A Survey on Deep Clustering: From the Prior Perspective

We notice that several surveys on deep clustering have been proposed in recent years. Briefly, Min et al [66] categorizes deep clustering methods according to the network architecture. Dong et al [18] focuses on applications of deep clustering. Ren et al [84] summarizes existing methods from the view of data types, such as single- and multi-view. Zhou et al [125] discusses various interactions between representation learning and clustering. Distinct from existing surveys, this work systematically provides a new perspective from the prior knowledge, which plays a more intrinsic and essential role in deep clustering.

Structure prior is mostly inspired by traditional clustering methods. Traditional cluster is mainly rooted in assumptions about the structural characteristics of clusters in data space. For example, K-means [64] aims to learn $k$ cluster centroids, which assumes that instances in each cluster form a spherical structure around its centroid. DBSCAN [19] is based on the assumption that a cluster in data space is a contiguous region of high point density, separated from other such clusters by regions of low point density. Spectral clustering [4] assumes data lies on a locally linear manifold so that the local neighborhoods’ relation should be preserved in latent space. Those methods partition instances according to the graph Laplacian. Agglomerative clustering [25] considers the hierarchical structure of data and performs clustering with merging and splitting. Motivated by the success of classic clustering methods, the early exploration of deep clustering mainly focuses on adapting mature structure priors as objective functions to optimize neural networks.

Given well-structured data in the latent space, ABDC [95] iteratively optimizes the data representation and clustering centers motivated by K-means. As the deep extension of classic spectral clustering, DEN [33], SpectralNet [91], and MvLNet [34, 35] compute the graph Laplacian in the latent space learned by auto-encoder [5] and SiameseNets [28, 90], respectively. Likewise, DCC [89] extends the core idea of RCC [88] by performing a relation matching based on the similarity between latent features. The auto-encoder is then optimized by minimizing the distance of paired instances in the latent space. PARTY [79] is the first deep subspace clustering method, which introduces the sparsity prior and self-representation property in subspace learning to optimize neural networks. Motivated by the hierarchical structure of clusters, JULE [110] achieves agglomerative deep clustering by progressively merging clusters and optimizing the features.

Distribution prior refers to instances of different semantics following distinct data distributions. Such a prior arouses the generative deep clustering paradigm, which employs variational autoencoder [43] (VAE) and generative adversarial network [24] (GAN) to learn the underlying distribution. Instances generated from similar distributions are then grouped together to achieve clustering.

VaDE [40] is the first deep generative clustering method, which computes different data distributions by fitting the Gaussian mixture model (GMM) in the latent space. To generate an instance, VaDE first samples a cluster distribution $p\left(c\right)$ to generate a latent vector $p\left(z\mid c\right)$, and then reconstructs the instance in the input space $p\left(x\mid z\right)$. The cluster assignment and neural network are jointly optimized by maximizing the log-likelihood of instance, i.e.,

Since directly computing Eq. 2 is intractable, the optimization is approximated by the evidence lower bound (ELBO) of variational inference objective, namely,

where $q(\mathrm{z},c\mid\mathrm{x})$ is variational posterior, which approximates the real posterior. The reparameterization trick introduced in VAE [43] is adopted to make the sampling process differentiable.

Though GMM could effectively distinguish distributions, Gaussian components are proved to be redundant, which harms the discriminability between different clusters [27]. As an improvement, ClusterGAN, DCGAN [69, 82] proposes to adopt GAN to implicitly learn the latent distributions. Specifically, in addition to the continuous latent variable $\mathbf{z}_{n}$, it introduces a one-hot encoding $\mathbf{z}_{c}$ to capture cluster distribution during the generation. The objective function of ClusterGAN is formulated as follows:

where $\mathbf{z}=(\mathbf{z}_{n},\mathbf{z}_{c})$ is the mixed latent variable, $\mathcal{E}$ is the inverse network which maps data from the raw to latent space, $\mathcal{H}\left(\cdot,\cdot\right)$ denotes the cross-entropy, and $\beta_{n}$, $\beta_{c}$ are the weight parameters. The first two terms are consistent with standard GAN. The last two clustering-specific terms encourage a more distinct cluster distribution, as well as map inputs to the latent space to achieve clustering.

In recent years, image augmentation methods [93] have gained widespread attention, grounded in the prior that augmentations of the same instance could preserve consistent semantic information. This augmentation-invariance character inspires exploration of how to leverage the positive pairs (i.e., different augmentations of the same image) with similar semantic information. Notably, mutual-information-based methods and contrastive-learning-based methods have emerged as pioneers in the realm of deep clustering. In this section, we delve into the fundamental concepts and related works of both mutual-information-based and contrastive-learning-based methods.

Firstly, mutual information is a measure of dependence between two continuous random variables $X$ and $Y$, formally,

where $p(x,y)$ is the joint probability mass function of $X$ and $Y$, $p(x)$ and $p(y)$ are the marginal probability mass functions of $X$ and $Y$ respectively. In the context of information theory, leveraging the mutual information between variables of positive instances could enhance the optimization of clustering-related information.

IMSAT [31] stands as a typical information-theoretic approach to deep clustering. Its fundamental concept includes enforcing invariance on pair-wise augmented instances and achieving unambiguous and uniform cluster assignments. Specifically, IMSAT encourages the representations of augmented instances to closely match the representations of the original instances, i.e.,

where $\mathbf{p}^{\prime}$ is the prediction representations of augmented instances. This aspect can be viewed as exploring the maximization of mutual information between data and its augmentations. Besides, IMSAT implements regularized information maximization for deep clustering inspired by RIM [44] to keep the cluster assignments unambiguous and uniform. Specifically, IMSAT seeks to maximize the mutual information between instances and their cluster assignments, expressed as:

where $H(\cdot)$ and $H(\cdot|\cdot)$ the entropy and conditional entropy, and $\mathbf{p}_{\cdot k}=\frac{1}{N}\sum_{i}\mathbf{p}_{ik}$. Increasing the first term (marginal entropy $H(Y)$) encourages uniform cluster assignments, i.e., the number of instances in each cluster tends to be the same. Conversely, decreasing the second term (conditional entropy $H(Y\mid X)$) encourages each instance to be unambiguously assigned to a certain cluster.

IIC [39] and Completer [57, 58] take a further step in exploring the mutual information between instances and their augmentations. The fundamental concept is to maximize the mutual information between the cluster assignments of pair-wise augmented instances. Specifically, IIC achieves semantically meaningful clustering and avoids trivial solutions by maximizing the mutual information between the cluster assignments,

where $\mathbf{z}$ and $\mathbf{z}^{\prime}$ are the representations of the original instance $x$ and its augmentation $\mathbf{x}^{\prime}$, respectively. The conditional joint distribution of $\mathbf{z}$ and $\mathbf{z}^{\prime}$ is given by the matrix $\mathbf{P}\in\mathbb{R}^{C\times C}$ which is constituted by,

where $\mathbf{P}_{cc^{\prime}}=P\left(z=c,z^{\prime}=c^{\prime}\right)$ denotes the element of $c$-th row and $c^{\prime}$-th column. Additionally, the marginals $\mathbf{P}_{c}=P(z=c)$ and $\mathbf{P}_{c^{\prime}}=P\left(z^{\prime}=c^{\prime}\right)$ can be obtained by summing over the rows and columns of this matrix. Notably, IIC stands out as one of the earliest deep frameworks designed entirely under the framework of information theory, distinguishing itself from IMSAT.

Similar to mutual-information-based methods, contrastive-learning-based methods treat instances augmented from the same instance as positive samples and the rest as negative samples. Let $\mathbf{z}_{2i}$ and $\mathbf{z}_{2i-1}$ represent two augmented representation of the $i$-th instance, the contrastive loss is formulated as:

where $\ell\left(i,j\right)$ represents the pairwise contrastive loss and $\tau$ controls the temperature of the softmax. The function $\operatorname{s}\left(\mathbf{z}_{i},\mathbf{z}_{j}\right)$ denotes the similarity between representations $\mathbf{z}_{i}$ and $\mathbf{z}_{j}$. This loss encourages representations of positive instances to be closer while being separated from negative instances, encouraging meaningful clustering patterns.

Notably, some theoretical works [78, 68, 59] have demonstrated that contrastive learning is equivalent to maximizing the mutual information from the instance level. Motivated by this observation, researchers have further explored the application of contrastive loss at the cluster level, proving beneficial for deep clustering. PICA [32] is one of the pioneer works of this domain. The fundamental concept behind it is to maximize the similarity between the cluster assignment of original and augmented data. This objective can be likened to conducting contrastive learning [60] at the cluster level. Motivated by PICA, CC [52] and DRC [123] conduct contrastive learning on both instance level and cluster level. Specifically, cluster-level contrastive loss helps learn discriminative cluster assignment, which is the key to the clustering task. Formally, the cluster-level contrastive loss is,

where $\mathbf{y}_{i}\in\mathbb{R}^{1\times N}$ is the cluster-level assignment and $\tau$ is the cluster-level temperature parameter. $H(\mathbf{Y})=H(\mathbf{Y}^{1})+H(\mathbf{Y}^{2})$ is the cluster assignment probabilities entropy of two augmentations. The inclusion of $H(\mathbf{Y})$ helps avoid the trivial solution where most instances are assigned to the same cluster. Notably, the utilization of contrastive learning at the cluster level in CC and DRC has inspired subsequent works in the field.

TCC [92] takes a step further in exploring the interaction between instance-level and cluster-level representations. The core idea is to leverage a unified representation combined of the cluster semantics and instances, enhancing the representation with cluster information to facilitate clustering tasks. Formally, for an instance representation $\mathbf{z}_{i}$, the enhanced representation is given by:

where $\mathbf{c}_{i}$ represents the cluster assignment of $i$-th instance after Gumbel Softmax. $\operatorname{NN}_{\theta}\left(\cdot\right)$ denotes a single fully connected network, which is the learnable cluster representation. Different from CC which performs contrastive loss on cluster assignment, TCC conducts contrastive loss on the unified representation to better capture cluster semantics. Inspired by TCC, some works [108, 50] explore the fusion of instance-level and cluster-level representation in various domains. and then conduct contrastive loss on the unified representation, which further explores its effectiveness.

Thanks to the advancements in self-supervised representation learning, the features acquired through discriminative pretext tasks can unveil high-level semantics in the latent space. This provides a crucial prior for clustering, as instances and their neighborhoods in the latent space are likely to belong to the same semantic cluster. Leveraging neighborhood-consistent semantics can further enhance clustering.

SCAN [97] first observes that similar instances will be mapped closely in latent space through self-supervised pretext tasks. Motivated by this observation, SCAN trains a cluster head based on the cluster neighborhood consistency within neighbors. Specifically, SCAN first obtains an encoder $f$ by a pretext task [23, 119, 104, 30]. It then optimizes a cluster head $h$ by requiring it to make consistent predictions between instances and their nearest neighbors:

Here $\mathcal{N}^{k}_{i}$ denotes the $k$-nearest neighbors of the $i$-th instance. The second term in Eq. 13 prevents $h$ from assigning all instances to a single cluster which is also used in Eq. 11.

NNM [15] and GCC [124] incorporate neighbor information into the framework of contrastive learning to group instances within neighborhoods. In particular, NNM aligns the clustering assignment of an instance with its neighbors through cluster-level contrastive learning:

where $\mathbf{q},~{}\mathbf{q}_{\mathcal{N}}\in\mathbb{R}^{C\times B}$ represent the transpose matrix of $\mathbf{p}$ and $\mathbf{p}_{\mathcal{N}}$, respectively. In contrast, GCC introduces the graph structure of the latent space to modify the vanilla instance-level contrastive loss. It constructs a normalized symmetric graph Laplacian $\mathbf{L}$ based on the $K$-nn graph:

Then, the loss function is given by the following form:

where $\tau$ is the temperature. The Graph Laplacian guides the model to attract instances within neighborhoods rather than just augmentation of themselves so that the influence of potential false negative samples [114, 112] can be mitigated. As a result, GCC can better minimize the intra-cluster variance and maximize the inter-cluster variance. The success of this approach has inspired numerous contrastive learning methods [38, 63] in various domains to leverage neighbor relationships that effectively address the false negative challenge.

As a prevalent paradigm of semi-supervised classification [48, 6, 94], pseudo-labeling has been extended to deep clustering in recent years. The fundamental assumption of pseudo-labeling is that the predictions on unlabeled data, especially the confident ones, can provide reliable supervision and guide model training. Motivated by this, recent deep clustering works leverage confident predictions to boost clustering performance.

DEC [106] is a pioneering work that utilizes labels generated by itself to simultaneously enhance feature representations and optimize clustering assignments. DEC initializes with a pre-trained auto-encoder and $C$ learnable cluster centroids. The soft assignment is calculated using the Student’s $t$-distribution, based on the distance between the representation $\mathbf{z}_{i}$ and centroid $\mathbf{c}_{j}$:

where $\alpha$ is the hyper-parameter and $\mathbf{q}_{ij}$ denotes the probability of assigning the instances $i$ to the cluster $j$. DEC refines the clusters by emphasizing the high-confidence assignments and making predictions more confident. Specifically, DEC uses the second power of $\mathbf{q}_{i}$ as a sharpened assignment to guide the training, i.e.,

where $\operatorname{freq}_{j}=\sum_{i}\mathbf{q}_{ij}$ is the soft cluster frequency and the sharpened assignment is normalized by $f_{j}$ to prevent feature collapse. Finally, a KL divergence loss between $\mathbf{p}$ and $\mathbf{q}$ minimizes the distances between the two distributions, i.e., $\mathcal{L}=\operatorname{KL}(\mathbf{p}|\mathbf{q})$.

Another notable method of pseudo-labeling is DeepCluster [8]. This approach employs K-means clustering on the learned representations to obtain cluster assignments as pseudo-labels. DeepCluster iteratively performs representation learning and clustering in a mutually beneficial manner to bootstrap each other. However, DeepCluster faces limitations in achieving outstanding performance, primarily due to the restricted semantics of the initial representation. Similar to DeepCluster, ProPos [36] proposes an EM framework of pseudo-labeling, iteratively performing K-means to obtain pseudo labels (E step) and the representation updating (M step). Notably, ProPos significantly outperforms DeepCluster and other methods because ProPos performs K-means on the learned feature of state-of-the-art self-supervised paradigm BYOL [26]. This observation has demonstrated that the semantics of the representation is vital to pseudo-label generation and clustering. Low-quality features would introduce potential noise in pseudo-labels, impact subsequent pseudo-label generation, and mislead representation learning, which accumulates the error in the process.

In addition to the progression of self-supervised paradigms, researchers are actively investigating strategies to alleviate the issue of error accumulation in pseudo-labeling. To be specific, the challenges in the realm of pseudo-labeling deep clustering remain two-fold: enhancing the accuracy of generating pseudo-labels and maximizing the utility of these pseudo-labels for effective clustering. On the one hand, inaccurate pseudo-labels pose a risk of degradation in clustering performance. On the other hand, determining how to effectively leverage these pseudo-labels for clustering is a critical consideration. These two challenges underscore the ongoing efforts in the pseudo-labeling learning of deep clustering.

The first challenge has been addressed by many works through carefully designing selection methods. For instance, SCAN [97] empirically observed that instances exhibiting highly confident predictions (i.e., $\max(\mathbf{p}_{i})\approx 1$) tend to be correctly clustered by the cluster head. Building on this insight, SCAN opts to choose instances with the most confident predictions as labeled data to fine-tune the model using the cross-entropy loss,

where $\eta$ is the threshold hyper-parameter to filter the uncertain instances. TCL [53] and SPICE [77] have devised more effective selection strategies to enhance the accuracy of pseudo-labeling. Specifically, TCL selects the most confident predictions as pseudo labels from each cluster $c$:

where $\operatorname{topK}(\cdot)$ returns the indices of the top $K$ confident instances and $\bigcup$ denotes the union of the pseudo labels from all clusters. Here $K=\gamma N/C$ and $\gamma$ is the selection ratio. The cluster-wise selection leads to more class-balanced pseudo labels compared to threshold-based criteria. It improves the clustering performance, especially for challenging classes.

SPICE introduces a prototype-based pseudo-labeling approach. Specifically, it first re-computes the centroids of each cluster only using the instances with confident predictions, then re-assign each instance with new pseudo labels according to the similarity to the new centroids, formally:

This operation helps mitigate the influence of potentially incorrect pseudo labels used in calculating centroids, which might accumulate errors in the iterative self-training process.

To address the second challenge, i.e., better utilizing the confident labels, TCL removes negative pairs with the same label in contrastive loss, preventing intra-class instances from pushing apart, i.e., the false negative issue. Meanwhile, SPICE and TCL adopt some semi-supervised classification techniques like FixMatch [94] that impose the pseudo-label consistency for strong augmentations of the same instance. The marvelous results achieved by these works show the effectiveness of combining reliable pseudo-labeling methods and semi-supervised paradigms in clustering.

Most clustering approaches focus on grouping data based on inherent characteristics such as structural priors, distribution priors, and augmentation invariance priors. Instead of pursuing internal priors from the data itself, some recent works [7, 54] attempt to introduce abundant external knowledge such as textual descriptions to guide clustering. These methods prove effective because the utilization of semantic information from natural language offers valuable supervisory signals that enhance the quality of clustering.

SIC [7] is one of the first works in incorporating external knowledge guidance into clustering. The fundamental concept revolves around generating image pseudo-labels from a textual space pre-trained by CLIP [83]. The process involves three main steps: i) Construction of Semantic Space: SIC selects meaningful texts resembling category names to build a semantic space. ii) Pseudo-labeling: Pseudo-labels are generated using text semantic centers $\mathbf{h}$ and image representations $\mathbf{z}_{i}$, formally,

where $c$ is the number of semantic centers, $\mathbf{h}_{l}$ is the $l$-th center of semantic centers, one-hot operator will generate a $c$-bit one-hot vector. The pseudo-labels is utilized to guide the clustering similar to SCAN [97],

where $CE\left(\cdot\right)$ is the cross entropy function. iii) Consistency learning: Enhancing clustering effect by enforcing the consistency between the images and their neighbors in the image space,

where $j$ is an instance index randomly selected from the nearest neighbors $\mathcal{N}_{k}\left(\mathbf{z}_{i}\right)$ of $i$-th instance. Note that, SIC essentially pulls image embeddings closer to embeddings in semantic space, while ignoring the improvement of text semantic embeddings.

TAC [54] focuses on leveraging textual semantics to enhance the feature discriminability. Specifically, it retrieves a text counterpart among representative nouns for each image, which improves K-means performance without any additional training. Besides, TAC proposes a mutual distillation paradigm to incorporate the image and text modalities, which further improves the clustering performance. The cross-modal mutual distillation strategy is formulated as follows:

where $\tau$ is the softmax temperature parameter, $\hat{\mathbf{p}}_{i},\hat{\mathbf{q}}_{i}\in\mathbb{R}^{1\times N}$ is the $i$-th column of image and text assignment matrix, $\hat{\mathbf{p}}_{i}^{\mathcal{N}},\hat{\mathbf{q}}_{i}^{\mathcal{N}}\in\mathbb{R}^{1\times N}$ is the $i$-th column of image and text random nearest neighbor matrix. The mutual distillation strategy has two advantages. On the one hand, it generates more discriminative clusters through cluster-level contrastive loss. On the other hand, it encourages consistent clustering assignments between each sample and its cross-modal neighbors, which bootstraps the clustering performance in both modalities.

For clustering evaluation, three metrics are commonly used to measure how the predicted cluster assignments $\tilde{y}$ match the ground truth labels $y$, including accuracy (ACC), normalized mutual information (NMI), and adjusted rand index (ARI). A higher value of the metrics corresponds to better clustering performance. The definitions of the three metrics are as follows:

• •

  ACC [1] indicates the correct rate of clustering predictions:

  $$\operatorname{ACC}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{1}\{y_{i}=\tilde{y}_{i}\},$$

  (26)

  where the Hungarian matching [46] is first applied to align the predictions and labels.

• •

  NMI [65] quantifies the mutual information between the predicted labels $\tilde{\mathbf{Y}}$ and ground truth labels $\mathbf{Y}$:

  $$\operatorname{NMI}=\frac{I(\tilde{\mathbf{Y}};\mathbf{Y})}{\frac{1}{2}[H(\tilde{\mathbf{Y}})+H(\mathbf{Y})]},$$

  (27)

  where $H(\mathbf{Y})$ denotes the entropy of $Y$ and $I(\tilde{\mathbf{Y}};\mathbf{Y})$ denotes the mutual information between $\tilde{\mathbf{Y}}$ and $\mathbf{Y}$.

• •

  ARI [37] is the normalization of the rand index (RI), which counts the number of instances pairs in the same cluster and different clusters:

  $$\operatorname{RI}=\frac{\operatorname{TP}+{\operatorname{TN}}}{\operatorname{C}^{2}_{N}},$$

  (28)

  where $\operatorname{TP}$ and $\operatorname{TN}$ refer to the number of true positive pairs and true negative pairs, $\operatorname{C}^{2}_{N}$ is the number of possible instance pairs. ARI is computed by adding the following normalization:

  $$\operatorname{ARI}=\frac{\operatorname{RI}-\mathbb{E}(\operatorname{RI})}{\operatorname{max}(\operatorname{RI})-\mathbb{E}\left(\operatorname{RI}\right)},$$

  (29)

  where $\mathbb{E}(\operatorname{RI})$ denotes the expectation of RI.

In the early stage, deep clustering methods are evaluated on relatively small and low-dimensional datasets (e.g. COIL-20 [72], YaleB [22]). Recently, with the rapid development of deep clustering methods, it has become more popular to evaluate clustering performance on more complex and challenging datasets. There are five widely used benchmark datasets:

• •

  CIFAR-10 [45] consists of 60,000 colored images from 10 different classes including airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck.

• •

  CIFAR-100 [45] contains 100 classes grouped into 20 superclasses. Each image comes with a “fine” class label and a “coarse” superclass label.

• •

  STL-10 [13] contains 13,000 labeled images from 10 object classes. Besides, it provides 100,000 unlabeled images for self-supervised learning to enhance the clustering performance.

• •

  ImageNet-10 [9] is a subset of the ImageNet dataset [17]. It contains 10 classes, each with 1,300 high-resolution images.

• •

  ImageNet-Dog [9] is another subset of ImageNet. It consists of images belonging to 15 dog breeds, which is suitable for fine-grained clustering tasks.

Apart from them, some recent works employ two more challenging large-scale datasets, Tiny-ImageNet [49] and ImageNet-1K [17], to evaluate the effectiveness and efficiency. A brief description of these datasets is summarized in Table 3.

The clustering performance on five widely used datasets is shown in Table 4. Thanks to the feature extraction ability of deep neural networks, early deep clustering methods based on structure and distribution priors achieve much better performance than the classic K-means. Then, a series of contrastive clustering methods significantly improve the performance by introducing additional priors through data augmentation. After that, more advanced methods boost the performance by further considering the neighborhood consistency (GCC compared with CC) and utilizing pseudo labels (SCAN compared with SCAN^∗). Notably, the performance gains of different priors are independent. For example, ProPos remarkably outperforms DEC and CC by additionally utilizing the augmentation invariance or pseudo-labeling priors, respectively. Very recently, external-knowledge-based methods achieved state-of-the-art performance, which proves the promising prospect of such a new deep clustering paradigm. In addition, clustering becomes more challenging when the category number grows (from CIFAR-10 to CIFAR-100) or the semantics becomes more complex (from CIFAR-10 to ImageNet-Dogs). Such results indicate that more challenging datasets such as full ImageNet-1K are expected to benchmark in future works.

The objective of fine-grained clustering is to discern subtle and intricate variations within data, which is particularly advantageous in research like the identification of biological subspecies [55, 56]. The primary challenge is that fine-grained classes exhibit a high degree of similarity, where distinctions often lie in coloration, markings, shape, or other subtle characteristics. In such scenarios, traditional coarse-grained clustering priors frequently prove inadequate. For instance, color and shape augmentations in augmentation invariance prior would become ineffective. Recently, C3-GAN [42] employs contrastive learning within adversarial training to generate lifelike images, enabling the nuanced capture of fine-grained details and ensuring the separability between clusters.

Many clustering methods typically require a predefined and fixed number of clusters. However, real-world datasets often present a challenge with an unknown number of clusters, reflecting situations closer to reality. Only a few works [11, 89, 122, 100] have been devoted to solving this problem. These methods often rely on calculating global similarity and introduce huge computational costs, especially in large-scale datasets. Therefore, efficiently determining the optimal value of cluster number $C$ remains an open challenge, often involving the incorporation of human priors. Among existing works, DeepDPM introduces Dirichlet Process Gaussian Mixture Models (DPGMM) [3] that utilize the Dirichlet Process as the prior distribution over mixture components. DeepDPM dynamically adjusts the number of clusters $C$ through split and merge operations guided by the Metropolis-Hastings framework [29].

Collecting Real-world datasets from diverse sources with various acquisition methods can enhance the generalization of machine learning models. However, these datasets frequently manifest inherent biases, notably in sensitive attributes such as gender, race, and ethnicity. These biases would introduce disparities among individuals and minority groups, leading to cluster partitions that deviate from the underlying objective characteristics of the data. The pursuit of fairness is particularly pertinent in applications where unbiased and equitable analyses are crucial, such as employment, healthcare, and education. To tackle this challenge, fair clustering seeks to mitigate the influence of these biases given the biased attributes for each sample.

To address this daunting task, Chierichetti et al first introduces a data pre-processing method known as fairlet decomposition. Recent advancements address this issue on large-scale data through adversarial training [51] and mutual information maximization [116]. Notably, Zeng et al designs a novel metric that assesses both clustering quality and fairness from the perspective of information theory. Despite these developments, there is still room for improvement, and the establishment of better evaluation metrics is a continuing area of this research.

Multi-view data [107, 62] is common in real-world situations where information is captured from a variety of sensors or observed from multiple angles. This data is inherently rich, offering diverse yet consistent information. For example, an RGB view would provide color details while the depth view reveals spatial information, which represents the complementary aspects of the views. Simultaneously, there exists a level of view consistency as the same object possesses common attributes across different views. To deal with multi-view data, multi-view clustering [16, 61] is proposed to exploit both the complementary and consistent characters. The goal is to integrate information from all views to produce a unified and insightful clustering result.

Over recent years, several deep-learning approaches [2, 99, 121, 80] have been developed to address this challenge. Binary multi-view clustering Zhang et al [120] simultaneously refines binary cluster structures alongside discrete data representations, ensuring cohesive clustering. In pursuit of view consistency, Lin et al [57, 58] maximize mutual information across views, thus aligning common properties. SURE [114] aims to strengthen the consistency of shared features between views by utilizing robust contrastive loss. Recently, Li et al [50] performs bound contrastive loss to preserve the view complementary at the cluster level. These innovative methodologies demonstrate the significant strides made in the field of multi-view analysis, where clustering continues to play a pivotal role in enhancing the synergistic exploitation of multi-view data.