In this paper, we consider nonparametric clustering of $M$ independent and identically distributed (i.i.d.) data sequences generated from {\em unknown} distributions. The distributions of the $M$ data sequences belong to $K$ underlying distribution clusters. Existing results on exponentially consistent nonparametric clustering algorithms, like single linkage-based (SLINK) clustering and $k$-medoids distribution clustering, assume that the maximum intra-cluster distance ($d_L$) is smaller than the minimum inter-cluster distance ($d_H$). First, in the fixed sample size (FSS) setting, we show that exponential consistency can be achieved for SLINK clustering under a less strict assumption, $d_I < d_H$, where $d_I$ is the maximum distance between any two sub-clusters of a cluster that partition the cluster. Note that $d_I < d_L$ in general. Thus, our results show that SLINK is exponentially consistent for a larger class of problems than previously known. In our simulations, we also identify examples where $k$-medoids clustering is unable to find the true clusters, but SLINK is exponentially consistent. Then, we propose a sequential clustering algorithm, named SLINK-SEQ, based on SLINK and prove that it is also exponentially consistent. Simulation results show that the SLINK-SEQ algorithm requires fewer expected number of samples than the FSS SLINK algorithm for the same probability of error.
The Generative Adversarial Network (GAN) was recently introduced in the literature as a novel machine learning method for training generative models. It has many applications in statistics such as nonparametric clustering and nonparametric conditional independence tests. However, training the GAN is notoriously difficult due to the issue of mode collapse, which refers to the lack of diversity among generated data. In this paper, we identify the reasons why the GAN suffers from this issue, and to address it, we propose a new formulation for the GAN based on randomized decision rules. In the new formulation, the discriminator converges to a fixed point while the generator converges to a distribution at the Nash equilibrium. We propose to train the GAN by an empirical Bayes-like method by treating the discriminator as a hyper-parameter of the posterior distribution of the generator. Specifically, we simulate generators from its posterior distribution conditioned on the discriminator using a stochastic gradient Markov chain Monte Carlo (MCMC) algorithm, and update the discriminator using stochastic gradient descent along with simulations of the generators. We establish convergence of the proposed method to the Nash equilibrium. Apart from image generation, we apply the proposed method to nonparametric clustering and nonparametric conditional independence tests. A portion of the numerical results is presented in the supplementary material.
Clustering by fast search and find of density peaks (DPC) (Since, 2014) has been proven to be a promising clustering approach that efficiently discovers the centers of clusters by finding the density peaks. The accuracy of DPC depends on the cutoff distance ($d_c$), the cluster number ($k$) and the selection of the centers of clusters. Moreover, the final allocation strategy is sensitive and has poor fault tolerance. The shortcomings above make the algorithm sensitive to parameters and only applicable for some specific datasets. To overcome the limitations of DPC, this paper presents an improved probability propagation algorithm for density peak clustering based on the natural nearest neighborhood (DPC-PPNNN). By introducing the idea of natural nearest neighborhood and probability propagation, DPC-PPNNN realizes the nonparametric clustering process and makes the algorithm applicable for more complex datasets. In experiments on several datasets, DPC-PPNNN is shown to outperform DPC, K-means and DBSCAN.
With advances in neural recording techniques, neuroscientists are now able to record the spiking activity of many hundreds of neurons simultaneously, and new statistical methods are needed to understand the structure of this large-scale neural population activity. Although previous work has tried to summarize neural activity within and between known populations by extracting low-dimensional latent factors, in many cases what determines a unique population may be unclear. Neurons differ in their anatomical location, but also, in their cell types and response properties. To identify populations directly related to neural activity, we develop a clustering method based on a mixture of dynamic Poisson factor analyzers (mixDPFA) model, with the number of clusters and dimension of latent factors for each cluster treated as unknown parameters. To analyze the proposed mixDPFA model, we propose a Markov chain Monte Carlo (MCMC) algorithm to efficiently sample its posterior distribution. Validating our proposed MCMC algorithm through simulations, we find that it can accurately recover the unknown parameters and the true clustering in the model, and is insensitive to the initial cluster assignments. We then apply the proposed mixDPFA model to multi-region experimental recordings, where we find that the proposed method can identify novel, reliable clusters of neurons based on their activity, and may, thus, be a useful tool for neural data analysis.
Deep Learning (DL) has shown great promise in the unsupervised task of clustering. That said, while in classical (i.e., non-deep) clustering the benefits of the nonparametric approach are well known, most deep-clustering methods are parametric: namely, they require a predefined and fixed number of clusters, denoted by K. When K is unknown, however, using model-selection criteria to choose its optimal value might become computationally expensive, especially in DL as the training process would have to be repeated numerous times. In this work, we bridge this gap by introducing an effective deep-clustering method that does not require knowing the value of K as it infers it during the learning. Using a split/merge framework, a dynamic architecture that adapts to the changing K, and a novel loss, our proposed method outperforms existing nonparametric methods (both classical and deep ones). While the very few existing deep nonparametric methods lack scalability, we demonstrate ours by being the first to report the performance of such a method on ImageNet. We also demonstrate the importance of inferring K by showing how methods that fix it deteriorate in performance when their assumed K value gets further from the ground-truth one, especially on imbalanced datasets. Our code is available at https://github.com/BGU-CS-VIL/DeepDPM.
In recent years, deep-learning-based speech emotion recognition models have outperformed classical machine learning models. Previously, neural network designs, such as Multitask Learning, have accounted for variations in emotional expressions due to demographic and contextual factors. However, existing models face a few constraints: 1) they rely on a clear definition of domains (e.g. gender, noise condition, etc.) and the availability of domain labels; 2) they often attempt to learn domain-invariant features while emotion expressions can be domain-specific. In the present study, we propose the Nonparametric Hierarchical Neural Network (NHNN), a lightweight hierarchical neural network model based on Bayesian nonparametric clustering. In comparison to Multitask Learning approaches, the proposed model does not require domain/task labels. In our experiments, the NHNN models generally outperform the models with similar levels of complexity and state-of-the-art models in within-corpus and cross-corpus tests. Through clustering analysis, we show that the NHNN models are able to learn group-specific features and bridge the performance gap between groups.
Image segmentation aims at identifying regions of interest within an image, by grouping pixels according to their properties. This task resembles the statistical one of clustering, yet many standard clustering methods fail to meet the basic requirements of image segmentation: segment shapes are often biased toward predetermined shapes and their number is rarely determined automatically. Nonparametric clustering is, in principle, free from these limitations and turns out to be particularly suitable for the task of image segmentation. This is also witnessed by several operational analogies, as, for instance, the resort to topological data analysis and spatial tessellation in both the frameworks. We discuss the application of nonparametric clustering to image segmentation and provide an algorithm specific for this task. Pixel similarity is evaluated in terms of density of the color representation and the adjacency structure of the pixels is exploited to introduce a simple, yet effective method to identify image segments as disconnected high-density regions. The proposed method works both to segment an image and to detect its boundaries and can be seen as a generalization to color images of the class of thresholding methods.
We describe an adaptation of the simulated annealing algorithm to nonparametric clustering and related probabilistic models. This new algorithm learns nonparametric latent structure over a growing and constantly churning subsample of training data, where the portion of data subsampled can be interpreted as the inverse temperature beta(t) in an annealing schedule. Gibbs sampling at high temperature (i.e., with a very small subsample) can more quickly explore sketches of the final latent state by (a) making longer jumps around latent space (as in block Gibbs) and (b) lowering energy barriers (as in simulated annealing). We prove subsample annealing speeds up mixing time N^2 -> N in a simple clustering model and exp(N) -> N in another class of models, where N is data size. Empirically subsample-annealing outperforms naive Gibbs sampling in accuracy-per-wallclock time, and can scale to larger datasets and deeper hierarchical models. We demonstrate improved inference on million-row subsamples of US Census data and network log data and a 307-row hospital rating dataset, using a Pitman-Yor generalization of the Cross Categorization model.
In this publication, we combine two Bayesian non-parametric models: the Gaussian Process (GP) and the Dirichlet Process (DP). Our innovation in the GP model is to introduce a variation on the GP prior which enables us to model structured time-series data, i.e. data containing groups where we wish to model inter- and intra-group variability. Our innovation in the DP model is an implementation of a new fast collapsed variational inference procedure which enables us to optimize our variationala pproximation significantly faster than standard VB approaches. In a biological time series application we show how our model better captures salient features of the data, leading to better consistency with existing biological classifications, while the associated inference algorithm provides a twofold speed-up over EM-based variational inference.
Clustering is an essential problem in machine learning and data mining. One vital factor that impacts clustering performance is how to learn or design the data representation (or features). Fortunately, recent advances in deep learning can learn unsupervised features effectively, and have yielded state of the art performance in many classification problems, such as character recognition, object recognition and document categorization. However, little attention has been paid to the potential of deep learning for unsupervised clustering problems. In this paper, we propose a deep belief network with nonparametric clustering. As an unsupervised method, our model first leverages the advantages of deep learning for feature representation and dimension reduction. Then, it performs nonparametric clustering under a maximum margin framework -- a discriminative clustering model and can be trained online efficiently in the code space. Lastly model parameters are refined in the deep belief network. Thus, this model can learn features for clustering and infer model complexity in an unified framework. The experimental results show the advantage of our approach over competitive baselines.
Density mode clustering is a nonparametric clustering method. The clusters are the basins of attraction of the modes of a density estimator. We study the risk of mode-based clustering. We show that the clustering risk over the cluster cores --- the regions where the density is high --- is very small even in high dimensions. And under a low noise condition, the overall cluster risk is small even beyond the cores, in high dimensions.
We present a new fast online clustering algorithm that reliably recovers arbitrary-shaped data clusters in high throughout data streams. Unlike the existing state-of-the-art online clustering methods based on k-means or k-medoid, it does not make any restrictive generative assumptions. In addition, in contrast to existing nonparametric clustering techniques such as DBScan or DenStream, it gives provable theoretical guarantees. To achieve fast clustering, we propose to represent each cluster by a skeleton set which is updated continuously as new data is seen. A skeleton set consists of weighted samples from the data where weights encode local densities. The size of each skeleton set is adapted according to the cluster geometry. The proposed technique automatically detects the number of clusters and is robust to outliers. The algorithm works for the infinite data stream where more than one pass over the data is not feasible. We provide theoretical guarantees on the quality of the clustering and also demonstrate its advantage over the existing state-of-the-art on several datasets.
We present a novel hierarchical distance-dependent Bayesian model for event coreference resolution. While existing generative models for event coreference resolution are completely unsupervised, our model allows for the incorporation of pairwise distances between event mentions -- information that is widely used in supervised coreference models to guide the generative clustering processing for better event clustering both within and across documents. We model the distances between event mentions using a feature-rich learnable distance function and encode them as Bayesian priors for nonparametric clustering. Experiments on the ECB+ corpus show that our model outperforms state-of-the-art methods for both within- and cross-document event coreference resolution.
Structural regularities in man-made environments reflect in the distribution of their surface normals. Describing these surface normal distributions is important in many computer vision applications, such as scene understanding, plane segmentation, and regularization of 3D reconstructions. Based on the small-variance limit of Bayesian nonparametric von-Mises-Fisher (vMF) mixture distributions, we propose two new flexible and efficient k-means-like clustering algorithms for directional data such as surface normals. The first, DP-vMF-means, is a batch clustering algorithm derived from the Dirichlet process (DP) vMF mixture. Recognizing the sequential nature of data collection in many applications, we extend this algorithm to DDP-vMF-means, which infers temporally evolving cluster structure from streaming data. Both algorithms naturally respect the geometry of directional data, which lies on the unit sphere. We demonstrate their performance on synthetic directional data and real 3D surface normals from RGB-D sensors. While our experiments focus on 3D data, both algorithms generalize to high dimensional directional data such as protein backbone configurations and semantic word vectors.
This paper presents a new approach to non-parametric cluster analysis called Adaptive Weights Clustering (AWC). The idea is to identify the clustering structure by checking at different points and for different scales on departure from local homogeneity. The proposed procedure describes the clustering structure in terms of weights \( w_{ij} \) each of them measures the degree of local inhomogeneity for two neighbor local clusters using statistical tests of "no gap" between them. % The procedure starts from very local scale, then the parameter of locality grows by some factor at each step. The method is fully adaptive and does not require to specify the number of clusters or their structure. The clustering results are not sensitive to noise and outliers, the procedure is able to recover different clusters with sharp edges or manifold structure. The method is scalable and computationally feasible. An intensive numerical study shows a state-of-the-art performance of the method in various artificial examples and applications to text data. Our theoretical study states optimal sensitivity of AWC to local inhomogeneity.
Inpatient care is a large share of total health care spending, making analysis of inpatient utilization patterns an important part of understanding what drives health care spending growth. Common features of inpatient utilization measures include zero inflation, over-dispersion, and skewness, all of which complicate statistical modeling. Mixture modeling is a popular approach that can accommodate these features of health care utilization data. In this work, we add a nonparametric clustering component to such models. Our fully Bayesian model framework allows for an unknown number of mixing components, so that the data determine the number of mixture components. When we apply the modeling framework to data on hospital lengths of stay for patients with lung cancer, we find distinct subgroups of patients with differences in means and variances of hospital days, health and treatment covariates, and relationships between covariates and length of stay.
The $k$-means algorithm is arguably the most popular nonparametric clustering method but cannot generally be applied to datasets with incomplete records. The usual practice then is to either impute missing values under an assumed missing-completely-at-random mechanism or to ignore the incomplete records, and apply the algorithm on the resulting dataset. We develop an efficient version of the $k$-means algorithm that allows for clustering in the presence of incomplete records. Our extension is called $k_m$-means and reduces to the $k$-means algorithm when all records are complete. We also provide initialization strategies for our algorithm and methods to estimate the number of groups in the dataset. Illustrations and simulations demonstrate the efficacy of our approach in a variety of settings and patterns of missing data. Our methods are also applied to the analysis of activation images obtained from a functional Magnetic Resonance Imaging experiment.
Motivated by problems in data clustering, we establish general conditions under which families of nonparametric mixture models are identifiable, by introducing a novel framework involving clustering overfitted \emph{parametric} (i.e. misspecified) mixture models. These identifiability conditions generalize existing conditions in the literature, and are flexible enough to include for example mixtures of Gaussian mixtures. In contrast to the recent literature on estimating nonparametric mixtures, we allow for general nonparametric mixture components, and instead impose regularity assumptions on the underlying mixing measure. As our primary application, we apply these results to partition-based clustering, generalizing the notion of a Bayes optimal partition from classical parametric model-based clustering to nonparametric settings. Furthermore, this framework is constructive so that it yields a practical algorithm for learning identified mixtures, which is illustrated through several examples on real data. The key conceptual device in the analysis is the convex, metric geometry of probability measures on metric spaces and its connection to the Wasserstein convergence of mixing measures. The result is a flexible framework for nonparametric clustering with formal consistency guarantees.