The joint optimization of representation learning and clustering in the embedding space has experienced a breakthrough in recent years. In spite of the advance, clustering with representation learning has been limited to flat-level categories, which oftentimes involves cohesive clustering with a focus on instance relations. To overcome the limitations of flat clustering, we introduce hierarchically clustered representation learning (HCRL), which simultaneously optimizes representation learning and hierarchical clustering in the embedding space. Specifically, we place a nonparametric Bayesian prior on embeddings to handle dynamic mixture hierarchies under the variational autoencoder framework, and to adopt the generative process of a hierarchical-versioned Gaussian mixture model. Compared with a few prior works focusing on unifying representation learning and hierarchical clustering, HCRL is the first model to consider a generation of deep embeddings from every component of the hierarchy, not just leaf components. This generation process enables more meaningful separations and mergers of clusters via branches in a hierarchy. In addition to obtaining hierarchically clustered embeddings, we can reconstruct data by the various abstraction levels, infer the intrinsic hierarchical structure, and learn the level-proportion features. We conducted evaluations with image and text domains, and our quantitative analyses showed competent likelihoods and the best accuracies compared with the baselines. Clustering is one of the most traditional and frequently used machine learning tasks. Clustering models are designed to represent intrinsic data structures, such as latent Dirichlet allocation BID2 . The recent development of representation learning has contributed to generalizing model feature engineering, which also enhances data representation BID1 . Therefore, representation learning has been merged into the clustering models, e.g., variational deep embedding (VaDE) (Jiang et al., 2017) . Besides merging representation learning and clustering, another critical line of research is structuring the clustering result, e.g., hierarchical clustering. This paper introduces a unified model enabling nonparametric Bayesian hierarchical clustering with neural-network-based representation learning.Autoencoder (Rumelhart et al., 1985) is a typical neural network for unsupervised representation learning and achieves a non-linear mapping from a high-dimensional input space to a lowdimensional embedding space by minimizing reconstruction errors. To turn the low-dimensional embeddings into random variables, a variational autoencoder (VAE) (Kingma & Welling, 2014) places a Gaussian prior on the embeddings. The autoencoder, whether it is probabilistic or not, has a limitation in reflecting the intrinsic hierarchical structure of data. For instance, VAE assuming a single Gaussian prior needs to be expanded to suggest an elaborate clustering structure.Due to the limitations of modeling the cluster structure with autoencoders, prior works combine the autoencoder and the clustering algorithm. While some early cases pipeline just two models, e.g., Huang et al. (2014) , a typical merging approach is to model an additional loss, such as a clustering loss, in the autoencoders (Xie et al., 2016; Guo et al., 2017; Yang et al., 2017; Nalisnick et al., 2016; BID4 Jiang et al., 2017) . These suggestions exhibit gains from unifying the encoding and the clustering, yet they remain at the parametric and flat-structured clustering. A more recent development releases the previous constraints by using the nonparametric Bayesian approach. Figure 1: Example of hierarchically clustered embeddings on MNIST with three levels of hierarchy, the reconstructed digits from the hierarchical Gaussian mixture components, and the extracted level proportion features. We marked the mean of a Gaussian mixture component with the colored square, and the digit written inside the square refers to the unique index of the mixture component.For example, the infinite mixture of VAEs (IMVAE) BID0 explores the infinite space for VAE mixtures by looking for an adequate embedding space through sampling, such as the Chinese restaurant process (CRP). Whereas IMVAE remains at the flat-structured clustering, VAEnested CRP (VAE-nCRP) (Goyal et al., 2017) captures a more complex structure, i.e., a hierarchical structure of the data, by adopting the nested Chinese restaurant process (nCRP) prior (Griffiths et al., 2004) into the cluster assignment of the Gaussian mixture model. This paper proposes hierarchically clustered representation learning (HCRL) that is a joint model of 1) nonparametric Bayesian hierarchical clustering, and 2) representation learning with neural networks. HCRL extends a previous work on merging flat clustering and representation learning, i.e., VaDE, by incorporating inter-cluster relation modelings. Unlike a previous work of VAE-nCRP, HCRL learns the full spectrum of hierarchical clusterings, such as the level assignment and the level proportion of generating a component hierarchy. These level assignments and proportions were not modeled in VAE-nCRP, so each data instance cannot be analyzed from the perspective of generalization and specialization in a hierarchy. On the contrary, by adding level assignment and proportion modeling, a data instance can be generated from an internal component of the hierarchy, which is limited to the leaf component in VAE-nCRP. Hierarchical mixture density estimation (Vasconcelos & Lippman, 1999) , where all internal and leaf components are directly modeled to generate data, is a flexible framework for hierarchical mixture modeling, such as hierarchical topic modeling (Mimno et al., 2007; Griffiths et al., 2004) , with regard to the learning of the internal components.Specifically, HCRL jointly optimizes soft-divisive hierarchical clustering in an embedding space from VAE via two mechanisms. First, HCRL includes a hierarchical-versioned Gaussian mixture model (HGMM) with a mixture of hierarchically organized Gaussian distributions. Then, HCRL sets the prior of embeddings by adopting the generative processes of HGMM. Second, to handle a dynamic hierarchy structure dealing with the clusters of unequal sizes, we explore the infinite hierarchy space by exploiting an nCRP prior. These mechanisms are fused as a unified objective function; this is done rather than concatenating the two distinct models of clustering and autoencoding. The quantitative evaluations focus on density estimation quality and hierarchical clustering accuracy, which shows that HCRL has competent likelihoods and the best accuracies compared with the baselines. When we observe our results qualitatively, we visualize 1) the hierarchical clusterings, 2) the embeddings under the hierarchy modeling, and 3) the reconstructed images from each Gaussian mixture component, as shown in FIG3 . These experiments were conducted by crossing the data domains of texts and images, so our benchmark datasets include MNIST, CIFAR-100, RCV1 v2, and 20Newsgroups. In this paper, we have introduced a hierarchically clustered representation learning framework for the hierarchical mixture density estimation on deep embeddings. HCRL aims at encoding the relations among clusters as well as among instances to preserve the internal hierarchical structure of data. The main differentiated features of HCRL are 1) the crucial assumption regarding the internal mixture components for having the ability to generate data directly, and 2) the unbalanced autoencoding neural architecture for the level proportion modeling as the encoding structure, and the probabilistic model as the decoding structure. From the modeling and the evaluation, we found that HCRL enables the improvements due to the high flexibility modeling compared with the baselines.
We introduce hierarchically clustered representation learning (HCRL), which simultaneously optimizes representation learning and hierarchical clustering in the embedding space. The paper proposes using the nested CRP as a clustering model rather than a topic model Presents a novel hierarchical clustering method over an embedding space where both the embedding space and the heirarchical clustering are simultaneously learnt