculation steps that are present in the long answer, these omitted steps are considered fast thinking, and the corresponding parts in the short answer should be enclosed within <fast_think></fast_think>. If the short answer contains the same reasoning or calculation steps as the long answer, these parts are considered slow thinking and should be enclosed within <slow_think></slow_think>. Fast thinking parts typically involve intuitive judgments, skipped steps, or direct conclusions, whereas slow thinking parts involve full reasoning or calculations that align with those in the long answer. The output should be the short answer with the appropriate <fast_think></fast_think> and </slow_think></slow_think> tags added.
+
+Problem#:
+
+Long Answer#:
+
+Short Answer#:
+
+Annotated Answe#:
+
+# B Experimental Setting
+
+# B.1 Training and Evaluation Details
+
+In the SFT stage, the learning rate is  $1 \times 10^{-5}$ , the batch size is 8, the number of epochs is 3. For evaluation, we use Qwen2.5 [38] tokenizer to calculate the number of tokens in the responses generated by each model for a fair comparison. All the experiments are conducted on 16 NVIDIA A100 GPUs.
+
+# B.2 Baseline Details
+
+We present descriptions of the three baseline methods SFT_Shortest, SimPO_Shortest, and SimPO_DAST from DAST in Table 1. We adopt their evaluation results as reported in the original DAST paper [19].
+
+- SFT_Shmost: Supervised fine-tuning using only the shortest correct sampled response of each problem as training data.  
+- SimPO_Shmost: SimPO with contrastive instance pairs, which take the shortest correct sampled response of each problem as positive instance and the longest as negative instance.  
+- SimPO_DAST: SimPO with contrastive instance pairs from  $D_{pre}$  constructed in DAST.
+
+# B.3 Prompt for ACPO Training
+
+We present the prompts used in the ACPO training process below.
+
+# Prompt for ACPO Training
+
+You are a helpful AI Assistant that provides well-reasoned and detailed responses. You first think about the reasoning process as an internal monologue and then provide the user with the answer. Respond in the following format: <think>..</think><answer>..</answer>. In the reasoning process, you think fast in simple steps and slowly in complex steps, and put the slow thinking content in <slow Think></slow Think> and fast thinking content in <fast Think></fast Think>. Let's think step by step and output the final answer within boxed{}.
+
+# C License for existing assets
+
+We list the licenses for the datasets and models used in our experiments.
+
+# C.1 Models
+
+- DeepSeek-R1-Distill-Qwen-1.5B: Apache 2.0 License.  
+- DeepSeek-R1-Distill-Qwen-7B: Apache 2.0 License.  
+- DeepSeek-R1-Distill-Llama-8B: Llama3.1 License.
+
+# C.2 Datasets
+
+- The GSM8K [29] dataset: MIT License.  
+- The MATH [30] dataset: MIT License.  
+- The DeepScaleR-Preview-Dataset [26]: MIT License.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We made sure our abstraction and introduction accurately reflect paper's main contribution and scope.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We include a standalone limitation section in the Appendix 7. We have discussed both the methodological limitations and performance bottlenecks of our approach.
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
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+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
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+# Answer: [NA]
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+Justification: Our paper does not include theoretical results.
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+# Guidelines:
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+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
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+
+# 5. Open access to data and code
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+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
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+
+# 6. Experimental setting/details
+
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+
+Answer: [Yes]
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+Justification: Yes, we disclose all the training and test details in Section 4 and Appendix B.
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+Guidelines:
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+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: Our evaluation experiment is conducted by sampling 16 responses and computing the pass@1 score to demonstrate the performance.
+
+Guidelines:
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+
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We provide the computer resources in Appendix B.
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+
+# 9. Code of ethics
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+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
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+Justification: Our method is a training technique for large language models which does not have negative societal impact that needs to be addressed.
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+
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+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: This paper poses no such risks.
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
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+Justification: We list licenses for used datasets and models in Appendix C.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
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+
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
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+
+# 13. New assets
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+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
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+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
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+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
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+# 16. Declaration of LLM usage
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+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [Yes]
+
+Justification: We describe the usage of LLMs in experiments in Section 4.
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+Guidelines:
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+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.

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+# Incomplete Multi-view Clustering via Hierarchical Semantic Alignment and Cooperative Completion
+
+Xiaojian Ding\*, Lin Zhao, Xian Li, Xiaoying Zhu  
+School of Computer and Artificial Intelligence, Nanjing University of Finance and Economics, Nanjing, China
+
+# Abstract
+
+Incomplete multi-view data, where certain views are entirely missing for some samples, poses significant challenges for traditional multi-view clustering methods. Existing deep incomplete multi-view clustering approaches often rely on static fusion strategies or two-stage pipelines, leading to suboptimal fusion results and error propagation issues. To address these limitations, this paper proposes a novel incomplete multi-view clustering framework based on Hierarchical Semantic Alignment and Cooperative Completion (HSACC). HSACC achieves robust cross-view fusion through a dual-level semantic space design. In the low-level semantic space, consistency alignment is ensured by maximizing mutual information across views. In the high-level semantic space, adaptive view weights are dynamically assigned based on the distributional affinity between individual views and an initial fused representation, followed by weighted fusion to generate a unified global representation. Additionally, HSACC implicitly recovers missing views by projecting aligned latent representations into high-dimensional semantic spaces and jointly optimizes reconstruction and clustering objectives, enabling cooperative learning of completion and clustering. Experimental results demonstrate that HSACC significantly outperforms state-of-the-art methods on five benchmark datasets. Ablation studies validate the effectiveness of the hierarchical alignment and dynamic weighting mechanisms, while parameter analysis confirms the model's robustness to hyperparameter variations. The code is available at https://github.com/XiaojianDing/2025-NeurIPS-HSACC.
+
+# 1 Introduction
+
+Incomplete multi-view data, where certain views are entirely missing for some samples, presents significant challenges for traditional multi-view clustering methods [1]. Due to sensor limitations, occlusions, data acquisition conditions, or storage costs, incomplete multi-view data is widely present in the fields of computer vision, multimedia, and image processing. Since missing views disrupt cross-view correlations, amplify noise interference, and introduce biases, there is a necessity to develop Incomplete Multi-View Clustering (IMVC) methods that jointly address view completion and feature learning to ensure robust clustering performance under incomplete conditions.
+
+Existing IMVC methods can be broadly classified into two categories: traditional approaches and deep learning-based approaches. Traditional IMVC methods usually infer the missing parts based on the available information [2, 3], and subsequently employ specific machine learning techniques for multi-view clustering, such as non-negative matrix factorization methods [4, 5], kernel methods [6, 7], subspace learning methods [8, 9], and graph methods [10, 11]. These shallow IMVC methods commonly suffer from limited linear modeling capacity and sensitivity to complex missing patterns,
+
+hindering their ability to fully exploit latent correlations and complementary information in incomplete multi-view data [7]. Recent advancements in deep learning have led to growing interest in IMVC methods, given their robust generalization ability and high scalability. The commonly used deep IMVC methods include (1) autoencoder-based approaches [12], (2) GAN-based methods [13], (3) GCN-based techniques [14], and (4) contrastive learning-based frameworks [15]. The core of these methods lies in the alignment and fusion of latent representations, which is achieved through various mechanisms. The ultimate goal is to enable effective complementarity and synergy of multi-view information in the latent space. Deep networks can adaptively compensate for missing views by implicitly inferring absent information.
+
+Despite significant advancements in existing deep IMVC methods, these methods still suffer from at least two key limitations. First, traditional methods that rely on static fusion strategies (e.g., uniform weighting) fail to adapt to address the distributional differences between views. This results in suboptimal fusion outcomes due to the inability to dynamically balance the specific contributions of each view. While some approaches employ dynamic fusion strategies (such as calculating view weights based on variance [16] or sharing weights across all views [17]), they lack a hierarchical semantic separation, failing to clearly distinguish between low-level consistency alignment and high-level semantic fusion. Consequently, this often leads to the loss of multi-granularity information, limiting the ability to capture both view-invariant and view-specific features. Second, the conventional two-stage process, where data completion is followed by clustering, suffers from error propagation due to the independent optimization of the completion and clustering objectives. Although some studies have attempted to unify data recovery and clustering into a single framework for joint optimization [18, 19], they overlook the varying significance of different views in the fused representation within specific tasks or distributions. As a result, the completion process fails to fully exploit information from high-quality views, ultimately limiting the overall effectiveness of the method.
+
+To address the aforementioned issues, we propose a novel IMVC framework through Hierarchical Semantic Alignment and Cooperative Completion (HSACC). HSACC first establishes low-level semantic consistency by maximizing mutual information across views, ensuring alignment of shared patterns. In the high-level semantic space, it computes adaptive view weights using distributional affinity between individual views and an initial fused representation, followed by weighted fusion to generate a unified global representation. A discrepancy minimization objective further harmonizes the global representation with view-specific semantics, enhancing cross-view coherence. Then, HSACC implicitly recovers missing views by projecting aligned latent representations into high-dimensional semantic spaces, leveraging learned discriminative features to guide completion. Finally, the completed views and latent representations are jointly optimized through reconstruction and clustering objectives. In summary, our contributions can be summarized as follows:
+
+- We propose a novel hierarchical semantic alignment and dynamic weighted fusion mechanism that, through dual-level semantic space design and adaptive weight allocation, significantly enhances the robustness and discriminability of cross-view fusion.  
+- We propose a unified framework combining joint optimization and dynamic weighting, which eliminates error propagation in traditional two-stage pipelines by implicitly recovering missing views through discriminative latent representations.  
+- Experimental results conducted on multiple benchmark datasets demonstrate the superiority of our proposed HSACC over the state of the art IMVC methods.
+
+# 2 Related Work
+
+# 2.1 Deep Incomplete Multi-View Clustering
+
+Leveraging the powerful nonlinear feature extraction capabilities of deep neural networks, deep IMVC approaches have achieved remarkable performance. Existing methods can be systematically categorized into four main groups: (1) Autoencoder-based methods. These methods employ shared latent spaces and cross-reconstruction mechanisms during training to capture inter-view dependencies. During inference, they reconstruct the representations of missing views from available ones, thereby effectively mitigating data incompleteness [12, 20, 21]. (2) GAN-based methods. By exploiting the strong generative capabilities of Generative Adversarial Networks (GANs), these approaches synthesize features or reconstruct data for missing views from observed view representations. This
+
+strategy preserves the completeness of multi-view representations, with representative works including [13, 22, 23, 24]. (3) Graph Convolutional Network-based methods. These methods build graph structures among samples and utilize graph convolution operations to aggregate information from neighboring complete views. Through message propagation, they effectively impute missing view data, as demonstrated in [14, 25, 26]. (4) Contrastive learning-based methods. By incorporating contrastive loss to maximize agreement between positive pairs across different views, these approaches leverage complementary inter-view information to infer representations for missing views [15, 18, 27].
+
+# 2.2 Contrastive Learning
+
+The core principle of contrastive learning [28, 29, 30] is to learn robust feature representations by contrasting similarities between positive and negative samples. This paradigm is particularly effective for incomplete multi-view data, as it enhances representation robustness through cross-view alignment. For instance, PVC-SSN [15] addresses incomplete multi-view clustering via self-supervised contrastive learning, where a self-supervised module enhances the discriminative capacity of the learned representations. Similarly, COMP [18] achieves both cross-view consistency and missing view recovery by maximizing mutual information across views while minimizing conditional entropy. Prolmp [27] introduces a prototype-based imputation framework that learns view-specific prototypes and sample relationships to recover missing view data, while contrastive learning is employed to strengthen both sample and prototype representations. The ADCL method [31] proposes a direct contrastive learning scheme by performing contrastive alignment on sub-vectors of latent features, effectively preventing dimensional collapse. Finally, MICA [32] explores the internal dependencies of multi-view data through a multi-level imputation strategy combined with instance- and cluster-level contrastive learning, further improving representation integrity under incomplete settings.
+
+![](images/96d36d7442e2e3419037983f63aa0e11dfcf17b366c021719d4d0b59ababe4e2.jpg)  
+Figure 1: The framework consists of three modules: (a) view reconstruction, where autoencoders extract latent representations and reconstruct the inputs; (b) multi-view representation learning, which aligns views by maximizing mutual information, learns adaptive weighting to fuse complementary information, and minimizes distribution discrepancies; (c) data recovery and clustering, where missing views are completed and clustering is performed based on the completed representations.
+
+# 3 Method
+
+As shown in Figure 1, given two views  $\mathbf{X}^1$  and  $\mathbf{X}^2$ , we first map them to  $\mathbf{Z}^1$  and  $\mathbf{Z}^2$  via autoencoders, and then extract semantic information at two levels. In the low-level semantic space, the model maximizes the mutual information between views to ensure consistency alignment. In the high-level semantic space,  $\mathbf{Z}^1$  and  $\mathbf{Z}^2$  are concatenated to form the initial fused representation  $\mathbf{R}$ , and cross-view distribution alignment is introduced to measure the matching degree between each view representation
+
+$\mathbf{Z}^v$  and the fused representation  $\mathbf{R}$ , to adaptively estimate view-specific weights. Then, using the computed weights, a weighted fusion of all view representations is performed to obtain a unified high-level shared representation  $\mathbf{H}$ , and an alignment objective is further designed to minimize the distribution discrepancy between  $\mathbf{H}$  and each view. Furthermore, based on the current learned latent representations  $\mathbf{Z}^1$  and  $\mathbf{Z}^2$ , the model maps them to a high-dimensional semantic space through an MLP to obtain  $\mathbf{Q}^1$  and  $\mathbf{Q}^2$ , and uses these results to complete the missing views, obtaining the complete view representations for subsequent clustering. The joint optimization of representation learning and data completion mutually reinforces both processes, thereby improving the overall clustering performance.
+
+# 3.1 Notation
+
+In this paper,  $\{\mathbf{X}^v\in \mathbb{R}^{N\times d_v}\}_{v = 1}^V$  represents a multi-view data set with  $V$  views, where  $\mathbf{X}^v = \{\mathbf{x}_1^v,\dots,\mathbf{x}_l^v,\dots,\mathbf{x}_N^v\}$ ,  $d_v$  represents the feature dimension of the  $v$ -th view. Without loss of generality, we consider a two-view dataset as an example, where  $V = 2$ . Here,  $N$  denotes the total number of samples, and  $\mathbf{x}_i^v$  indicates the  $i$ -th sample in the  $v$ -th view. Additionally, we define an indicator matrix  $\mathbf{A}\in \{0,1\}^{N\times V}$ , whose elements are defined as follows:
+
+$$
+\mathbf {A} _ {i v} = \left\{ \begin{array}{l l} 1, & \text {i f t h e i - t h s a m p l e i s a v a i l a b l e i n t h e v - t h v i e w} \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {1}
+$$
+
+In this matrix, each column represents the availability status of all instances in a specific view.
+
+# 3.2 The Loss Function
+
+View Reconstruction Loss We construct view-specific autoencoders, where  $E^v$  and  $D^v$  denote the encoder and decoder for the  $v$ -th view, respectively. Given the input  $\mathbf{X}^v$ , the encoder extracts the latent representation  $\mathbf{Z}^v = E^v(\mathbf{X}^v) \in \mathbb{R}^{N \times D}$ , which is then reconstructed through the decoder as  $\hat{\mathbf{X}}^v = D^v(\mathbf{Z}^v)$ . The reconstruction loss is computed using Mean Squared Error (MSE), which enforces the preservation of essential structural information in the latent space. The overall reconstruction loss is defined as follows:
+
+$$
+L _ {R E C} = \sum_ {v} \| \mathbf {X} ^ {v} - D ^ {v} \left(E ^ {v} \left(\mathbf {X} ^ {v}\right)\right) \| ^ {2}. \tag {2}
+$$
+
+Cross-view Consistency Loss In deep multi-view learning, maximizing the mutual information between features of different views is crucial for improve representation consistency [18]. To this end, we estimate mutual information via feature-level similarity. This approach explicitly models the joint and marginal distributions of the features from different views. Let the latent representations of the  $i$ -th sample in view 1 and view 2 be  $\mathbf{z}_i^1$  and  $\mathbf{z}_i^2$ , respectively, where  $z_{i,m}^{1}$  and  $z_{i,n}^{2}$  represent the  $m$ -th and  $n$ -th scalar features. The similarity between the features of this sample across the two views is defined as  $p_{m,n}^{(i)} = z_{i,m}^{1} \cdot z_{i,n}^{2}$ . By aggregating the similarities across all samples, we obtain the joint probability distribution  $\mathbf{P} \in \mathbb{R}^{D \times D}$  between view 1 and view 2. The  $(m,n)$ -th element of the matrix is defined as:
+
+$$
+P _ {(m, n)} = \frac {1}{N} \sum_ {i = 1} ^ {N} p _ {m, n} ^ {(i)} = \frac {1}{N} \sum_ {i = 1} ^ {N} z _ {i, m} ^ {1} \cdot z _ {i, n} ^ {2}. \tag {3}
+$$
+
+By summing over the rows and columns of the joint probability distribution  $\mathbf{P} \in \mathbb{R}^{D \times D}$ , we obtain the marginal probability distributions  $\mathbf{P}^{(1)}, \mathbf{P}^{(2)} \in \mathbb{R}^D$  for each view, with elements defined as  $P_{(m)} = \sum_{n=1}^{D} P_{(m,n)}$  and  $P_{(n)} = \sum_{m=1}^{D} P_{(m,n)}$ . Based on the above distributions, we define the mutual information loss between views as:
+
+$$
+L _ {M M I} = - I \left(\mathbf {Z} ^ {1}; \mathbf {Z} ^ {2}\right) = - \sum_ {m = 1} ^ {D} \sum_ {n = 1} ^ {D} P _ {(m, n)} \ln \frac {P _ {(m , n)}}{P _ {(m)} \cdot P _ {(n)}}, \tag {4}
+$$
+
+where  $I(\mathbf{Z}^1; \mathbf{Z}^2)$  measures the information correlation between  $\mathbf{Z}^1$  and  $\mathbf{Z}^2$ . Minimizing this loss is equivalent to maximizing their mutual information.
+
+Distribution Alignment Loss In this section, we use the Maximum Mean Discrepancy (MMD) to simultaneously calculate view weights and optimize feature representations. Specifically, We first measure the distribution difference between each view and the initial fusion representation to assign weights, then perform weighted fusion to obtain a higher-level common representation and minimize its discrepancy with each view, thereby enhancing the consistency and complementarity of cross-view representations.
+
+# Step 1: Estimating View Weights
+
+For the given two view representations  $\mathbf{Z}^1$  and  $\mathbf{Z}^2$ , we first calculate their initial fusion representation  $\mathbf{R}$  as  $\mathbf{R} = \frac{\mathbf{Z}^1 + \mathbf{Z}^2}{2}$ . Subsequently, to evaluate the contribution of each view, we compute the distribution discrepancy between each view and the initial fused representation  $\mathbf{R}$  based on similarity measures. Under the assumption of a linear kernel (i.e.,  $k(x,y) = x^{T}y$ ), we calculate the internal similarity of each view, the internal similarity of the fused representation, and their mutual similarity via dot products. This allows us to quantify the discrepancy between the view-specific representation and the initial fused representation, as follows:
+
+$$
+D \left(\mathbf {Z} ^ {v}, \mathbf {R}\right) = \frac {1}{N ^ {2}} \left(\sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} \mathbf {z} _ {i} ^ {v} \cdot \mathbf {z} _ {j} ^ {v} + \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} \mathbf {r} _ {i} \cdot \mathbf {r} _ {j} - 2 \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} \mathbf {z} _ {i} ^ {v} \cdot \mathbf {r} _ {j}\right), \tag {5}
+$$
+
+where  $N$  is the number of samples,  $\mathbf{z}_i^v$  and  $\mathbf{r}_i$  represent the  $i$ -th sample of the  $v$ -th view and the initial fusion representation  $\mathbf{R}$ , respectively.
+
+Considering that a view whose latent representation is highly consistent with the initial fused representation should contribute more to the final decision, we assign it a higher weight. Conversely, a view exhibiting a greater discrepancy is assigned a lower weight. This design enables the dynamic adjustment of each view's contribution. Accordingly, we define a weight function  $W^{v} = f(D(\mathbf{Z}^{v},\mathbf{R}))$  to calculate the weights, where  $f(D)$  is defined as:
+
+$$
+f (D) = \frac {\exp (- D \left(\mathbf {Z} ^ {v} , \mathbf {R}\right))}{\sum_ {v = 1} ^ {V} \exp (- D \left(\mathbf {Z} ^ {v} , \mathbf {R}\right))}. \tag {6}
+$$
+
+In multi-view learning tasks, since each view  $\mathbf{Z}^v$  may carry complementary information, we perform weighted fusion of all view-specific representations using the learned weights  $W^v$ , in order to integrate diverse perspectives into the high-level common representation  $\mathbf{H}$ , where  $\mathbf{H} = \sum_{v=1}^{V} W^v \cdot \mathbf{Z}^v$ .
+
+# Step 2: Distribution alignment between H and  $\mathbf{Z}^v$
+
+Motivated by domain adaptation [33], to further enhance the information interaction between views and promote the completion of missing views, we aim to align the distribution of single-view representations  $\mathbf{Z}^v$  with the high-level common representation  $\mathbf{H}$ . Specifically, let  $\mathcal{P}$  denote the distribution of the view-specific representations  $\mathbf{Z}^v = \{\mathbf{z}_1^v,\mathbf{z}_2^v,\dots ,\mathbf{z}_i^v\}_{i = 1}^N$ , and let  $\mathcal{Q}$  denote the distribution of the high-level common representations  $\mathbf{H} = \{\mathbf{h}_1,\mathbf{h}_2,\ldots ,\mathbf{h}_j\}_{j = 1}^N$ . We measure their discrepancy in the high-level semantic space by computing the distance between  $\mathcal{P}$  and  $\mathcal{Q}$ :
+
+$$
+L _ {M M D} = \sum_ {v = 1} ^ {V} \mathrm {M M D} ^ {2} \left(\mathcal {P} _ {\mathbf {Z} ^ {v}}, \mathcal {Q} _ {\mathbf {H}}\right) = \sum_ {v = 1} ^ {V} \left\| \mathbb {E} _ {x \sim \mathcal {P} _ {\mathbf {Z} ^ {v}}} [ \phi (x) ] - \mathbb {E} _ {y \sim \mathcal {Q} _ {\mathbf {H}}} [ \phi (y) ] \right\| _ {\mathcal {H}} ^ {2}. \tag {7}
+$$
+
+Here,  $L_{MMD}$  denotes the overall discrepancy loss between all views and the high-level common representation  $\mathbf{H}$ , where  $\mathcal{H}$  is the Reproducing Kernel Hilbert Space (RKHS). The terms  $\mathbb{E}_{x\sim \mathcal{P}_{\mathbf{Z}^v}}[\phi (x)]$  and  $\mathbb{E}_{y\sim \mathcal{Q}_{\mathbf{H}}}[\phi (y)]$  represent the mean embeddings of the  $v$ -th view and the common representation  $\mathbf{H}$  in the RKHS, respectively. Here,  $\phi (\cdot)$  is a feature mapping function that projects samples into the RKHS. Since it is difficult to compute the expectations directly, we approximate them using sample means. Taking  $\mathbb{E}_{x\sim \mathcal{P}_{\mathbf{Z}^v}}[\phi (x)]$  as an example:
+
+$$
+\mathbb {E} _ {x \sim \mathcal {P} _ {\mathbf {Z} ^ {v}}} [ \phi (x) ] \approx \frac {1}{N} \sum_ {i = 1} ^ {N} \phi \left(\mathbf {z} _ {i} ^ {v}\right) = \frac {1}{N} \mathbf {1} _ {N} ^ {T} \Phi \left(\mathbf {Z} ^ {v}\right), \tag {8}
+$$
+
+where  $\Phi (\mathbf{Z}^v) = [\phi (\mathbf{z}_1^v)^\top ,\phi (\mathbf{z}_2^v)^\top ,\ldots ,\phi (\mathbf{z}_N^v)^\top ]^\top$  denote the feature mapping of all samples in view  $\mathbf{Z}^v$ , where  $\mathbf{1}_N$  is a vector of length  $N$  with all elements equal to 1. Similarly, the expectation  $\mathbb{E}_{y\sim \mathcal{Q}_{\mathrm{H}}}\left[\phi (y)\right]$  can be approximated. In RKHS, the inner product of feature mappings can be computed using the kernel function  $k(x,y) = \langle \phi (x),\phi (y)\rangle_{\mathcal{H}}$ . To simplify the computation, we construct kernel matrices  $\mathbf{K}_{\mathbf{Z}^v\mathbf{Z}^v}$ ,  $\mathbf{K}_{\mathbf{HH}}$ , and  $\mathbf{K}_{\mathbf{Z}^v\mathbf{H}}$ , each of size  $N\times N$ , representing the internal similarities of
+
+the view-specific representations, the internal similarities of the high-level common representation, and the similarities between them, respectively. By substituting Eq. (8) into Eq. (7) and expanding it using kernel matrices, we obtain the final distribution alignment loss:
+
+$$
+L _ {M M D} = \sum_ {v = 1} ^ {V} \frac {1}{N ^ {2}} \mathbf {1} _ {N} ^ {T} \mathbf {K} _ {\mathbf {Z} ^ {v} \mathbf {Z} ^ {v}} \mathbf {1} _ {N} + \frac {1}{N ^ {2}} \mathbf {1} _ {N} ^ {T} \mathbf {K} _ {\mathbf {H H}} \mathbf {1} _ {N} - \frac {2}{N ^ {2}} \mathbf {1} _ {N} ^ {T} \mathbf {K} _ {\mathbf {Z} ^ {v} \mathbf {H}} \mathbf {1} _ {N}. \tag {9}
+$$
+
+Inference Consistency Loss In incomplete multi-view clustering, we design cross-view inference modules  $f_{\mathrm{MLP1}}$  and  $f_{\mathrm{MLP2}}$  to learn nonlinear mappings in the latent space for implicitly completing missing views. Specifically, the latent representations  $\mathbf{Z}^1$  and  $\mathbf{Z}^2$  are projected into a high-dimensional semantic space, yielding  $\mathbf{Q}^1 = f_{\mathrm{MLP1}}(\mathbf{Z}^1) = f_{\mathrm{MLP1}}(E^1(\mathbf{X}^1))$  and  $\mathbf{Q}^2 = f_{\mathrm{MLP2}}(\mathbf{Z}^2) = f_{\mathrm{MLP2}}(E^2(\mathbf{X}^2))$ , which leverage information from the complete views to infer the semantics of missing views without relying on explicit interpolation or generation. To measure the semantic consistency between the inferred results and the original latent representations, we introduce the inference loss  $L_{INF}$  to supervise the accuracy of the cross-view mapping:
+
+$$
+L _ {I N F} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left\| \mathbf {z} _ {i} ^ {2} - \mathbf {q} _ {i} ^ {1} \right\| _ {2} ^ {2} + \frac {1}{N} \sum_ {i = 1} ^ {N} \left\| \mathbf {z} _ {i} ^ {1} - \mathbf {q} _ {i} ^ {2} \right\| _ {2} ^ {2}. \tag {10}
+$$
+
+Here,  $\mathbf{z}_i^v$  is the true latent representation of the  $i$ -th sample in the  $v$ -th view, and  $\mathbf{q}_i^v$  is the inference result obtained by the inference module for the corresponding view. By minimizing this loss, effective inference between views can be achieved, completing better completion.
+
+Extending to Multi-View For  $V$  views, we first construct independent autoencoders for each view to extract latent representations, and compute the cross-view consistency loss over all view pairs  $(v_{1}, v_{2})$  as  $L_{MMI} = \frac{2}{V(V - 1)} \sum_{v_{1} < v_{2}} L_{MMI}(\mathbf{Z}^{v_{1}}, \mathbf{Z}^{v_{2}})$ , after which the latent representations are fused to obtain the global representation  $\mathbf{H}$ , followed by distribution alignment with each view through the loss  $L_{MMD}$ . Finally, in cross-view inference, the latent representation of view  $v_{1}$  is projected by an MLP to infer the semantics of view  $v_{2}$ , and the inference consistency loss is defined as  $L_{INF} = \frac{1}{N} \sum_{i=1}^{N} \frac{1}{V(V - 1)} \sum_{v_{1} \neq v_{2}} \| \mathbf{z}_{i}^{v_{2}} - \mathbf{q}_{i}^{v_{1} \rightarrow v_{2}} \|_{2}^{2}$ , to ensure consistency between the inferred results and the true latent representations.
+
+# 3.3 Objective Function and Optimization Algorithm
+
+Thus, our overall objective function can be expressed as:
+
+$$
+L = \lambda_ {1} L _ {R E C} + \lambda_ {2} L _ {I N F} + \lambda_ {3} L _ {M M I} + \lambda_ {4} L _ {M M D}. \tag {11}
+$$
+
+The  $\lambda_{1},\lambda_{2},\lambda_{3}$ , and  $\lambda_{4}$  are trade-off parameters in the loss function. Ultimately, we optimize Eq. (11) to recover the missing views, thereby obtaining complete view representations. These representations are then concatenated and subjected to  $k$ -means clustering to obtain the final clustering results. The overall training procedure of HSACC is summarized in Algorithm 1.
+
+Algorithm 1: Incomplete Multi-View Clustering via Hierarchical Semantic Alignment and Cooperative Completion  
+Input: Incomplete multi-view dataset  $\{\mathbf{X}^v\}_{v = 1}^V$  with indicator matrix A, epoch  $E,E_{1}$  Parameters: Trade-off coefficients  $\lambda_1,\lambda_2,\lambda_3,\lambda_4$    
+Output: Clustering results   
+for epoch  $= 1$  to  $E$  do Learn view-specific representations via Eqs. (2) and (4); Concatenate the learned  $\mathbf{Z}^v$  to obtain R; Calculate the view weight  $W^{v}$  using Eq. (6); Fuse based on the weight  $W^{v}$  to obtain H; Calculate the distribution discrepancy between H and  $\mathbf{Z}^v$  via Eq. (9); if epoch  $\geq E_{1}$  then Obtain all  $\mathbf{Q}^{v_1\to v_2}$  via cross-view inference MLP to complete the views; Update representations via joint optimization using Eq. (11);   
+Perform  $k$  -means clustering algorithm on the concatenated views;
+
+# 4 Experiments
+
+# 4.1 Experimental Setting
+
+Datasets To evaluate the effectiveness of the proposed method, we selected five representative datasets. LandUse_21 [34] contains 2,100 remote sensing images from 21 categories. Noisy MNIST [35] consists of noisy handwritten digit images, with the original images as View 1 and Gaussian-noised images as View 2. Caltech101-20 [36] contains 2,386 images from 20 categories using HOG and GIST features. Hdigit [37] contains 10,000 handwritten digit images from 10 categories. 100leaves [38] contains 1,600 samples from 100 categories.
+
+Comparing Methods We conducted comparative experiments with nine state-of-the-art incomplete multi-view clustering methods. RPCIC [39] addresses the IMVC problem through cross-view contrastive learning and robust prototype discriminative learning. MRL_CAL [19] tackles IMVC by combining contrastive learning with adversarial learning, aiming to learn multi-level features to enhance clustering performance. ICMVC [14] leverages contrastive learning with high-confidence guidance to handle incomplete multi-view data. MCAC [40] employs attention-based contrastive learning to enhance consistent representation across views and effectively handle missing data. PROLMP [27] restores missing views by learning view-specific prototypes and instance-prototype relationships, and further enhances feature representation via contrastive learning. DCP [41] adopts dual contrastive prediction and an information-theoretic framework to achieve both data recovery and view-consistency learning. DSIMVC [42] dynamically completes missing views through a bi-level optimization framework and automatically selects high-quality imputed samples for training. SURE [43] handles partially missing samples in IMVC via noise-robust contrastive learning and a class-level identification framework. COMP [18] simultaneously achieves data recovery and view-consistency learning by integrating contrastive learning and a dual-prediction mechanism.
+
+Experimental Configuration All experiments were conducted on an NVIDIA RTX 4070 GPU using PyTorch 2.3.1. Our method employs a unified fully connected autoencoder, with the encoder structured as Input-1024-1024-1024-Output and the decoder mirrored accordingly. During the data recovery phase, a multilayer perceptron (MLP) with hidden layer dimensions of 256-128-256 is used for cross-view inference and feature reconstruction. In our experiments, each dataset underwent  $E$  training epochs (e.g.,  $E = 500$ ), and the computation of the inference loss was introduced starting from the  $E_{1}$ -th epoch (e.g.,  $E_{1} = 100$ ). The learning rate was set to 0.0001, and the batch size was 256. We conducted experiments under different missing rates to evaluate the performance of various comparative methods.
+
+# 4.2 Experimental Results and Analysis
+
+Performance Comparison Table 1 presents the clustering results of various IMVC methods under different missing rates, where the best and second-best results are highlighted in bold and underline, respectively. It can be observed that our method consistently outperforms the other nine approaches across all datasets. Among them, DSIMVC and DCP are two state-of-the-art imputation-based IMVC methods. However, the experimental results demonstrate that our method still achieves superior performance compared to both. In particular, on the Caltech101-20 dataset with a missing rate of 0.5, HSACC improves the ACC and ARI metrics by  $5.3\%$  and  $8.57\%$ , respectively, compared to the second-best method. Moreover, on the Noisy MNIST dataset, when the missing rate increases from 0.3 to 0.7, the accuracy of HSACC decreases by only  $6.92\%$ , whereas ICMVC suffers a drop of  $35.19\%$ . These results indicate that HSACC is capable of accurately capturing the intrinsic structural features within each view from incomplete multi-view data and effectively reconstructing the missing parts using the learned representations, thereby exhibiting excellent robustness and generalization ability.
+
+Ablation Study To evaluate the effectiveness of each loss component, we conducted ablation experiments on the Caltech101-20 dataset, focusing on four modules:  $L_{REC}$ ,  $L_{MMI}$ ,  $L_{MMD}$ , and  $L_{INF}$ . As shown in Table 2, the results indicate that removing any of these components leads to a significant drop in model performance. Specifically, the baseline model M-1, which only includes  $L_{REC}$ , achieves an ACC, NMI, and ARI of  $42.07\%$ ,  $41.06\%$ , and  $28.38\%$ , respectively. By introducing the cross-view consistency loss  $L_{MMI}$ , model M-9 shows substantial improvements
+
+Table 1: Performance comparison under different missing rates on five datasets.  
+
+<table><tr><td rowspan="2">Missing Rate</td><td rowspan="2">Method</td><td colspan="3">LandUse_21</td><td colspan="3">Noisy MNIST</td><td colspan="3">Caltech101-20</td><td colspan="3">Hdigi</td><td colspan="3">100leaves</td></tr><tr><td>ACC</td><td>NMI</td><td>ARI</td><td>ACC</td><td>NMI</td><td>ARI</td><td>ACC</td><td>NMI</td><td>ARI</td><td>ACC</td><td>NMI</td><td>ARI</td><td>ACC</td><td>NMI</td><td>ARI</td></tr><tr><td rowspan="10">0.3</td><td>RPCIC</td><td>21.29</td><td>26.52</td><td>8.21</td><td>53.19</td><td>52.18</td><td>39.21</td><td>41.49</td><td>58.30</td><td>34.97</td><td>91.06</td><td>82.49</td><td>81.23</td><td>53.62</td><td>76.32</td><td>38.20</td></tr><tr><td>MRL_CAL</td><td>19.76</td><td>21.21</td><td>6.85</td><td>15.73</td><td>2.80</td><td>1.18</td><td>20.03</td><td>38.25</td><td>12.54</td><td>12.30</td><td>0.69</td><td>0.23</td><td>12.88</td><td>50.63</td><td>7.31</td></tr><tr><td>ICMVC</td><td>26.14</td><td>30.01</td><td>13.13</td><td>95.79</td><td>91.90</td><td>93.07</td><td>35.88</td><td>61.16</td><td>26.65</td><td>16.34</td><td>8.64</td><td>3.56</td><td>20.49</td><td>76.62</td><td>30.88</td></tr><tr><td>MCAC</td><td>20.21</td><td>21.50</td><td>7.08</td><td>95.17</td><td>91.36</td><td>91.99</td><td>32.62</td><td>46.97</td><td>22.64</td><td>29.49</td><td>20.02</td><td>11.40</td><td>35.26</td><td>64.88</td><td>20.58</td></tr><tr><td>PROLMP</td><td>20.33</td><td>21.81</td><td>8.37</td><td>88.80</td><td>78.47</td><td>76.88</td><td>32.47</td><td>53.08</td><td>25.33</td><td>93.17</td><td>87.46</td><td>84.81</td><td>51.54</td><td>77.34</td><td>38.29</td></tr><tr><td>DCP</td><td>26.84</td><td>30.40</td><td>13.84</td><td>72.98</td><td>77.22</td><td>58.93</td><td>71.51</td><td>70.53</td><td>79.06</td><td>94.31</td><td>93.33</td><td>89.43</td><td>32.21</td><td>75.34</td><td>29.62</td></tr><tr><td>DSIMVC</td><td>21.43</td><td>25.25</td><td>8.04</td><td>63.70</td><td>60.70</td><td>51.40</td><td>27.57</td><td>49.72</td><td>21.03</td><td>94.30</td><td>91.38</td><td>91.30</td><td>29.78</td><td>61.91</td><td>17.53</td></tr><tr><td>SURE</td><td>24.80</td><td>29.04</td><td>11.10</td><td>95.11</td><td>91.05</td><td>91.88</td><td>49.70</td><td>65.26</td><td>40.96</td><td>49.74</td><td>38.18</td><td>23.94</td><td>46.51</td><td>71.21</td><td>30.04</td></tr><tr><td>COMP</td><td>26.85</td><td>31.50</td><td>12.87</td><td>84.76</td><td>81.89</td><td>77.00</td><td>71.55</td><td>69.61</td><td>79.00</td><td>93.84</td><td>85.66</td><td>86.86</td><td>38.39</td><td>72.15</td><td>28.00</td></tr><tr><td>Ours</td><td>27.39</td><td>31.80</td><td>14.27</td><td>95.81</td><td>92.33</td><td>93.80</td><td>72.34</td><td>71.48</td><td>79.36</td><td>94.89</td><td>93.78</td><td>92.06</td><td>54.28</td><td>77.57</td><td>39.21</td></tr><tr><td rowspan="10">0.5</td><td>RPCIC</td><td>19.24</td><td>22.49</td><td>5.83</td><td>51.63</td><td>47.77</td><td>35.75</td><td>46.19</td><td>62.70</td><td>38.53</td><td>83.58</td><td>73.18</td><td>66.95</td><td>40.13</td><td>71.69</td><td>27.63</td></tr><tr><td>MRL_CAL</td><td>18.52</td><td>21.16</td><td>6.29</td><td>14.98</td><td>3.20</td><td>1.22</td><td>31.52</td><td>38.24</td><td>24.25</td><td>12.62</td><td>0.82</td><td>0.28</td><td>12.12</td><td>49.35</td><td>7.03</td></tr><tr><td>ICMVC</td><td>24.76</td><td>27.20</td><td>11.29</td><td>86.84</td><td>82.33</td><td>80.00</td><td>33.70</td><td>58.22</td><td>25.06</td><td>16.44</td><td>8.67</td><td>3.64</td><td>51.39</td><td>74.71</td><td>33.83</td></tr><tr><td>MCAC</td><td>17.91</td><td>18.52</td><td>6.14</td><td>91.03</td><td>84.21</td><td>84.98</td><td>32.56</td><td>33.17</td><td>16.78</td><td>24.08</td><td>14.75</td><td>6.93</td><td>27.59</td><td>58.43</td><td>12.14</td></tr><tr><td>PROLMP</td><td>18.26</td><td>19.75</td><td>6.41</td><td>79.01</td><td>85.11</td><td>83.12</td><td>33.16</td><td>52.87</td><td>24.60</td><td>86.05</td><td>78.45</td><td>68.16</td><td>39.86</td><td>69.78</td><td>26.56</td></tr><tr><td>DCP</td><td>27.34</td><td>31.56</td><td>14.34</td><td>86.50</td><td>81.21</td><td>78.26</td><td>71.51</td><td>71.60</td><td>78.56</td><td>94.10</td><td>86.28</td><td>87.40</td><td>35.50</td><td>74.68</td><td>30.33</td></tr><tr><td>DSIMVC</td><td>21.53</td><td>25.29</td><td>8.34</td><td>83.93</td><td>76.72</td><td>73.95</td><td>28.18</td><td>48.94</td><td>21.00</td><td>94.40</td><td>89.37</td><td>90.40</td><td>27.29</td><td>59.14</td><td>15.10</td></tr><tr><td>SURE</td><td>25.68</td><td>30.45</td><td>11.99</td><td>92.34</td><td>84.99</td><td>84.31</td><td>48.78</td><td>58.52</td><td>42.04</td><td>66.59</td><td>17.66</td><td>15.47</td><td>30.52</td><td>58.81</td><td>10.46</td></tr><tr><td>COMP</td><td>25.24</td><td>31.69</td><td>13.53</td><td>78.68</td><td>74.28</td><td>67.37</td><td>70.13</td><td>69.04</td><td>76.10</td><td>91.36</td><td>80.73</td><td>81.85</td><td>28.40</td><td>62.32</td><td>15.68</td></tr><tr><td>Ours</td><td>28.11</td><td>31.89</td><td>14.60</td><td>92.55</td><td>85.92</td><td>85.50</td><td>76.81</td><td>73.15</td><td>87.13</td><td>95.57</td><td>90.68</td><td>90.61</td><td>52.17</td><td>74.78</td><td>35.30</td></tr><tr><td rowspan="10">0.7</td><td>RPCIC</td><td>17.71</td><td>18.73</td><td>4.45</td><td>47.24</td><td>43.19</td><td>30.47</td><td>41.53</td><td>57.32</td><td>38.94</td><td>49.63</td><td>42.52</td><td>31.86</td><td>35.19</td><td>67.95</td><td>21.22</td></tr><tr><td>MRL_CAL</td><td>16.14</td><td>16.83</td><td>4.92</td><td>15.37</td><td>2.42</td><td>1.01</td><td>24.43</td><td>30.13</td><td>13.61</td><td>12.59</td><td>0.92</td><td>0.38</td><td>9.94</td><td>44.78</td><td>4.66</td></tr><tr><td>ICMVC</td><td>20.71</td><td>23.79</td><td>8.33</td><td>60.60</td><td>62.62</td><td>49.92</td><td>32.51</td><td>58.04</td><td>25.32</td><td>16.53</td><td>8.94</td><td>3.67</td><td>43.43</td><td>67.84</td><td>25.21</td></tr><tr><td>MCAC</td><td>17.20</td><td>18.52</td><td>5.03</td><td>82.91</td><td>73.41</td><td>70.02</td><td>41.08</td><td>42.96</td><td>36.68</td><td>24.78</td><td>16.48</td><td>7.55</td><td>27.96</td><td>58.19</td><td>11.35</td></tr><tr><td>PROLMP</td><td>14.74</td><td>14.68</td><td>3.68</td><td>59.23</td><td>46.45</td><td>34.17</td><td>31.77</td><td>50.79</td><td>23.29</td><td>82.97</td><td>73.36</td><td>63.36</td><td>34.27</td><td>65.00</td><td>19.27</td></tr><tr><td>DCP</td><td>24.89</td><td>27.04</td><td>11.27</td><td>85.03</td><td>78.10</td><td>77.12</td><td>70.76</td><td>69.80</td><td>75.23</td><td>90.98</td><td>84.02</td><td>85.12</td><td>34.98</td><td>67.86</td><td>24.96</td></tr><tr><td>DSIMVC</td><td>20.87</td><td>23.69</td><td>7.76</td><td>58.10</td><td>55.50</td><td>44.30</td><td>27.33</td><td>46.77</td><td>19.98</td><td>91.05</td><td>85.67</td><td>84.67</td><td>25.12</td><td>56.97</td><td>12.84</td></tr><tr><td>SURE</td><td>25.36</td><td>28.74</td><td>11.12</td><td>83.12</td><td>77.63</td><td>74.08</td><td>38.86</td><td>53.08</td><td>25.77</td><td>44.50</td><td>31.52</td><td>22.56</td><td>21.90</td><td>52.76</td><td>6.93</td></tr><tr><td>COMP</td><td>24.25</td><td>29.11</td><td>10.43</td><td>68.72</td><td>65.59</td><td>55.88</td><td>69.28</td><td>67.58</td><td>75.04</td><td>87.45</td><td>73.86</td><td>74.29</td><td>34.53</td><td>66.85</td><td>24.23</td></tr><tr><td>Ours</td><td>27.56</td><td>30.41</td><td>13.85</td><td>88.89</td><td>79.05</td><td>77.73</td><td>72.02</td><td>70.39</td><td>76.30</td><td>91.50</td><td>85.86</td><td>85.91</td><td>43.47</td><td>68.49</td><td>25.99</td></tr></table>
+
+across all three metrics, with an ACC of  $68.69\%$ , NMI of  $67.63\%$ , and ARI of  $73.68\%$ . By further incorporating the distribution alignment loss  $L_{MMD}$  in model M-12, the ACC increases to  $71.29\%$  and both NMI and ARI also improve correspondingly. When all four loss modules are utilized (M-15), the model achieves the highest performance, with all metrics surpassing those of models using only partial components. This clearly demonstrates that each loss module plays a crucial role in improving the model's performance.
+
+# 4.3 Model Analysis
+
+Parameter Analysis In this section, we analyze the sensitivity of the four hyperparameters  $\lambda_1$ ,  $\lambda_2$ ,  $\lambda_3$ , and  $\lambda_4$  in the total loss function of our proposed method. Experiments are conducted on the Caltech101-20 dataset with a missing rate of 0.5, and each hyperparameter is evaluated over the range  $\{0.01, 0.1, 1, 10, 100\}$ . As shown in Figure 2, smaller values of  $\lambda_1$  and  $\lambda_2$  lead to better performance, with the optimal value being 0.1, while the best values for  $\lambda_3$  and  $\lambda_4$  are 10 and 1, respectively. Although varying these hyperparameters has some effect on model performance, the fluctuations are minor, indicating that the model is relatively robust to hyperparameter changes.
+
+Visualization As shown in Figure 3, we perform t-SNE visualizations of the learned common representations on the Caltech101-20 and Noisy MNIST datasets. Figures (a)-(d) and (e)-(h) show the clustering results at different training epochs for the two datasets, respectively. As training progresses, the clustering structures in the feature space become increasingly clear, with more distinct inter-class boundaries and more compact intra-class distributions, demonstrating stronger discriminative ability and improved clustering performance.
+
+Convergence Analysis To verify the convergence of the proposed method, we conduct experiments on multiple datasets to observe the changes in loss values and clustering metrics over training epochs. As shown in Figure 4, the corresponding curves on the Caltech101-20 and Noisy MNIST and LandUse_21 datasets indicate that as the number of training epochs increases, the loss values decrease
+
+![](images/0c6f2f8aff883a3b445cc80ffe449ac129a1e2a4e9cbad85148ce82e3511cfd9.jpg)  
+(a) Analysis of parameters  $\lambda_{1}$  and  $\lambda_{2}$
+
+![](images/986059cf6627fecf8fe00eff7a39a7f4692f70a9a514114f25c40bb8d4713857.jpg)
+
+![](images/77143e68740faf4c4bb5818020a69bbbe4a8dadc85cfcb30dfcd912da71d7903.jpg)
+
+![](images/3d3f0e18fe7674bc0ede5656c4eee2e945b0fa6f6bc8fdbbad63929e69c0ca89.jpg)  
+(b) Analysis of parameters  $\lambda_3$  and  $\lambda_4$
+
+![](images/c5739a1ea642a0047a57bbab20b4695d0fa08f9ba60c6c4127d2b630000d5e04.jpg)  
+Figure 2: Parameter sensitivity analysis on Caltech101-20 dataset with missing rate 0.5.
+
+![](images/5568a7a58a62f4addafd1c470b51fc69b8461c900bc9be85fa4083e4804bd81e.jpg)
+
+Table 2: Performance comparison across different loss combinations on Caltech101-20 dataset.  
+
+<table><tr><td>Model</td><td>M-1</td><td>M-2</td><td>M-3</td><td>M-4</td><td>M-5</td><td>M-6</td><td>M-7</td><td>M-8</td><td>M-9</td><td>M-10</td><td>M-11</td><td>M-12</td><td>M-13</td><td>M-14</td><td>M-15</td></tr><tr><td>\( L_{REC} \)</td><td>✓</td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td>✓</td><td>✓</td><td></td><td>✓</td><td>✓</td></tr><tr><td>\( L_{MMI} \)</td><td></td><td>✓</td><td></td><td></td><td></td><td></td><td></td><td>✓</td><td>✓</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>\( L_{MMD} \)</td><td></td><td></td><td>✓</td><td></td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td>✓</td><td>✓</td><td>✓</td><td></td><td>✓</td></tr><tr><td>\( L_{INF} \)</td><td></td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>✓</td><td></td><td></td><td>✓</td><td>✓</td><td></td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>ACC</td><td>42.07</td><td>54.92</td><td>36.90</td><td>43.96</td><td>42.45</td><td>51.52</td><td>55.97</td><td>67.40</td><td>68.69</td><td>71.60</td><td>50.80</td><td>71.29</td><td>72.31</td><td>73.15</td><td>76.81</td></tr><tr><td>NMI</td><td>41.06</td><td>54.70</td><td>44.70</td><td>31.71</td><td>28.38</td><td>53.68</td><td>57.85</td><td>67.97</td><td>67.63</td><td>71.12</td><td>53.12</td><td>68.48</td><td>71.65</td><td>71.25</td><td>73.15</td></tr><tr><td>ARI</td><td>28.38</td><td>51.66</td><td>23.50</td><td>29.61</td><td>31.05</td><td>38.05</td><td>51.22</td><td>71.08</td><td>73.68</td><td>80.30</td><td>34.68</td><td>78.27</td><td>78.72</td><td>79.88</td><td>87.13</td></tr></table>
+
+significantly, while the clustering metrics steadily improve. These results further demonstrate the convergence and stability of the proposed method across different datasets.
+
+![](images/2688829bd7f80d206ed60baf5dca5dc8e6f8e7759063136712f499231560fcb4.jpg)  
+(a) Caltech101-20
+
+![](images/dfe22342f90e4852e1a0384c26581910b01b40519ed999372de39fe30aa0128e.jpg)  
+(b) Noisy MNIST  
+Figure 4: Clustering metrics (ACC, NMI, ARI) and loss curves over training epochs on the Caltech101-20, Noisy MNIST, and LandUse_21 datasets.
+
+![](images/caf886665750213d64526a12db7d91100c975ca365469fb0aa6d496c95fdb1f6.jpg)  
+(c) LandUse_21
+
+# 4.4 Experiments With More Than Two Views
+
+To validate scalability, we will add experiments on 5-view Mfeat dataset. Table 3 compares the training time and clustering metrics (ACC, NMI, ARI) of multi-view clustering with different numbers of views (from 2 to 5). As the number of views increases, the training time rises from 52.80 seconds for 2 views to 186.99 seconds for 5 views, while the clustering performance steadily improves, achieving the highest ACC (88.64), NMI (88.14), and ARI (81.09) with 5 views.
+
+![](images/a6c7291e21160e3cb62c0bf95efbd09632d2d3b4832564b1d57342690111fc52.jpg)  
+(a) 50 epoch
+
+![](images/fe7fc793070ad1be5eff746f86acc1f72cc1f5f4d980466710b1299f1f4c07d5.jpg)  
+(b) 150 epoch
+
+![](images/5ae71aa00f45abcd9fa853c323e4db877cf15dc90adf665809cc1246f2800481.jpg)  
+(c) 250 epoch
+
+![](images/dd7f09fd832ebb124b8a703fca9aeae9601ade11cc284218a152782c9e3ad07d.jpg)  
+(d) 600 epoch
+
+![](images/1b38d4c86f2264efebc245bc950bf92ac17fba05664d3a1ef8883554b0edabad.jpg)  
+(e) 50 epoch
+
+![](images/166e2e1f9a27863db88d2d1bb8229d1441e20801f9e9beb1fdd613e2bcd3ef79.jpg)  
+(f) 100 epoch
+
+![](images/0fd7f06a6261bf21269a27cc044a3cf29e4961ffbe6a0e89316ce50840afb549.jpg)  
+(g) 350 epoch  
+Figure 3: t-SNE visualizations of clustering results on the Caltech101-20 and Noisy MNIST datasets with increasing training iteration. (a)-(d) show the results on the Caltech101-20 dataset. (e)-(h) show the results on the Noisy MNIST dataset.
+
+![](images/d9499e1eb87db29b65210409ebbcd109188b33a06eaf84bdce7882b86c652f64.jpg)  
+(h) 500 epoch
+
+Table 3: Performance of HSACC on Mfeat dataset with different numbers of views.  
+
+<table><tr><td>Views Number</td><td>Training Time (s)</td><td>ACC</td><td>NMI</td><td>ARI</td></tr><tr><td>2</td><td>52.80</td><td>74.96</td><td>74.58</td><td>59.35</td></tr><tr><td>3</td><td>98.64</td><td>77.68</td><td>78.11</td><td>62.17</td></tr><tr><td>4</td><td>138.70</td><td>85.91</td><td>85.55</td><td>77.11</td></tr><tr><td>5</td><td>186.99</td><td>88.64</td><td>88.14</td><td>81.09</td></tr></table>
+
+# 5 Conclusion
+
+In this paper, we propose a novel framework, HSACC, based on hierarchical semantic alignment and cooperative completion. By designing a dual-level semantic space and employing a joint optimization strategy, the proposed method maps the learned latent representations into high-dimensional semantic spaces to implicitly infer missing view information, thereby providing more complete representations for subsequent clustering tasks. Extensive experimental results demonstrate that HSACC consistently outperforms existing mainstream methods across multiple datasets, verifying its effectiveness and superiority. In future work, we plan to extend this framework to more complex multi-modal incomplete data scenarios, and further enhance its generalization ability and computational efficiency in real-world large-scale applications.
+
+# 6 Acknowledgements
+
+This work was supported by the Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (2024SJYB0220), and the Scientific Research Innovation Project of Nanjing University of Finance and Economics (XKYC2202507).
+
+# References
+
+[1] H. Gao, Y. Peng, and S. Jian, "Incomplete multi-view clustering," in Intelligent Information Processing VIII: 9th IFIP TC 12 International Conference, IIP 2016, Melbourne, VIC, Australia, November 18-21, 2016, Proceedings 9. Springer, 2016, pp. 245-255.  
+[2] J. Yin and S. Sun, "Incomplete multi-view clustering with reconstructed views," IEEE Transactions on Knowledge and Data Engineering, vol. 35, no. 3, pp. 2671-2682, 2023.  
+[3] X. Liu, M. Li, C. Tang, J. Xia, J. Xiong, L. Liu, M. Kloft, and E. Zhu, "Efficient and effective regularized incomplete multi-view clustering," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 43, no. 8, pp. 2634-2646, 2021.
+
+[4] H. Zhao, Z. Ding, and Y. Fu, "Multi-view clustering via deep matrix factorization," in Proceedings of the AAAI conference on artificial intelligence, vol. 31, no. 1, 2017.  
+[5] J. Wen, G. Xu, Z. Tang, W. Wang, L. Fei, and Y. Xu, “Graph regularized and feature aware matrix factorization for robust incomplete multi-view clustering,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 34, no. 5, pp. 3728–3741, 2024.  
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+[7] J. Guo and J. Ye, "Anchors bring ease: An embarrassingly simple approach to partial multi-view clustering," in Proceedings of the AAAI conference on artificial intelligence, vol. 33, no. 01, 2019, pp. 118-125.  
+[8] S. Deng, J. Wen, C. Liu, K. Yan, G. Xu, and Y. Xu, "Projective incomplete multi-view clustering," IEEE Transactions on Neural Networks and Learning Systems, vol. 35, no. 8, pp. 10539-10551, 2024.  
+[9] N. Xu, Y. Guo, X. Zheng, Q. Wang, and X. Luo, “Partial multi-view subspace clustering,” in Proceedings of the 26th ACM International conference on multimedia, 2018, pp. 1794–1801.  
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+
+# NeurIPS Paper Checklist
+
+The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
+
+Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
+
+- You should answer [Yes], [No], or [NA].  
+- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.  
+- Please provide a short (1–2 sentence) justification right after your answer (even for NA).
+
+The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
+
+The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
+
+IMPORTANT, please:
+
+- Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist",  
+- Keep the checklist subsection headings, questions/answers and guidelines below.  
+- Do not modify the questions and only use the provided macros for your answers.
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: The scope and contributions of this paper are clearly stated in the abstract and introduction.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: The limitations are discussed in the Conclusion section.
+
+# Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: Our work is empirical in nature and does not contain formal theoretical results or proofs.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: The paper provides sufficient details on the algorithm, datasets, and experimental setup to support reproducibility of the main results.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
+
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
+
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: We will upload the code package as supplementary material, and the latest GitHub link will be provided upon paper acceptance.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.
+
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: The paper provides the details of data preprocessing and hyperparameter settings in the Experiments section, while the optimization process is included in the appendix.\*\*
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Justification: Most compared baseline methods do not include statistical significance in their experiments; therefore, we follow the same practice and do not report it in our work.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: The computer resources is stated in the Experiments section.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: This paper conforms with the NeurIPS Code of Ethics.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: The proposed algorithm has no direct societal impact. All datasets used in this paper are publicly available, and the algorithm is solely focused on producing clustering assignments.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: All datasets used in this paper are publicly available, so no specific safeguards are required.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: Original papers and datasets are properly cited.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: No new assets are introduced in the paper.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.
+
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: No crowdsourcing or human subjects were involved in this work.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve any research with human subjects or crowdsourcing, and therefore does not require IRB or equivalent ethical approval.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The core methodology does not involve the use of large language models (LLMs).
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Technical Appendices and Supplementary Material
+
+# A.1 Visualization
+
+As shown in Figure 5, we perform t-SNE visualizations of the learned common representations on the LandUse_21 and Hdigit and 100leaves datasets. Figures (a)-(d) show the clustering results of the LandUse_21 dataset at different training epochs, Figures (e)-(h) present the clustering results of the Hdigit dataset at different training epochs, while Figures (i)-(l) show the clustering results of the 100leaves dataset at different training epochs. It can be observed that as the number of training epochs increases, the learned common representations exhibit increasingly clearer clustering structures in the feature space. The boundaries between different categories become more distinct, and the intra-class sample distributions become more compact, demonstrating stronger discriminative ability and improved clustering performance.
+
+![](images/edefe6649a93765abb8f6a07b3d8df5f023b154349115727df0f572eebe1bb86.jpg)  
+(a) 50 epoch
+
+![](images/468c7b919f8b52823ac4e43a98c14981f39c0b84f6c8864ea02a7f9d0b9bfe67.jpg)  
+(b) 100 epoch
+
+![](images/cc6853145d45bfc10ba6006d7507c4c8dfbdb4055482f34586ab781768144d2d.jpg)  
+(c) 200 epoch
+
+![](images/c3e32d04690412a044129c33d83bf5eba4ecd23f933e7efa7ec0b826a71568f8.jpg)  
+(d) 500 epoch
+
+![](images/4021aec8ae74ff21125a018aba27b1151e6ac415fe95a8a3e1b5091d3380755a.jpg)
+
+![](images/769339a04a2bae4b8393cd759b440b37332f831a95d545664bc57d355b25f6b1.jpg)
+
+![](images/4049f5e18e6e76f6ae6a5cd34d612cd6edfdfed2b79fe6fada0a7f404fa6b1b6.jpg)
+
+![](images/37bd3b479c27cccbc6f0c9c7e007202b54009c7606acdd60c35ab394300e11f2.jpg)
+
+![](images/52b8178d7d037ac94adaa5f444d7ac0527c0ae4ba2c0ff53f20178c0df90f127.jpg)  
+(e) 50 epoch  
+(i) 50 epoch  
+Figure 5: t-SNE visualizations of clustering results on the LandUse_21 and Hdigit and 100leaves datasets with increasing training iteration. (a)-(d) show the results on the LandUse_21 dataset. (e)-(h) show the results on the Hdigit dataset. (i)-(j) show the results on the 100leaves dataset.
+
+![](images/4699ac2b28a863023c5303792ded2bdea987b4c6a02a0c013256bffa89c3cf39.jpg)  
+(f) 100 epoch  
+(j) 100 epoch
+
+![](images/1093413de8299a4c3ae2d00262a27cd1eb7b55d02f782e7a5c254efacf2e97d3.jpg)  
+(g) 150 epoch  
+(k) 150 epoch
+
+![](images/9d0ef6477ab3c3dc3560e93d8a4c637f19b560e44e72006b7cfbdf12c59e6e2d.jpg)  
+(h) 200 epoch  
+(1) 450 epoch

2025/Incomplete Multi-view Clustering via Hierarchical Semantic Alignment and Cooperative Completion/images.zip

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2025/Incomplete Multi-view Deep Clustering with Data Imputation and Alignment/e7860326-e4c1-4de4-8db7-83dbf37bfc65_origin.pdf

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2025/Incomplete Multi-view Deep Clustering with Data Imputation and Alignment/full.md

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+# Incomplete Multi-view Deep Clustering with Data Imputation and Alignment
+
+Jiyuan Liu $^{1}$ , Xinwang Liu $^{1*}$ , Xinhang Wan $^{1}$ , Ke Liang $^{1}$ , Weixuan Liang $^{1}$ , Sihang Zhou $^{1}$ , Huijun Wu $^{1}$  and Kehua Guo $^{2}$
+
+$^{1}$  National University of Defense Technology, Changsha, Hunan, China. 410072.
+
+$^{2}$  Central South University, Changsha, Hunan, China. 410083.
+
+liujiyuan13@nudt.edu.cn
+
+# Abstract
+
+Incomplete multi-view deep clustering is an emerging research hot-pot to incorporate data information of multiple sources or modalities when parts of them are missing. Most of existing approaches encode the available data observations into multiple view-specific latent representations and subsequently integrate them for the next clustering task. However, they ignore that the latent representations are unique to a fixed set of data samples in all views. Meanwhile, the pair-wise similarities of missing data observations are also failed to utilize in latent representation learning sufficiently, leading to unsatisfactory clustering performance. To address these issues, we propose an incomplete multi-view deep clustering method with data imputation and alignment. Assuming that each data sample corresponds to a same latent representation among all views, it projects the latent representations into feature spaces with neural networks. As a result, not only the available data observations are reconstructed, but also the missing ones can be imputed accordingly. Moreover, a linear alignment measurement of linear complexity is defined to compute the pair-wise similarities of all data observations, especially including those of the missing. By executing the above two procedures iteratively, the discriminative latent representations can be learned and used to group the data into categories with off-the-shelf clustering algorithms. In experiment, the proposed method is validated on a set of benchmark datasets and achieves state-of-the-art performances.
+
+# 1 Introduction
+
+With the development of electronic device and information technology, the data observations are widely accumulated from different sources and modalities, collectively referred to as multi-view data in academic communities. For instance, when making advertisement recommendation to Internet users, the content provider would collect their personal details from multiple aspects in advance, such as living habit, shopping record, etc. [1] In the diagnosis and treatment of Alzheimer's disease, multiple types of data are collected and analyzed along with a set of medical examinations, including blood test, Cerebrospinal Fluid (CSF) examination, Magnetic Resonance Imaging (MRI), Computed Tomography (CT), Positron Emission Tomography (PET), etc. [2] Therefore, how to integrate the multi-view data effectively and efficiently is a critical problem. In this background, multi-view clustering can explore the consensus and complementary information among different data views and improve the performance over single-view clustering by large margins, catching a large volume of attentions from researchers [3].
+
+Since most of the existing multi-view clustering are derived from single-view algorithms, they can be grouped into five categories accordingly, i.e., multiple kernel clustering [4, 5, 6, 5], multi-view subspace clustering [7, 8, 9], multi-view spectral clustering [10, 11], multi-view matrix factorization [12, 13] and multi-view deep clustering [14, 15]. Inside, multi-view deep clustering is emerging in recent years along with the rapid development of neural network techniques. Among the advances, Cui et al. propose to extract high-level features from data observations of each view with multiple view-specific fully connected auto-encoders, then design the dual contrasting losses to maximize the distance between different clusters and enhance the compactness within clusters [16]. Yu et al. employ graph convolutional encoder on each data view [17]. Subsequently, the contrastive learning technique and block diagonalization constraint are adopted to guide representation matrix learning, while representation learning is utilized into clustering result generation. Similarly, Chen et al. learn view-invariant data representations with auto-encoders and compute results by contrasting the cluster assignments among different views [18]. Usually, multi-view deep clustering approaches achieve better performance than the others due to the superior capability of neural networks to generate high-quality latent representations on data observations.
+
+Due to sensor failure or restricted conditions, the collected multi-view data are often incomplete where one or more views are missing completely. In such setting, the researchers propose a number of incomplete multi-view deep clustering approaches to make the most of available data observations rather than discarding the data samples with missing views [19, 20]. Mostly, they first encode the available data observations into multiple view-specific latent representations and subsequently integrate them by designing different losses on them. For instance, Chao et al. adopt the contrastive loss to maximize representation similarities of the same data sample in different views while minimize those of different data samples [21]. At the same time, Lin et al. modify the contrastive loss to maximize mutual information across all data views, promoting the learning of informative and consistent latent representations [22].
+
+Although the existing methods achieves satisfactory performance, they ignore the fact that the latent representations of a fixed set of data samples are unique and invariant to different views. Meanwhile, they only concentrate on the integration of available data observations but overlook the pair-wise similarity information among missing data observations, limiting the further improvement of clustering performance. To address these issues, we propose a novel Incomplete Multi-view Deep Clustering with Data Imputation and Alignment (IMDC-DIA). Specifically, it assumes that all views share a same latent representation with respect to a certain data sample. Contrary to the mainstream methods of encoding data observations into latent representations, the proposed approach projects the unique latent representations to feature space of each view with multiple independent neural networks. As a consequence, not only the available data observations are reconstructed, but also the missing ones can be imputed accordingly. Moreover, a linear alignment measurement is defined and analyzed accordingly. Then, we adopt it to align the latent representations with the data observations of each view, so as to utilize the pair-wise similarities of all data observations, especially including those of the missing. By executing the above data imputation and alignment coherently and iteratively, the discriminative latent representations can be learned and used to group the data into categories with off-the-shelf clustering algorithms. To validate effectiveness of the proposed IMDC-DIA method, we conduct extensive experiments on a set of benchmark datasets and compare it with the most recent advances in literature. Corresponding results show it achieves state-of-the-art clustering performance in almost all settings, well validating its effectiveness and superiority.
+
+# 2 Related work
+
+# 2.1 Multi-view deep clustering
+
+As a representative of multi-view clustering, multi-view deep clustering is one of the most effective technique to group data samples with integrating their multi-view information. Typically, it achieves better performances than the other multi-view clustering methods, since the adopted neural networks can extract higher-quality latent representations on the data observations of each view. Among them, the Multiple Layer Perceptron (MLP) network is the most widely used in literature, such as those in [16, 23]. Besides, the other types of neural networks are also employed to further improve the quantity of latent representations. For example, Yu et al. generate the view-specific latent representations with Graph Convolutional Network (GCN) encoder on each view, successfully capturing the structural data information along with learning attribute features [17]. Observing the poor scalability of MLP,
+
+Zhu et al. transform the feature vector into image and use Convolutional Neural Networks (CNN) subsequently, achieving satisfactory clustering performance [24]. Yang et al. adopt the Transformer structure by utilizing self-attention mechanism, enabling the encoder to better retain complementary information across different views and remove the exclusive information [25].
+
+Apart from the neural network structure, plenty of constraints and loss functions are developed in existing researches. Inside, contrastive loss is the most popular one of them. Commonly, it regards the latent representations of a data sample in different views as positive pairs while the rest as negative pairs, and maximizes the similarities of the former while minimizes those of the latter [25, 23]. At the same time, Cui et al. integrate the pseudo-labels in training stage and consider the latent representations of data samples in a same temporary cluster as positive pair, improving the clustering performance effectively [16]. In addition, a large number of researches employ the self-representation loss derive from the self-representation constraint in classical multi-view clustering [24, 26]. It assumes each latent representation to be a linear sum of the others and minimizes their differences. Nevertheless, some of the other popular constraints and loss functions would be the adversarial similarity constraint [27], the discriminative constraint [15] and the self-supervised loss [17].
+
+# 2.2 Incomplete multi-view deep clustering
+
+To deal with incomplete data in real-world scenarios, incomplete multi-view deep clustering is emerging to be a promising approach and explored in recent literature. Some of them only rely on the available data observations to compute the clustering results [28, 29]. They usually encode the available data observations into latent representations with multiple neural networks with each in one data view. On the basis, Wen et al. propose a graph embedding strategy to simultaneously capture the high-level features and local structure of each data view, while a self-paced strategy is also utilized to select the most confident samples in model training, reducing the negative influence of outliers [29]. Differently, Xu et al. learn latent representations by incorporating the Mixture-of-Gaussians prior information to enhance their clustering-friendly structure and develop a Product-of-Experts approach to efficiently aggregate them [30]. Xu et al. utilize all view-specific latent representations into a consensus one with an adaptive feature projection module, avoiding to impute the missing data [31]. Then, the correlated common cluster information is explored by maximizing its mutual information, while the distribution alignment is achieved by minimizing its mean discrepancy coherently.
+
+Different from the above approaches, the other incomplete multi-view deep clustering methods try to recover the missing data along with the latent representation learning and data clustering [21, 32, 33]. For instance, Wang et al. learn the low-dimensional latent representations with view-specific encoder networks and explicitly generate the missing data with generative adversarial networks at the same time [33]. Lin et al. unify the cross-view consistency learning and data recovery techniques by maximizing the mutual information of different views and minimizing the conditional entropy through dual prediction, respectively [22, 34]. Liu et al. propose a two-stage autoencoder network with recurrent graph reconstruction mechanism to extract high-level latent representations and recover the missing data synchronously [35].
+
+# 3 Methodology
+
+In this section, the proposed IMDC-DIA method is introduced, where its framework is visualized in Fig. 1. Briefly, it aims to learn the high-quality latent representations which are subsequently fed to off-the-shelf clustering algorithms so as to group the data accurately. To accomplish this, the latent representations are assumed to be unique to all views. On the basis, three main components are concerned, i.e., data reconstruction, imputation and alignment. On the left of Fig. 1, the proposed IMDC-DIA method projects the unique latent representations to feature space of each view with multiple independent neural networks. Thereby, not only the available data observations are reconstructed (data reconstruction), but also the missing ones can be imputed and completed (data imputation) accordingly. In data alignment, a linear alignment measurement is defined and, as shown on the right of Fig. 1, is adopted to align the unique data representations with the completed data observations of each view.
+
+![](images/5c942c73a2f9d91f4a3a5a1054f7080334072d74934d51fc75db51b317dcfafd.jpg)  
+Figure 1: Framework of the proposed IMDC-DIA method. For the ease of expression, only two data views are presented, but arbitrary number of views are supported. Also, any types of neural networks, apart from the fully-connected, can be employed, only if they project the latent representations into data feature space.
+
+# 3.1 Formulation
+
+Denoting  $N, V$  and  $D_v$  as the number of data samples, the number of views and the feature dimension of  $v$ -th view, the multi-view data observations can be specified to  $\{\mathbf{x}_i^{(v)}\}_{i,v=1}^{N,V}$  in which  $\mathbf{x}_i^{(v)} \in \mathbb{R}^{D_v}$ . Corresponding availability indicators are  $\{\mathbf{a}_i\}_{i=1}^N$  where  $\mathbf{a}_i$  is a discrete vector in  $\{0,1\}^V$ . Here, the data observation of  $v$ -th view  $\mathbf{x}_i^{(v)}$  is available if the  $v$ -th element  $\mathbf{a}_{i,v}$  is 1, while missing if 0. In such setting, incomplete multi-view deep clustering methods aim to group the data into clusters with the available observations where the cluster number  $K$  is known in advance.
+
+# 3.1.1 Data reconstruction and imputation
+
+Instead of encoding the available data observations into multiple view-specific latent representations, the proposed IMDC-DIA method assumes that the data latent structure is intrinsic and corresponding latent representations should be unique to all data views. Denoting the latent representations to be  $\{\mathbf{z}_i\}_{i=1}^N$  where  $\mathbf{z}_i \in \mathbb{R}^D$ , it proposes to reconstruct the multi-view data observations by projecting the latent representations into feature spaces with  $V$  independent neural networks as
+
+$$
+\hat {\mathbf {x}} _ {i} ^ {(v)} = g _ {\Theta_ {v}} \left(\mathbf {z} _ {i}\right), \tag {1}
+$$
+
+where  $\Theta_v$  represents parameters of the  $v$ -th neural network. It is worth to note that the latent representation  $\mathbf{z}_i$  is not fixed but a variable which can be optimized in model training. Correspondingly, we can obtain the reconstruction loss by minimizing the differences between the generated data and the available data observations via
+
+$$
+L _ {r} = \frac {1}{N} \sum_ {i = 1} ^ {N} \frac {1}{V _ {i}} \sum_ {v = 1} ^ {V} \mathbf {a} _ {i, v} \| \hat {\mathbf {x}} _ {i} ^ {(v)} - \mathbf {x} _ {i} ^ {(v)} \| _ {2} ^ {2}, \tag {2}
+$$
+
+in which  $V_{i}$  refers to the number of available views of  $i$ -th data sample. Nevertheless, it can be observed in aforementioned reconstruction process that not only the available data observations are reconstructed, but also the missing ones are imputed accordingly. Therefore, the multi-view data can be completed by filling the missing with those imputed in Eq. (1). As a result, the complete data can be formulated as
+
+$$
+\bar {\mathbf {x}} _ {i} ^ {(v)} = \left\{ \begin{array}{l l} \mathbf {x} _ {i} ^ {(v)} & \text {i f} \quad \mathbf {a} _ {i, v} = 1, \\ \hat {\mathbf {x}} _ {i} ^ {(v)} & \text {i f} \quad \mathbf {a} _ {i, v} = 0. \end{array} \right. \tag {3}
+$$
+
+# 3.1.2 Data alignment
+
+On the basis of the complete multi-view data of Eq. (3), the proposed IMDC-DIA method also incorporate the pair-wise similarities of all data observations, especially those of the missing, to regularize the latent representations. Primarily, the Linear Alignment of two matrices is defined in Definition 1.
+
+Definition 1 (Linear Alignment) Given two arbitrary matrices  $\mathbf{X}_1$  and  $\mathbf{X}_2$  of size  $n \times d_1$  and  $n \times d_2$  respectively, the linear alignment is computed as
+
+$$
+L A = \frac {L _ {F} ^ {2} \left(\mathbf {X} _ {1} ^ {\top} \mathbf {X} _ {2}\right)}{L _ {F} \left(\mathbf {X} _ {1} ^ {\top} \mathbf {X} _ {1}\right) \cdot L _ {F} \left(\mathbf {X} _ {2} ^ {\top} \mathbf {X} _ {2}\right)}, \tag {4}
+$$
+
+in which  $L_{F}(\cdot)$  denotes the Frobenius norm of matrix. Corresponding computation complexity is linear to  $n$ .
+
+Denoting vector  $\mathbf{x}_i$  to the  $i$ -th row of an arbitrary matrix  $\mathbf{X}$ , the pair-wise linear similarity between its  $i$ -th and  $j$ -th rows is measured by their dot-products that
+
+$$
+k _ {i, j} = \mathbf {x} _ {i} \mathbf {x} _ {j} ^ {\top}, \quad w. r. t. i, j \in \{1, 2, \dots , N \} \tag {5}
+$$
+
+With considering two arbitrary matrices  $\mathbf{X}_1$  and  $\mathbf{X}_2$  coherently, their pair-wise linear similarities should be
+
+$$
+k _ {i, j} ^ {1} = \mathbf {x} _ {i} ^ {1} \mathbf {x} _ {j} ^ {1 \top} \quad \text {a n d} \quad k _ {i, j} ^ {2} = \mathbf {x} _ {i} ^ {2} \mathbf {x} _ {j} ^ {2 \top} \tag {6}
+$$
+
+On this basis, the similarity consistency of these two matrices can be measured by the sum of their dot-products as
+
+$$
+c = \frac {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {1} k _ {i , j} ^ {2}}{\sqrt {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {1} k _ {i , j} ^ {1}} \sqrt {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {2} k _ {i , j} ^ {2}}} \tag {7}
+$$
+
+where the denominator is a scalar term to ensure the obtained consistency in range  $[-1, 1]$ .
+
+Theorem 1 The linear alignment of two arbitrary matrices measures the consistency of the pair-wise linear similarities of their rows.
+
+Proof 1 The consistency of Eq. (7) can be transformed to
+
+$$
+\begin{array}{l} c = \frac {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {1} k _ {i , j} ^ {2}}{\sqrt {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {1} k _ {i , j} ^ {1}} \sqrt {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {2} k _ {i , j} ^ {2}}} = \frac {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {1} k _ {j , i} ^ {2}}{\sqrt {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {1} k _ {i , j} ^ {1}} \sqrt {\sum_ {i , j = 1} ^ {N} k _ {i , j} ^ {2} k _ {i , j} ^ {2}}} \\ = \frac {<   \mathbf {K} _ {1} , \mathbf {K} _ {2} > _ {F}}{\sqrt {<   \mathbf {K} _ {1} , \mathbf {K} _ {1} > _ {F} <   \mathbf {K} _ {2} , \mathbf {K} _ {2} > _ {F}}} = \frac {\operatorname {T r} [ (\mathbf {X} _ {1} \mathbf {X} _ {1} ^ {\top}) (\mathbf {X} _ {2} \mathbf {X} _ {2} ^ {\top}) ]}{\sqrt {\operatorname {T r} [ (\mathbf {X} _ {1} \mathbf {X} _ {1} ^ {\top}) (\mathbf {X} _ {1} \mathbf {X} _ {1} ^ {\top}) ] \cdot \operatorname {T r} [ (\mathbf {X} _ {2} \mathbf {X} _ {2} ^ {\top}) (\mathbf {X} _ {2} \mathbf {X} _ {2} ^ {\top}) ]}} \tag {8} \\ = \frac {\mathrm {T r} [ (\mathbf {X} _ {1} ^ {\top} \mathbf {X} _ {2}) ^ {\top} (\mathbf {X} _ {1} ^ {\top} \mathbf {X} _ {2}) ]}{\sqrt {\mathrm {T r} [ (\mathbf {X} _ {1} ^ {\top} \mathbf {X} _ {1}) (\mathbf {X} _ {1} ^ {\top} \mathbf {X} _ {1}) ] \cdot \mathrm {T r} [ (\mathbf {X} _ {2} ^ {\top} \mathbf {X} _ {2}) (\mathbf {X} _ {2} ^ {\top} \mathbf {X} _ {2}) ]}} = \frac {\| \mathbf {X} _ {1} ^ {\top} \mathbf {X} _ {2} \| _ {F} ^ {2}}{\| \mathbf {X} _ {1} ^ {\top} \mathbf {X} _ {1} \| _ {F} \cdot \| \mathbf {X} _ {2} ^ {\top} \mathbf {X} _ {2} \| _ {F}} = L A, \\ \end{array}
+$$
+
+in which the second equation holds for
+
+$$
+k _ {j, i} ^ {2} = \mathbf {x} _ {2, j} \mathbf {x} _ {2, i} ^ {\top} = \mathbf {x} _ {2, i} \mathbf {x} _ {2, j} ^ {\top} = k _ {i, j} ^ {2}, \tag {9}
+$$
+
+while the third holds for
+
+$$
+\mathbf {K} _ {1} = \left[ \begin{array}{c c c c} \mathbf {x} _ {1, 1} \mathbf {x} _ {1, 1} ^ {\top} & \mathbf {x} _ {1, 1} \mathbf {x} _ {1, 2} ^ {\top} & \dots & \mathbf {x} _ {1, 1} \mathbf {x} _ {1, N} ^ {\top} \\ \mathbf {x} _ {1, 2} \mathbf {x} _ {1, 1} ^ {\top} & \mathbf {x} _ {1, 2} \mathbf {x} _ {1, 2} ^ {\top} & \dots & \mathbf {x} _ {1, 2} \mathbf {x} _ {1, N} ^ {\top} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf {x} _ {1, N} \mathbf {x} _ {1, 1} ^ {\top} & \mathbf {x} _ {1, N} \mathbf {x} _ {1, 2} ^ {\top} & \dots & \mathbf {x} _ {1, N} \mathbf {x} _ {1, N} ^ {\top} \end{array} \right] = \left[ \begin{array}{c c c c} k _ {1, 1} ^ {1} & k _ {1, 2} ^ {1} & \dots & k _ {1, 3} ^ {1} \\ k _ {2, 2} ^ {1} & k _ {2, 2} ^ {1} & \dots & k _ {2, 3} ^ {1} \\ \dots & \dots & \ddots & \dots \\ k _ {N, 1} ^ {1} & k _ {N, 2} ^ {1} & \dots & k _ {N, N} ^ {1} \end{array} \right] a n d
+$$
+
+$$
+\mathbf {K} _ {2} = \left[ \begin{array}{c c c c} \mathbf {x} _ {2, 1} \mathbf {x} _ {2, 1} ^ {\top} & \mathbf {x} _ {2, 1} \mathbf {x} _ {2, 2} ^ {\top} & \dots & \mathbf {x} _ {2, 1} \mathbf {x} _ {2, N} ^ {\top} \\ \mathbf {x} _ {2, 2} \mathbf {x} _ {2, 1} ^ {\top} & \mathbf {x} _ {2, 2} \mathbf {x} _ {2, 2} ^ {\top} & \dots & \mathbf {x} _ {2, 2} \mathbf {x} _ {2, N} ^ {\top} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf {x} _ {2, N} \mathbf {x} _ {2, 1} ^ {\top} & \mathbf {x} _ {2, N} \mathbf {x} _ {2, 2} ^ {\top} & \dots & \mathbf {x} _ {2, N} \mathbf {x} _ {2, N} ^ {\top} \end{array} \right] = \left[ \begin{array}{c c c c} k _ {1, 1} ^ {2} & k _ {1, 2} ^ {2} & \dots & k _ {1, 3} ^ {2} \\ k _ {2, 2} ^ {2} & k _ {2, 2} ^ {2} & \dots & k _ {2, 3} ^ {2} \\ \dots & \dots & \ddots & \dots \\ k _ {N, 1} ^ {2} & k _ {N, 2} ^ {2} & \dots & k _ {N, N} ^ {2} \end{array} \right]
+$$
+
+This completes the proof.
+
+According to Theorem 1, the linear alignment of two arbitrary matrices measures the consistency of the pair-wise linear similarities of their rows. To consider the pair-wise similarities of all data observations, especially including those of the missing ones, we propose to maximize the linear alignment between the latent representations and the completed data observations of  $v$ -th view, i.e.,  $\max LA(\mathbf{Z},\bar{\mathbf{X}}^{(v)})$ . Therefore, corresponding loss can be formulated to
+
+$$
+L _ {a} ^ {(v)} = - L A (\mathbf {Z}, \bar {\mathbf {X}} ^ {(v)}) = - \frac {L _ {F} ^ {2} \left(\mathbf {Z} ^ {\top} \bar {\mathbf {X}} ^ {(v)}\right)}{L _ {F} \left(\mathbf {Z} ^ {\top} \mathbf {Z}\right) \cdot L _ {F} \left(\bar {\mathbf {X}} ^ {(v) \top} \bar {\mathbf {X}} ^ {(v)}\right)}, \tag {11}
+$$
+
+in which  $\mathbf{Z}$  and  $\bar{\mathbf{X}}^{(v)}$  are the matrix form of  $\{\mathbf{z}_i\}_{i=1}^N$  and  $\{\bar{\mathbf{x}}_i^{(v)}\}_{i=1}^N$ , respectively. Furthermore, to consider the data quality of each view, an additional set of weight parameters  $\{w_v\}_{v=1}^V$  are introduced to balance their effect on latent representations. As a consequence, the overall data alignment loss can be written to
+
+$$
+L _ {a} = \sum_ {v = 1} ^ {V} w _ {v} L _ {a} ^ {(v)}, \quad s. t. \sum_ {v = 1} ^ {V} w _ {v} ^ {2} = 1. \tag {12}
+$$
+
+It is worth to note that  $\{w_v\}_{v=1}^V$  are variables to be optimized in model learning rather than hyperparameters specified in advance.
+
+# 3.1.3 Loss function
+
+As seen in the framework of Fig. 1, the proposed IMDC-DIA method is composed of three main components, including the data reconstruction, imputation and alignment. By utilizing the reconstruction loss of Eq. (1), missing data imputation of Eq. (3) and data alignment loss of Eq. (12), the overall loss function is written to
+
+$$
+L = L _ {r} + \beta L _ {a}, \quad s. t. \sum_ {v = 1} ^ {V} w _ {v} ^ {2} = 1 \tag {13}
+$$
+
+in which  $\beta$  is a trade-off parameter and should be specified before model training.
+
+# 3.2 Optimization
+
+According to the overall loss of Eq. (13), there are three variables to optimize in model training, including the neural network parameter  $\{\Theta_v\}_{v=1}^V$ , the unique latent representation  $\{\mathbf{z}_i\}_{i=1}^N$  and weight parameter  $\{w_v\}_{v=1}^V$ . Each of them can be optimized in the following.
+
+Optimization of neural network parameter  $\{\Theta_v\}_{v = 1}^V$  . Same to most of deep learning methods, the neural network parameter can be optimized with gradient descent strategy. In experiment, we adopt the popular Adaptive Moment Estimation (Adam) optimizer.
+
+Optimization of latent representation  $\{\mathbf{z}_i\}_{i = 1}^N$ . Different from the existing multi-view clustering methods, such as [13], the proposed IMDC-DIA method is difficult to find the close-form solution of data latent representation. Therefore, the gradient descent strategy with Adam optimizer is adopted in its optimization.
+
+Optimization of weight parameter  $\{w_v\}_{v=1}^V$ . With fixing the others,  $L_a^{(v)}$  is given and minimizing the overall loss of Eq. (13) equals to
+
+$$
+\max  _ {\{w _ {v} \} _ {v = 1} ^ {V}} \sum_ {v = 1} ^ {V} w _ {v} \left(- L _ {a} ^ {(v)}\right), \quad s. t. \sum_ {v = 1} ^ {V} w _ {v} ^ {2} = 1. \tag {14}
+$$
+
+According to o Cauchy-Schwarz inequality [36],
+
+$$
+\sum_ {v = 1} ^ {V} w _ {v} \left(- L _ {a} ^ {(v)}\right) \leq \sqrt {\left(\sum_ {v = 1} ^ {V} w _ {v} ^ {2}\right) \left(\sum_ {v = 1} ^ {V} L _ {a} ^ {(v) 2}\right)} = \sqrt {\sum_ {v = 1} ^ {V} L _ {a} ^ {(v) 2}}, \tag {15}
+$$
+
+where the equality holds for when
+
+$$
+\frac {w _ {1}}{- L _ {a} ^ {(1)}} = \frac {w _ {2}}{- L _ {a} ^ {(2)}} = \dots = \frac {w _ {V}}{- L _ {a} ^ {(V)}}. \tag {16}
+$$
+
+Unifying the constraint  $\sum_{v=1}^{V} w_v^2 = 1$  in Eq. (14), the solution can be computed to
+
+$$
+w _ {v} ^ {*} = - L _ {a} ^ {(v)} / \sqrt {\sum_ {v ^ {\prime} = 1} ^ {V} L _ {a} ^ {(v ^ {\prime}) 2}}. \tag {17}
+$$
+
+Overall, the aforementioned three parameters are optimized alternately and the latent representations can be achieved until the convergence or maximal epoch. Next,  $k$ -means algorithm is applied on the computed latent representations  $\{\mathbf{z}_i\}_{i=1}^N$  to group multi-view data into categories. Furthermore, the pseudo-code of the optimization procedure is summarized in Alg. 1.
+
+Algorithm 1 Incomplete Multi-view Deep Clustering with Data Imputation and Alignment  
+Input: incomplete multi-view data  $\{\mathbf{x}_i^{(v)}\}_{i,v = 1}^{N,V}$  with availability indicator  $\{\mathbf{a}_i\}_{i = 1}^N$  , cluster number  $k$    
+Output: data labels   
+1: initialize latent representation  $\{\mathbf{z}_i\}_{i = 1}^N$  and network parameters  $\{\Theta_v\}_{v = 1}^V$  .   
+2:  $t = 0$    
+3: while  $t <   \mathrm{epoch}$  do   
+4: #forward   
+5: compute  $L_{r}$  with Eq. (2);   
+6: complete multi-view data with Eq. (3);   
+7: compute  $\{L_a^{(v)}\}_{v = 1}^V$  with Eq. (11);   
+8: update the view weights  $\{w_v\}_{v = 1}^V$  with Eq. (17);   
+9: compute the overall loss  $L$  with Eq. (13);   
+10: # back propagation   
+11: update latent representation  $\{\mathbf{z}_i\}_{i = 1}^N$  and network parameters  $\{\Theta_v\}_{v = 1}^V$  .   
+12:  $t = t + 1$    
+13: end while   
+14: compute the data labels with  $k$  -means on latent representations  $\{\mathbf{z}_i\}_{i = 1}^N$
+
+# 3.3 Computation complexity
+
+Specifically, the data reconstruction and imputation of Eq. (1), (2) and (3) only require projecting the latent representations into feature spaces with parameterized neural networks, hence introducing a  $\mathcal{O}(N)$  complexity. Also, corresponding updates on latent representations and network parameters are of  $\mathcal{O}(N)$  complexity. Besides, the complexity of data alignment loss  $L_{a}^{(v)}$  in Eq. (12) is mainly on the computation of  $L_F^2 (\mathbf{Z}^\top \bar{\mathbf{X}}^{(v)})$ ,  $L_{F}(\mathbf{Z}^{\top}\mathbf{Z})$  and  $L_{F}(\bar{\mathbf{X}}^{(v)\top}\bar{\mathbf{X}}^{(v)})$  which are all of  $\mathcal{O}(N)$  complexity. Nevertheless, the optimization of weight parameter  $\{w_v\}_{v = 1}^V$  via Eq. (17) is of  $\mathcal{O}(V)$  complexity. In summary, the overall complexity of the proposed IMDC-DIA method is linear to the number of data samples, i.e.,  $\mathcal{O}(N)$ .
+
+# 4 Experiment
+
+# 4.1 Experiment setting
+
+To validate the proposed IMDC-DIA method, we conduct extensive experiments on four benchmark datasets, including HandWritten $^2$  [37], Caltech5V $^3$  [38], Flower17 $^4$  [39] and MSRCV1 $^5$  [40]. On the basis, we generate incomplete datasets by following the common strategy [41] in literature. Specifically, assuming the missing ratio to be  $m$ ,  $m$  percent of data samples are selected to remove at least one views randomly. In the following experiments,  $m$  is set in  $\{0.1, 0.3, 0.5, 0.7, 0.9\}$ . Meanwhile, five recent incomplete multi-view deep clustering approaches are considered in comparison, including DITA-IMVC [42], DSIMVC [41], DCP [34], DVIMVC [30] and CPSPAN [43]. Note that, we
+
+Table 1: Performance comparison between the proposed IMDC-DIA and recent incomplete deep multi-view clustering approaches. The avg. column refers to performance averages of all missing ratios. Note that, the best results are marked in bold, while the second-best with underline.  
+
+<table><tr><td colspan="2">Metric</td><td colspan="6">ACC</td><td colspan="6">NMI</td></tr><tr><td colspan="2">Missing ratio</td><td>0.1</td><td>0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>avg.</td><td>0.1</td><td>0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>avg.</td></tr><tr><td rowspan="6">HandWritten</td><td>DITA-IMVC</td><td>75.48</td><td>78.92</td><td>81.37</td><td>81.02</td><td>55.00</td><td>74.36</td><td>75.81</td><td>78.05</td><td>77.62</td><td>75.54</td><td>52.16</td><td>71.84</td></tr><tr><td>DSIMVC</td><td>79.55</td><td>80.73</td><td>78.83</td><td>77.48</td><td>50.87</td><td>73.49</td><td>78.57</td><td>77.76</td><td>74.93</td><td>71.97</td><td>48.53</td><td>70.35</td></tr><tr><td>DCP</td><td>81.95</td><td>75.73</td><td>77.23</td><td>71.77</td><td>13.07</td><td>63.95</td><td>84.37</td><td>78.95</td><td>79.22</td><td>74.39</td><td>0.93</td><td>63.57</td></tr><tr><td>DVIMC</td><td>86.42</td><td>83.05</td><td>45.48</td><td>25.20</td><td>18.92</td><td>51.81</td><td>87.64</td><td>84.78</td><td>56.58</td><td>35.43</td><td>17.96</td><td>56.48</td></tr><tr><td>CPSPAN</td><td>90.42</td><td>91.25</td><td>91.08</td><td>90.27</td><td>86.73</td><td>89.95</td><td>83.84</td><td>84.37</td><td>84.01</td><td>83.55</td><td>82.62</td><td>83.68</td></tr><tr><td>IMDC-DIA</td><td>96.37</td><td>93.68</td><td>91.80</td><td>90.65</td><td>87.93</td><td>92.09</td><td>91.80</td><td>93.68</td><td>91.80</td><td>90.65</td><td>87.93</td><td>91.17</td></tr><tr><td rowspan="6">CaltechSV</td><td>DITA-IMVC</td><td>79.10</td><td>75.76</td><td>68.02</td><td>58.90</td><td>40.29</td><td>64.41</td><td>67.56</td><td>64.78</td><td>58.84</td><td>52.40</td><td>29.29</td><td>54.57</td></tr><tr><td>DSIMVC</td><td>76.64</td><td>73.14</td><td>66.36</td><td>57.38</td><td>44.95</td><td>63.69</td><td>68.02</td><td>63.11</td><td>56.71</td><td>49.23</td><td>33.99</td><td>54.21</td></tr><tr><td>DCP</td><td>44.50</td><td>46.95</td><td>46.45</td><td>44.67</td><td>16.98</td><td>39.91</td><td>45.22</td><td>52.31</td><td>50.69</td><td>45.07</td><td>0.54</td><td>38.77</td></tr><tr><td>DVIMC</td><td>88.64</td><td>81.36</td><td>84.93</td><td>80.86</td><td>68.81</td><td>80.92</td><td>80.52</td><td>73.38</td><td>76.27</td><td>73.56</td><td>62.40</td><td>73.23</td></tr><tr><td>CPSPAN</td><td>83.29</td><td>81.10</td><td>77.21</td><td>76.79</td><td>79.02</td><td>79.48</td><td>73.81</td><td>71.30</td><td>68.73</td><td>68.43</td><td>69.26</td><td>70.31</td></tr><tr><td>IMDC-DIA</td><td>86.05</td><td>85.05</td><td>83.33</td><td>85.02</td><td>81.29</td><td>84.15</td><td>86.05</td><td>85.05</td><td>83.33</td><td>85.02</td><td>81.29</td><td>84.15</td></tr><tr><td rowspan="6">Flower17</td><td>DITA-IMVC</td><td>18.75</td><td>17.52</td><td>13.77</td><td>13.16</td><td>12.84</td><td>15.21</td><td>18.15</td><td>16.60</td><td>10.23</td><td>8.65</td><td>8.66</td><td>12.46</td></tr><tr><td>DSIMVC</td><td>18.63</td><td>16.86</td><td>13.50</td><td>13.36</td><td>13.36</td><td>15.14</td><td>17.51</td><td>14.78</td><td>9.29</td><td>9.27</td><td>9.22</td><td>12.01</td></tr><tr><td>DCP</td><td>26.15</td><td>25.49</td><td>22.21</td><td>15.05</td><td>10.78</td><td>19.94</td><td>26.18</td><td>25.94</td><td>23.47</td><td>15.37</td><td>3.69</td><td>18.93</td></tr><tr><td>DVIMC</td><td>36.69</td><td>31.23</td><td>26.84</td><td>23.41</td><td>21.69</td><td>27.97</td><td>35.43</td><td>29.82</td><td>24.30</td><td>20.93</td><td>20.08</td><td>26.11</td></tr><tr><td>CPSPAN</td><td>36.27</td><td>39.34</td><td>31.15</td><td>37.35</td><td>36.00</td><td>36.02</td><td>37.13</td><td>39.65</td><td>30.88</td><td>38.15</td><td>35.97</td><td>36.36</td></tr><tr><td>IMDC-DIA</td><td>52.03</td><td>46.47</td><td>44.22</td><td>41.54</td><td>39.14</td><td>44.68</td><td>52.03</td><td>46.47</td><td>44.22</td><td>41.54</td><td>39.14</td><td>44.68</td></tr><tr><td rowspan="6">MSRCV1</td><td>DITA-IMVC</td><td>76.03</td><td>74.60</td><td>70.95</td><td>66.83</td><td>55.87</td><td>68.86</td><td>67.62</td><td>63.02</td><td>60.61</td><td>54.96</td><td>43.32</td><td>57.91</td></tr><tr><td>DSIMVC</td><td>78.57</td><td>76.98</td><td>70.48</td><td>71.59</td><td>64.76</td><td>72.48</td><td>68.37</td><td>65.81</td><td>62.26</td><td>59.19</td><td>51.78</td><td>61.48</td></tr><tr><td>DCP</td><td>17.94</td><td>20.00</td><td>22.06</td><td>22.22</td><td>22.70</td><td>20.98</td><td>7.11</td><td>7.54</td><td>7.05</td><td>4.95</td><td>4.62</td><td>6.25</td></tr><tr><td>DVIMC</td><td>74.60</td><td>61.43</td><td>56.67</td><td>43.17</td><td>38.25</td><td>54.82</td><td>66.68</td><td>56.40</td><td>54.91</td><td>42.02</td><td>30.08</td><td>50.02</td></tr><tr><td>CPSPAN</td><td>85.08</td><td>86.51</td><td>85.40</td><td>84.29</td><td>77.62</td><td>83.78</td><td>75.86</td><td>77.25</td><td>75.51</td><td>74.09</td><td>69.06</td><td>74.35</td></tr><tr><td>IMDC-DIA</td><td>91.27</td><td>92.22</td><td>86.03</td><td>85.08</td><td>83.02</td><td>87.52</td><td>91.27</td><td>92.22</td><td>86.03</td><td>85.08</td><td>83.02</td><td>87.52</td></tr></table>
+
+directly adopt the public codes available on the authors' websites without modification. To ensure the comparison fairness, we grid-search their parameters and report the best. So does the proposed IMDC-DIA $^6$  method with setting its only parameter  $\beta$  in  $\{0.01, 0.1, 1, 10, 100\}$ . By following the literature, four most common metrics are adopted to measure the clustering performance, including accuracy (ACC), normalized mutual information (NMI), purity (PUR) and adjusted rand index (ARI). Nevertheless, all methods are executed multiple times and their averages are reported to remove randomness effect.
+
+# 4.2 Effectiveness
+
+In order to validate effectiveness of the proposed IMDC-DIA method, we compare it with the recent advances on incomplete deep multi-view clustering in literature. Corresponding results can be found in Table 1. It is obvious that the proposed IMDC-DIA outperforms the recent advances in almost all missing ratios on all datasets. Concretely, it achieves the improvements of  $5.95\%$ ,  $2.43\%$ ,  $0.72\%$ ,  $0.38\%$ ,  $1.20\%$  in accuracy and  $4.16\%$ ,  $8.90\%$ ,  $7.79\%$ ,  $7.10\%$ ,  $5.31\%$  in NMI on HandWritten;  $15.34\%$ ,  $7.13\%$ ,  $13.07\%$ ,  $4.19\%$ ,  $3.14\%$  in accuracy and  $14.90\%$ ,  $6.82\%$ ,  $13.34\%$ ,  $3.39\%$ ,  $3.17\%$  in NMI on Flower17;  $6.19\%$ ,  $5.71\%$ ,  $0.63\%$ ,  $0.79\%$ ,  $5.40\%$  in accuracy and  $15.41\%$ ,  $14.97\%$ ,  $10.52\%$ ,  $10.99\%$ ,  $13.96\%$  in NMI, respectively. Although slight decreases in 0.1 and 0.3 missing ratios on Caltech5V are observed, i.e.,  $2.59\%$  and  $1.60\%$ , the proposed IMDC-DIA obtains better performances on the other missing ratios consistently. Nevertheless, with averaging the results in all missing ratios, we can see that it improves the accuracy by  $2.14\%$ ,  $2.27\%$ ,  $3.14\%$ ,  $3.74\%$  and the NMI by  $7.49\%$ ,  $10.92\%$ ,  $8.32\%$ ,  $13.17\%$  on the four benchmark datasets. To be summarized, the aforementioned observations
+
+Table 2: Performance comparison between the proposed IMDC-DIA method and its variants of modifying and removing the data imputation and alignment components. The avg. column refers to performance averages of all missing ratios, while the avg. row reports the performances in the setting of completing missing data with average values. Note that, the best results are marked in bold, while the second-best with underline.  
+
+<table><tr><td colspan="2">Metric</td><td colspan="6">ACC</td><td colspan="6">NMI</td></tr><tr><td colspan="2">Missing ratio</td><td>0.1</td><td>0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>avg.</td><td>0.1</td><td>0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>avg.</td></tr><tr><td rowspan="4">HandWritten w/o imp.</td><td>avg.</td><td>75.58</td><td>66.17</td><td>56.47</td><td>50.63</td><td>41.78</td><td>58.13</td><td>75.30</td><td>64.92</td><td>55.95</td><td>51.55</td><td>42.18</td><td>57.98</td></tr><tr><td>zero</td><td>83.85</td><td>68.87</td><td>61.33</td><td>48.23</td><td>44.72</td><td>61.40</td><td>76.90</td><td>64.85</td><td>56.41</td><td>49.60</td><td>42.58</td><td>58.07</td></tr><tr><td>rand</td><td>79.98</td><td>69.73</td><td>58.37</td><td>55.05</td><td>44.33</td><td>61.49</td><td>78.33</td><td>68.31</td><td>57.06</td><td>52.81</td><td>42.45</td><td>59.79</td></tr><tr><td>none</td><td>96.13</td><td>82.55</td><td>87.82</td><td>78.70</td><td>55.83</td><td>80.21</td><td>91.37</td><td>82.39</td><td>82.03</td><td>74.80</td><td>49.31</td><td>75.98</td></tr><tr><td colspan="2">w/o align.</td><td>93.62</td><td>86.65</td><td>90.57</td><td>81.27</td><td>82.78</td><td>86.98</td><td>87.78</td><td>81.89</td><td>81.82</td><td>76.88</td><td>73.44</td><td>80.36</td></tr><tr><td colspan="2">IMDC-DIA</td><td>96.37</td><td>93.68</td><td>91.80</td><td>90.65</td><td>87.93</td><td>92.09</td><td>91.80</td><td>86.68</td><td>83.31</td><td>82.05</td><td>77.39</td><td>84.25</td></tr><tr><td rowspan="4">CaltechSV w/o imp.</td><td>avg.</td><td>74.14</td><td>73.19</td><td>63.86</td><td>52.07</td><td>46.98</td><td>62.05</td><td>72.26</td><td>66.11</td><td>56.68</td><td>47.16</td><td>39.89</td><td>56.42</td></tr><tr><td>zero</td><td>80.74</td><td>73.60</td><td>67.38</td><td>61.81</td><td>52.12</td><td>67.13</td><td>74.06</td><td>63.71</td><td>57.33</td><td>52.74</td><td>43.37</td><td>58.24</td></tr><tr><td>rand</td><td>78.76</td><td>68.31</td><td>62.05</td><td>58.10</td><td>47.95</td><td>63.03</td><td>72.70</td><td>60.89</td><td>53.63</td><td>50.37</td><td>37.17</td><td>54.95</td></tr><tr><td>none</td><td>87.98</td><td>82.14</td><td>81.71</td><td>76.31</td><td>61.74</td><td>77.98</td><td>78.57</td><td>72.57</td><td>70.35</td><td>62.81</td><td>46.61</td><td>66.18</td></tr><tr><td colspan="2">w/o align.</td><td>80.21</td><td>81.90</td><td>80.07</td><td>83.50</td><td>81.60</td><td>81.46</td><td>73.42</td><td>72.84</td><td>71.04</td><td>72.08</td><td>67.51</td><td>71.38</td></tr><tr><td colspan="2">IMDC-DIA</td><td>86.05</td><td>85.05</td><td>83.33</td><td>85.02</td><td>81.29</td><td>84.15</td><td>76.39</td><td>75.28</td><td>72.22</td><td>72.28</td><td>68.14</td><td>72.86</td></tr><tr><td rowspan="4">Flower17 w/o imp.</td><td>avg.</td><td>47.23</td><td>39.80</td><td>32.70</td><td>29.78</td><td>25.02</td><td>34.91</td><td>46.18</td><td>38.56</td><td>33.21</td><td>27.93</td><td>24.29</td><td>34.03</td></tr><tr><td>zero</td><td>45.15</td><td>36.86</td><td>31.81</td><td>27.38</td><td>24.63</td><td>33.17</td><td>44.51</td><td>34.58</td><td>29.05</td><td>25.27</td><td>22.77</td><td>31.24</td></tr><tr><td>rand</td><td>45.64</td><td>35.20</td><td>30.15</td><td>25.98</td><td>23.75</td><td>32.14</td><td>45.17</td><td>33.54</td><td>27.65</td><td>23.60</td><td>20.75</td><td>30.14</td></tr><tr><td>none</td><td>42.01</td><td>35.93</td><td>25.64</td><td>13.11</td><td>17.23</td><td>26.78</td><td>42.74</td><td>34.80</td><td>22.81</td><td>7.58</td><td>11.43</td><td>23.87</td></tr><tr><td colspan="2">w/o align.</td><td>38.28</td><td>36.18</td><td>35.54</td><td>30.81</td><td>28.50</td><td>33.86</td><td>35.90</td><td>35.13</td><td>31.87</td><td>27.53</td><td>24.44</td><td>30.97</td></tr><tr><td colspan="2">IMDC-DIA</td><td>51.13</td><td>45.00</td><td>42.89</td><td>40.69</td><td>37.06</td><td>43.35</td><td>47.15</td><td>45.02</td><td>41.33</td><td>38.42</td><td>34.81</td><td>41.35</td></tr><tr><td rowspan="4">MRSCVI w/o imp.</td><td>avg.</td><td>72.70</td><td>57.46</td><td>51.27</td><td>42.86</td><td>36.67</td><td>52.19</td><td>64.86</td><td>51.51</td><td>43.94</td><td>38.28</td><td>30.14</td><td>45.75</td></tr><tr><td>zero</td><td>68.10</td><td>62.86</td><td>56.83</td><td>45.56</td><td>42.38</td><td>55.14</td><td>59.26</td><td>46.69</td><td>46.66</td><td>36.53</td><td>30.58</td><td>43.94</td></tr><tr><td>rand</td><td>66.35</td><td>55.08</td><td>48.89</td><td>44.60</td><td>43.17</td><td>51.62</td><td>59.30</td><td>44.20</td><td>40.67</td><td>32.61</td><td>31.49</td><td>41.65</td></tr><tr><td>none</td><td>78.89</td><td>62.06</td><td>58.25</td><td>31.90</td><td>39.84</td><td>54.19</td><td>75.10</td><td>58.82</td><td>50.66</td><td>17.53</td><td>23.40</td><td>45.10</td></tr><tr><td colspan="2">w/o align.</td><td>85.56</td><td>85.24</td><td>73.65</td><td>67.30</td><td>60.95</td><td>74.54</td><td>78.09</td><td>76.37</td><td>60.14</td><td>52.43</td><td>48.95</td><td>63.20</td></tr><tr><td colspan="2">IMDC-DIA</td><td>91.27</td><td>92.22</td><td>86.03</td><td>85.08</td><td>83.02</td><td>87.52</td><td>84.37</td><td>85.63</td><td>77.84</td><td>74.21</td><td>70.47</td><td>78.51</td></tr></table>
+
+well illustrate the effectiveness of the proposed IMDC-DIA method and its superiority over the recent advances in literature.
+
+# 4.3 Ablation study
+
+To demonstrate the rationality of our motivations and effectiveness of the proposed techniques, we further conduct an thorough ablation study in the following. Two settings are designed by modifying and removing the data imputation and alignment components. Specifically,
+
+1. The avg., zero and rand of w/o imp. substitute the data imputation into completing the missing data with averages of available data observations, zeros and random values, respectively. Meanwhile, the none of w/o imp. refers to only using the available data observations in data alignment module rather than completing the missing.  
+2. w/o align. removes the data alignment module, which can be easily implemented by setting parameter  $\beta$  to 0.
+
+By comparing results of the proposed IMDC-DIA method and its w/o imp. settings, the former achieves better clustering performances in almost missing ratios on all benchmark datasets. Although slight decreases in missing ratio 0.1 on Caltech5V are observed, i.e.,  $1.93\%$  accuracy and  $2.18\%$  NMI, the proposed IMDC-DIA obtains better performances on the other missing ratios, i.e.,  $2.91\%$ ,  $1.62\%$ ,  $8.71\%$ ,  $19.55\%$  accuracy and  $2.71\%$ ,  $1.87\%$ ,  $9.47\%$ $21.53\%$  NMI. In average, it improves the accuracy by  $11.88\%$ ,  $6.17\%$ ,  $8.44\%$ ,  $32.38\%$  and the NMI by  $8.27\%$ ,  $6.68\%$ ,  $7.32\%$ ,  $32.76\%$ . These observations well validate effectiveness of the proposed data imputation module. In addition, we
+
+![](images/ab8b94e482d16d644f799788ef482042a4932b62070f9531e5cdbf47b2406b51.jpg)  
+Figure 2: Accuracy variation with respect to parameter  $\beta$  in different missing ratios on the four benchmark datasets. The shaded area represents the accuracy variance correspondingly.
+
+![](images/8fef0b3444ebe4a2863515732736864c67439cbe29e46e5f999407a49403b17a.jpg)
+
+![](images/7e6f3ccbef2f73ae68b711f1de66984c11a441335cf21c024550560dfdc28d10.jpg)
+
+![](images/b5a921611000bcdd0d6e0cf12bf5be28c0ea6c137e5edd351735032137b00c25.jpg)
+
+find the proposed IMDC-DIA exhibits larger and larger performance improvements when increasing the missing ratio. Also, the results in none setting are better than those in avg., zero and rand settings in most cases. The two observations indicate that complementing the missing data without data-irrelevant values would introduce extra bias hence decreasing the performance.
+
+By comparing results of the proposed IMDC-DIA method and its w/o align. settings, similar observations can be obtained. Specifically,  $0.31\%$  accuracy decrease is presented in missing ratio 0.9 on Caltech5V, while different degrees of improvements are observed in the other missing ratios, i.e.,  $5.84\%$ ,  $3.15\%$ ,  $3.26\%$ ,  $1.52\%$  accuracy and  $2.97\%$ ,  $2.44\%$ ,  $1.18\%$ ,  $0.20\%$  NMI. Also, with averaging the results in all missing ratios, the proposed IMDC-DIA method achieves better accuracies by  $5.11\%$ ,  $2.69\%$ ,  $9.49\%$ $12.98\%$  and NMIs by  $3.89\%$ ,  $1.48\%$ ,  $10.38\%$ ,  $15.31\%$ , respectively. These well demonstrate that the defined linear alignment is conducive to the performance improvement.
+
+Globally, the proposed IMDC-DIA method achieves better clustering performances than its none of w/o imp. setting, i.e.,  $11.88\%$ ,  $6.17\%$ ,  $16.57\%$ ,  $33.33\%$  accuracies and  $8.27\%$ ,  $6.68\%$ ,  $17.48\%$ ,  $33.41\%$  NMs. This proves that incorporating the data similarities among missing data observations can largely enhance the learning of latent representations, hence obtaining better clustering results.
+
+# 4.4 Parameter analysis
+
+In the proposed IMDC-DIA method, the only parameter is the trade-off  $\beta$  between the data reconstruction loss and data alignment loss. To investigate its effect on performance, we collect the clustering accuracies when  $\beta$  set in range  $\{0.01, 0.1, 1, 10, 100\}$  and present their average curves and corresponding variances in Fig. 2. In general, the clustering accuracy decreases with a larger  $\beta$  in almost all missing ratios on all benchmark datasets. Specifically, it can be seen that the best accuracies are obtained mostly when setting  $\beta$  to 0.01 and sometimes when setting  $\beta$  to 0.1. Therefore, we recommend to set it from 0.01 to 0.1 in first priority.
+
+# 5 Conclusion
+
+In literature, existing incomplete multi-view deep clustering approaches overlook the fact that the latent representations should be unique for a specific set of data samples. Also, they fail to utilize the pair-wise similarities of missing data observations sufficiently. Therefore, this paper proposes an incomplete multi-view deep clustering method with data imputation and alignment by assuming each data sample corresponds to a same latent representation among all views. Inside, a novel linear alignment measurement of linear complexity is defined and incorporated to integrate the pair-wise similarities of all data observations, including those of the missing ones. Nevertheless, an alternate optimization strategy is proposed to learn high-quality latent representations for the next clustering task. In experiment, the proposed method is compared with recent advances, while its motivations and method design are validated. Also, its parameter effect is explored empirically.
+
+Although the proposed method achieves state-of-the-art results compared to recent advances, it may be partially limited by lacking sufficient constraints on the latent representations, preventing from further improvement of the clustering performance. In the future work, we will explore more feasible constraints. Meanwhile, the evaluation on missing data imputation is also a promising research direction on the basis of this work.
+
+# Acknowledgments and Disclosure of Funding
+
+This work is supported by the National Natural Science Foundation of China (No. 62306324, 62376279, U24A20333, 62325604, 62276271, 62421002), the Science and Technology Innovation Program of Hunan Province (No. 2024RC3128) and the National University of Defense Technology Research Foundation (No. ZK24-30).
+
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+
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+
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+
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+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
+
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: This paper does not release new assets.
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: This paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
+
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: We do not use LLM at all.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Convergence analysis
+
+In the optimization, the proposed IMDC-DIA method aims to minimize the loss  $L$  of Eq. (13). Inside, three variables are optimized alternately, including the neural network parameter  $\{\Theta_v\}_{v=1}^V$ , the unique latent representation  $\{\mathbf{z}_i\}_{i=1}^N$  and weight parameter  $\{w_v\}_{v=1}^V$ . With a little abuse of notation, let the loss be represented by
+
+$$
+L = \ell \left(\left\{\Theta_ {v} \right\} _ {v = 1} ^ {V}, \left\{\mathbf {z} _ {i} \right\} _ {i = 1} ^ {N}, \left\{w _ {v} \right\} _ {v = 1} ^ {V}\right). \tag {18}
+$$
+
+Taking the loss at  $t$  epoch to be
+
+$$
+L ^ {(t)} = \ell \left(\left\{\Theta_ {v} ^ {(t)} \right\} _ {v = 1} ^ {V}, \left\{\mathbf {z} _ {i} ^ {(t)} \right\} _ {i = 1} ^ {N}, \left\{w _ {v} ^ {(t)} \right\} _ {v = 1} ^ {V}\right), \tag {19}
+$$
+
+By fixing  $\{\mathbf{z}_i\}_{i = 1}^N$  and  $\{w_v\}_{v = 1}^V$  at  $\{\mathbf{z}_i^{(t)}\}_{i = 1}^N$  and  $\{w_v^{(t)}\}_{v = 1}^V$ , optimizing  $\{\Theta_v\}_{v = 1}^V$  at  $t + 1$  epoch induces to
+
+$$
+\ell \left(\left\{\Theta_ {v} ^ {(t)} \right\} _ {v = 1} ^ {V}, \left\{\mathbf {z} _ {i} ^ {(t)} \right\} _ {i = 1} ^ {N}, \left\{w _ {v} ^ {(t)} \right\} _ {v = 1} ^ {V}\right) \leq \ell \left(\left\{\Theta_ {v} ^ {(t + 1)} \right\} _ {v = 1} ^ {V}, \left\{\mathbf {z} _ {i} ^ {(t)} \right\} _ {i = 1} ^ {N}, \left\{w _ {v} ^ {(t)} \right\} _ {v = 1} ^ {V}\right). \tag {20}
+$$
+
+So do the optimizations of  $\{\mathbf{z}_i\}_{i=1}^N$  and  $\{w_v\}_{v=1}^V$ , resulting in
+
+$$
+\begin{array}{l} \ell \left(\left\{\Theta_ {v} ^ {(t)} \right\} _ {v = 1} ^ {V}, \left\{\mathbf {z} _ {i} ^ {(t)} \right\} _ {i = 1} ^ {N}, \left\{w _ {v} ^ {(t)} \right\} _ {v = 1} ^ {V}\right) \leq \ell \left(\left\{\Theta_ {v} ^ {(t + 1)} \right\} _ {v = 1} ^ {V}, \left\{\mathbf {z} _ {i} ^ {(t)} \right\} _ {i = 1} ^ {N}, \left\{w _ {v} ^ {(t)} \right\} _ {v = 1} ^ {V}\right) \tag {21} \\ \leq \ell (\{\Theta_ {v} ^ {(t + 1)} \} _ {v = 1} ^ {V}, \{\mathbf {z} _ {i} ^ {(t + 1)} \} _ {i = 1} ^ {N}, \{w _ {v} ^ {(t)} \} _ {v = 1} ^ {V}) \leq \ell (\{\Theta_ {v} ^ {(t + 1)} \} _ {v = 1} ^ {V}, \{\mathbf {z} _ {i} ^ {(t + 1)} \} _ {i = 1} ^ {N}, \{w _ {v} ^ {(t + 1)} \} _ {v = 1} ^ {V}), \\ \end{array}
+$$
+
+which can be simplified to
+
+$$
+L ^ {(t)} \leq L ^ {(t + 1)}. \tag {22}
+$$
+
+Nevertheless,
+
+$$
+L = L _ {r} + \beta L _ {a} \geq 0 + (- 1) = - 1. \tag {23}
+$$
+
+To be summarized, Eq. (22) indicates that the loss monotonically decreases, while Eq. (23) illustrates that the loss is lower bounded. Therefore, the optimization algorithm is convergent.
+
+# B Dataset detail
+
+The specifics of the used benchmark datasets are presented in Table 3. Also, they are briefly introduced as follows:
+
+1. HandWritten<sup>7</sup> [37]. It consists of 2,000 digital images of ten classes from 0 to 9 in which there are 200 images in each class. Here, six features are used, i.e., 216-D profile correlations, 76-D Fourier coefficients of the character shapes, 64-D Karhunen-Love coefficients, 6-D morphological features, 240-D pixel averages in  $2 \times 3$  windows and 47-D Zernike moments.  
+2. Caltech5V $^{8}$  [38]. It collects 1400 object images belonging to 7 classes. I experiment, five well-known conventional feature descriptors are adopted, including 48-D wavelet moments, 40-D CENTRIST, 254-D LBP, 512-D GIST and 928-D HOG, respectively.  
+3. Flower17 $^{9}$  [39]. It collects flower images of 17 categories with 80 images for each class. Corresponding features are 5376-D Color Histogram, 512-D GIST, 5376-D HOG  $(2 \times 2)$ , 5376-D HOG  $(3 \times 3)$ , 1239-D LBP, 5376-D SIFT and 5376-D SSIM features.  
+4. MSRCV1 $^{10}$  [40]. It collects 210 images of seven categories with each category provides 30 images. For each image, six features are extracted, including 1302-D CENTRIST, 48-D Color Moment, 512-D GIST, 100-D HOG, 256-D LBP and 210 SIFT, respectively.
+
+For HandWritten, Flower17 and MSRCV1, the creators share them publicly but do not issue licenses, while Caltech5V is issued the Creative Commons Attribution 4.0 International (CC BY 4.0) license.
+
+# C Incomplete multi-view data setting
+
+The incomplete multi-view data are generated by following the common strategy [41] in literature. Specifically, assuming the missing ratio to be  $m$ ,  $m$  percent of data samples are selected to remove at least one views randomly. In the following experiments,  $m$  is set in  $\{0.1, 0.3, 0.5, 0.7, 0.9\}$ . Meanwhile, the corresponding psuedo-code is outlined in Alg. 2.
+
+Table 3: Details of the used benchmark datasets. Note that, in brackets are the feature dimensions of each data view.  
+
+<table><tr><td>Dataset</td><td>Samples</td><td>Clusters</td><td>Number of Views (Dimensions)</td></tr><tr><td>HandWritten</td><td>2000</td><td>10</td><td>6 (216/76/64/6/240/47)</td></tr><tr><td>Caltech5V</td><td>1400</td><td>7</td><td>5 (48/40/254/512/928)</td></tr><tr><td>Flower17</td><td>1360</td><td>17</td><td>7 (512/5376/5376/1239/5376/5376)</td></tr><tr><td>MSRCV1</td><td>210</td><td>7</td><td>6 (1032/48/512/100/256/210)</td></tr></table>
+
+# Algorithm 2 Generation of Incomplete Multi-view Data
+
+Input: incomplete multi-view data  $\{\mathbf{x}_i^{(v)}\}_{i,v = 1}^{N,V}$ , missing ratio  $m$
+
+Output: availability indicator  $\{\mathbf{a}_i\}_{i = 1}^N$
+
+1: obtain the numbers of data samples and views, i.e.  $N$  and  $V$ ;  
+2: random sample incomplete data index  $S$  with missing ratio  $m$ ;  
+3: initialize availability indicator  $\mathbf{a}_i = \{1,2,\dots ,V\}$  w.r.t.  $i\in \{1,2,\dots ,N\}$  
+4: for  $p \in S$  do  
+5: random sample missing index  $\mathbf{m}_p\subset \{1,2,\dots ,V\}$  and  $\mathbf{m}_p = \emptyset$  
+6: set availability indicator  $\mathbf{a}_p = \{1,2,\dots ,V\} -\mathbf{m}_p$  
+7: end for
+
+# D More experiment settings
+
+In the experiments, we adopt fully-connected neural networks on all datasets where different numbers of neurons are adopted according to different feature dimensions, as shown in Table 4. Meanwhile, the unique data latent representation and parameters of neural netowrks are both optimized with gradient descent strategy with Adam optimizer whose learning rate is set to 0.001. Additionally, the codes of the proposed method and recent advances in comparison are executed on a server with 40 Intel(R) Xeon(R) Silver 4210R CPU @ 2.40GHz CPUs and 2 NVIDIA GeForce RTX 3090 GPUs. The software environment includes Python 3.10.13 and PyTorch 1.12.1 optimized with CUDA 11.4.
+
+# E Additional result
+
+Apart from the ACC and NMI results in Table 1 and 2, we also provide the experiment results in PUR and ARI metrics in Table 5 and 6, respectively. It can be observed that the PUR and ARI results follow similar trends with those in ACC and NMI. Nevertheless, to analyze the training process more deeply, we also record the loss value and corresponding performances at each training epoch, which are visualized in Fig. 3. It is obvious that the loss value continuously decreases and finally converges to the minimum, while the ACC and NMI increase to the top and fluctuate around their top values. These observations well illustrate the effectiveness of loss design in Eq. (13) and the proposed IMDC-DIA method.
+
+Table 4: The neuron numbers of the proposed IMDC-DIA method specific to different datasets.  
+
+<table><tr><td></td><td>HandWritten</td><td>Caltech5V</td><td>Flower17</td><td>MSRCV1</td></tr><tr><td>1st-view</td><td>(60, 64, 216)</td><td>(21, 16, 40)</td><td>(34, 512, 5376)</td><td>(21, 256, 1302)</td></tr><tr><td>2nd-view</td><td>(60, 64,76)</td><td>(21, 64, 254)</td><td>(34, 256, 512)</td><td>(21, 32, 48)</td></tr><tr><td>3rd-view</td><td>(60, 32, 64)</td><td>(21, 128, 928)</td><td>(34, 512, 5376)</td><td>(21, 64, 512)</td></tr><tr><td>4th-view</td><td>(60, 16, 6)</td><td>(21, 128, 512)</td><td>(34, 512, 5376)</td><td>(21, 32, 100)</td></tr><tr><td>5th-view</td><td>(60, 64, 240)</td><td>(21, 256, 1984)</td><td>(34, 256, 1239)</td><td>(21, 64, 256)</td></tr><tr><td>6th-view</td><td>(40, 32, 47)</td><td>-</td><td>(34, 512, 5376)</td><td>(21, 64, 210)</td></tr><tr><td>7th-view</td><td>-</td><td>-</td><td>(34, 512, 5376)</td><td>-</td></tr></table>
+
+Table 5: Performance (PUR and ARI) comparison between the proposed IMDC-DIA and recent incomplete deep multi-view clustering approaches. The avg. column refers to performance averages of all missing ratios. Note that, the best results are marked in bold, while the second-best with underline.  
+
+<table><tr><td colspan="2">Metric</td><td colspan="6">PUR</td><td colspan="6">ARI</td></tr><tr><td colspan="2">Missing ratio</td><td>0.1</td><td>0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>avg.</td><td>0.1</td><td>0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>avg.</td></tr><tr><td rowspan="6">HandWritten</td><td>DITA-IMVC</td><td>77.85</td><td>80.90</td><td>81.38</td><td>81.02</td><td>55.00</td><td>75.23</td><td>66.61</td><td>70.78</td><td>70.87</td><td>68.67</td><td>35.92</td><td>62.57</td></tr><tr><td>DSIMVC</td><td>81.13</td><td>80.73</td><td>79.88</td><td>77.50</td><td>51.18</td><td>74.08</td><td>71.69</td><td>70.19</td><td>67.27</td><td>63.91</td><td>32.36</td><td>61.08</td></tr><tr><td>DCP</td><td>83.97</td><td>78.08</td><td>79.40</td><td>73.40</td><td>13.42</td><td>65.65</td><td>75.32</td><td>61.27</td><td>62.61</td><td>53.62</td><td>0.02</td><td>50.57</td></tr><tr><td>DVIMC</td><td>86.53</td><td>83.12</td><td>45.53</td><td>25.27</td><td>19.45</td><td>51.98</td><td>82.38</td><td>77.91</td><td>35.65</td><td>14.20</td><td>5.74</td><td>43.18</td></tr><tr><td>CPSPAN</td><td>90.42</td><td>91.25</td><td>91.08</td><td>90.27</td><td>87.82</td><td>90.17</td><td>80.83</td><td>82.12</td><td>81.69</td><td>80.46</td><td>78.02</td><td>80.62</td></tr><tr><td>IMDC-DIA</td><td>96.37</td><td>93.68</td><td>91.80</td><td>90.65</td><td>87.93</td><td>92.09</td><td>92.09</td><td>93.68</td><td>91.80</td><td>90.65</td><td>87.93</td><td>91.23</td></tr><tr><td rowspan="6">CaltechSV</td><td>DITA-IMVC</td><td>79.10</td><td>75.76</td><td>68.02</td><td>60.64</td><td>41.31</td><td>64.97</td><td>61.06</td><td>57.26</td><td>48.75</td><td>40.62</td><td>18.10</td><td>45.16</td></tr><tr><td>DSIMVC</td><td>76.64</td><td>73.14</td><td>66.36</td><td>58.57</td><td>46.31</td><td>64.20</td><td>60.87</td><td>54.83</td><td>47.05</td><td>37.67</td><td>22.88</td><td>44.66</td></tr><tr><td>DCP</td><td>49.24</td><td>54.31</td><td>51.52</td><td>48.40</td><td>17.29</td><td>44.15</td><td>23.18</td><td>27.54</td><td>28.11</td><td>20.46</td><td>-0.06</td><td>19.85</td></tr><tr><td>DVIMC</td><td>88.64</td><td>81.36</td><td>84.93</td><td>81.38</td><td>68.83</td><td>81.03</td><td>77.33</td><td>67.12</td><td>71.87</td><td>68.58</td><td>52.92</td><td>67.56</td></tr><tr><td>CPSPAN</td><td>83.29</td><td>81.10</td><td>78.14</td><td>77.76</td><td>80.05</td><td>80.07</td><td>69.41</td><td>65.71</td><td>62.80</td><td>61.27</td><td>63.60</td><td>64.56</td></tr><tr><td>IMDC-DIA</td><td>86.05</td><td>85.05</td><td>83.33</td><td>85.02</td><td>81.29</td><td>84.15</td><td>86.05</td><td>85.05</td><td>83.33</td><td>85.02</td><td>81.29</td><td>84.15</td></tr><tr><td rowspan="6">Flower17</td><td>DITA-IMVC</td><td>20.61</td><td>19.53</td><td>15.10</td><td>13.80</td><td>13.41</td><td>16.49</td><td>5.63</td><td>4.81</td><td>2.20</td><td>2.10</td><td>1.96</td><td>3.34</td></tr><tr><td>DSIMVC</td><td>20.07</td><td>18.58</td><td>14.34</td><td>14.00</td><td>13.82</td><td>16.16</td><td>5.38</td><td>4.30</td><td>1.98</td><td>2.06</td><td>2.48</td><td>3.24</td></tr><tr><td>DCP</td><td>27.11</td><td>27.18</td><td>23.87</td><td>15.76</td><td>11.10</td><td>21.00</td><td>10.07</td><td>8.02</td><td>3.86</td><td>1.76</td><td>0.03</td><td>4.75</td></tr><tr><td>DVIMC</td><td>37.84</td><td>32.57</td><td>27.87</td><td>24.53</td><td>22.97</td><td>29.16</td><td>18.88</td><td>14.32</td><td>10.93</td><td>8.47</td><td>7.71</td><td>12.06</td></tr><tr><td>CPSPAN</td><td>38.11</td><td>40.86</td><td>32.55</td><td>39.68</td><td>37.23</td><td>37.69</td><td>17.47</td><td>19.36</td><td>13.19</td><td>19.06</td><td>16.89</td><td>17.19</td></tr><tr><td>IMDC-DIA</td><td>52.03</td><td>46.47</td><td>44.22</td><td>41.54</td><td>39.14</td><td>44.68</td><td>52.03</td><td>46.47</td><td>44.22</td><td>41.54</td><td>39.14</td><td>44.68</td></tr><tr><td rowspan="6">MSRCV1</td><td>DITA-IMVC</td><td>76.83</td><td>74.60</td><td>70.95</td><td>66.83</td><td>56.67</td><td>69.18</td><td>58.42</td><td>54.43</td><td>49.80</td><td>43.22</td><td>29.20</td><td>47.01</td></tr><tr><td>DSIMVC</td><td>78.57</td><td>76.98</td><td>70.79</td><td>71.59</td><td>65.40</td><td>72.67</td><td>59.93</td><td>57.68</td><td>51.54</td><td>50.56</td><td>41.58</td><td>52.26</td></tr><tr><td>DCP</td><td>18.10</td><td>20.63</td><td>22.54</td><td>22.54</td><td>23.17</td><td>21.40</td><td>-0.01</td><td>0.07</td><td>0.24</td><td>-0.16</td><td>-0.21</td><td>-0.01</td></tr><tr><td>DVIMC</td><td>74.60</td><td>61.43</td><td>56.83</td><td>43.81</td><td>38.25</td><td>54.98</td><td>57.67</td><td>45.83</td><td>41.15</td><td>27.52</td><td>19.05</td><td>38.24</td></tr><tr><td>CPSPAN</td><td>85.08</td><td>86.51</td><td>85.40</td><td>84.29</td><td>78.10</td><td>83.88</td><td>70.63</td><td>72.54</td><td>70.95</td><td>69.02</td><td>61.63</td><td>68.95</td></tr><tr><td>IMDC-DIA</td><td>91.27</td><td>92.22</td><td>86.03</td><td>85.08</td><td>83.02</td><td>87.52</td><td>91.27</td><td>92.22</td><td>86.03</td><td>85.08</td><td>83.02</td><td>87.52</td></tr></table>
+
+![](images/2fd211aa27d1c0eb0d57a182ea71142810363645f112112bd6c3484da328d324.jpg)  
+Figure 3: Loss and performance variation at each epoch on the four benchmark datasets. The blue curve represents loss value, while the yellow and green curves are accuracy and NMI, respectively.
+
+![](images/e8addb2cb4ac8156aa9b12d07e671d6b6ca8f13b937b630ecdaf6d3f38f51f03.jpg)
+
+![](images/c9282a41b5382d04df1b8e014f531c86bedf236a4a1949d418f642c24ae43a79.jpg)
+
+![](images/6d5f8b606f6fc713f1a5af2eaf533c11f6bea7a0057024e9633a97686bdf073a.jpg)
+
+Table 6: Performance (PUR and ARI) comparison between the proposed IMDC-DIA method and its variants of modifying and removing the data imputation and alignment components. The avg. column refers to performance averages of all missing ratios, while the avg. row reports the performances in the setting of completing missing data with average values. Note that, the best results are marked in bold, while the second-best with underline.  
+
+<table><tr><td colspan="2">Metric</td><td colspan="6">PUR</td><td colspan="6">ARI</td></tr><tr><td colspan="2">Missing ratio</td><td>0.1</td><td>0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>avg.</td><td>0.1</td><td>0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>avg.</td></tr><tr><td rowspan="4">HandWritten w/o imp.</td><td>avg</td><td>78.12</td><td>68.13</td><td>59.27</td><td>53.78</td><td>44.48</td><td>60.76</td><td>62.70</td><td>44.29</td><td>24.49</td><td>18.14</td><td>11.49</td><td>32.22</td></tr><tr><td>zero</td><td>83.98</td><td>70.98</td><td>62.73</td><td>51.35</td><td>46.63</td><td>63.14</td><td>71.39</td><td>47.75</td><td>32.75</td><td>21.58</td><td>15.48</td><td>37.79</td></tr><tr><td>rand</td><td>80.18</td><td>71.13</td><td>59.95</td><td>55.62</td><td>46.58</td><td>62.69</td><td>70.79</td><td>55.61</td><td>32.89</td><td>25.63</td><td>15.86</td><td>40.16</td></tr><tr><td>none</td><td>96.13</td><td>84.73</td><td>89.28</td><td>79.67</td><td>56.68</td><td>81.30</td><td>91.59</td><td>77.60</td><td>80.11</td><td>69.16</td><td>37.18</td><td>71.13</td></tr><tr><td colspan="2">w/o align.</td><td>93.62</td><td>88.18</td><td>90.57</td><td>83.77</td><td>84.12</td><td>88.05</td><td>86.73</td><td>78.23</td><td>80.55</td><td>71.71</td><td>69.92</td><td>77.43</td></tr><tr><td colspan="2">IMDC-DIA</td><td>96.37</td><td>93.68</td><td>91.80</td><td>90.65</td><td>87.93</td><td>92.09</td><td>92.09</td><td>86.37</td><td>82.60</td><td>80.53</td><td>75.42</td><td>83.40</td></tr><tr><td rowspan="4">Caltech5V w/o imp.</td><td>avg</td><td>78.50</td><td>74.48</td><td>64.81</td><td>54.62</td><td>48.86</td><td>64.25</td><td>60.29</td><td>46.02</td><td>30.32</td><td>17.49</td><td>12.95</td><td>33.41</td></tr><tr><td>zero</td><td>82.17</td><td>73.60</td><td>67.88</td><td>61.81</td><td>53.00</td><td>67.69</td><td>65.43</td><td>50.59</td><td>38.09</td><td>27.57</td><td>16.36</td><td>39.61</td></tr><tr><td>rand</td><td>79.24</td><td>68.67</td><td>62.43</td><td>58.93</td><td>48.24</td><td>63.50</td><td>67.77</td><td>51.12</td><td>37.88</td><td>28.01</td><td>17.16</td><td>40.39</td></tr><tr><td>none</td><td>87.98</td><td>82.45</td><td>81.71</td><td>76.31</td><td>61.93</td><td>78.08</td><td>76.63</td><td>68.91</td><td>68.49</td><td>58.79</td><td>38.25</td><td>62.22</td></tr><tr><td colspan="2">w/o align.</td><td>80.81</td><td>82.19</td><td>80.93</td><td>83.50</td><td>81.60</td><td>81.80</td><td>68.21</td><td>68.29</td><td>67.07</td><td>68.89</td><td>65.25</td><td>67.54</td></tr><tr><td colspan="2">IMDC-DIA</td><td>86.05</td><td>85.05</td><td>83.33</td><td>85.02</td><td>81.29</td><td>84.15</td><td>73.54</td><td>72.05</td><td>69.00</td><td>71.06</td><td>65.78</td><td>70.29</td></tr><tr><td rowspan="4">Flower17 w/o imp.</td><td>avg</td><td>48.70</td><td>41.72</td><td>34.07</td><td>30.59</td><td>26.20</td><td>36.25</td><td>25.93</td><td>14.78</td><td>11.40</td><td>8.11</td><td>5.40</td><td>13.12</td></tr><tr><td>zero</td><td>45.81</td><td>37.55</td><td>32.70</td><td>27.97</td><td>25.42</td><td>33.89</td><td>27.61</td><td>19.97</td><td>13.97</td><td>9.85</td><td>7.61</td><td>15.80</td></tr><tr><td>rand</td><td>46.37</td><td>35.71</td><td>30.96</td><td>26.79</td><td>24.46</td><td>32.86</td><td>28.77</td><td>18.25</td><td>12.66</td><td>9.27</td><td>6.30</td><td>15.05</td></tr><tr><td>none</td><td>42.92</td><td>36.84</td><td>26.62</td><td>13.85</td><td>18.14</td><td>27.67</td><td>22.89</td><td>18.09</td><td>9.91</td><td>0.68</td><td>2.98</td><td>10.91</td></tr><tr><td colspan="2">w/o align.</td><td>39.46</td><td>37.60</td><td>36.57</td><td>32.28</td><td>29.71</td><td>35.12</td><td>18.55</td><td>18.39</td><td>16.34</td><td>13.59</td><td>11.55</td><td>15.68</td></tr><tr><td colspan="2">IMDC-DIA</td><td>52.03</td><td>46.47</td><td>44.22</td><td>41.54</td><td>39.14</td><td>44.68</td><td>27.04</td><td>25.11</td><td>21.99</td><td>20.89</td><td>17.73</td><td>22.55</td></tr><tr><td rowspan="4">MSRCV1 w/o imp.</td><td>avg</td><td>72.86</td><td>59.52</td><td>52.22</td><td>44.60</td><td>38.10</td><td>53.46</td><td>52.89</td><td>27.30</td><td>20.49</td><td>10.75</td><td>7.80</td><td>23.85</td></tr><tr><td>zero</td><td>69.05</td><td>63.33</td><td>57.30</td><td>46.51</td><td>43.81</td><td>56.00</td><td>43.29</td><td>33.62</td><td>25.93</td><td>14.12</td><td>11.75</td><td>25.74</td></tr><tr><td>rand</td><td>67.46</td><td>56.83</td><td>51.59</td><td>46.03</td><td>44.92</td><td>53.37</td><td>48.75</td><td>31.53</td><td>26.17</td><td>15.75</td><td>12.57</td><td>26.96</td></tr><tr><td>none</td><td>79.68</td><td>62.38</td><td>62.22</td><td>33.97</td><td>41.43</td><td>55.94</td><td>66.16</td><td>46.72</td><td>39.52</td><td>8.04</td><td>14.72</td><td>35.03</td></tr><tr><td colspan="2">w/o align.</td><td>85.56</td><td>85.24</td><td>73.65</td><td>67.62</td><td>62.86</td><td>74.98</td><td>72.51</td><td>70.51</td><td>52.78</td><td>43.35</td><td>39.98</td><td>55.83</td></tr><tr><td colspan="2">IMDC-DIA</td><td>91.27</td><td>92.22</td><td>86.03</td><td>85.08</td><td>83.02</td><td>87.52</td><td>80.92</td><td>82.50</td><td>70.95</td><td>69.28</td><td>64.59</td><td>73.65</td></tr></table>

2025/Incomplete Multi-view Deep Clustering with Data Imputation and Alignment/images.zip

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