Spectral clustering techniques are valuable tools in signal processing and machine learning for partitioning complex data sets. The effectiveness of spectral clustering stems from constructing a non-linear embedding based on creating a similarity graph and computing the spectral decomposition of the Laplacian matrix. However, spectral clustering methods fail to scale to large data sets because of high computational cost and memory usage. A popular approach for addressing these problems utilizes the Nystrom method, an efficient sampling-based algorithm for computing low-rank approximations to large positive semi-definite matrices. This paper demonstrates how the previously popular approach of Nystrom-based spectral clustering has severe limitations. Existing time-efficient methods ignore critical information by prematurely reducing the rank of the similarity matrix associated with sampled points. Also, current understanding is limited regarding how utilizing the Nystrom approximation will affect the quality of spectral embedding approximations. To address the limitations, this work presents a principled spectral clustering algorithm that makes full use of the information obtained from the Nystrom method. The proposed method exhibits linear scalability in the number of input data points, allowing us to partition large complex data sets. We provide theoretical results to reduce the current gap and present numerical experiments with real and synthetic data. Empirical results demonstrate the efficacy and efficiency of the proposed method compared to existing spectral clustering techniques based on the Nystrom method and other efficient methods. The overarching goal of this work is to provide an improved baseline for future research directions to accelerate spectral clustering.

We introduce a new approach to clustering by using tangles, a tool that originates in mathematical graph theory. Given a collection of "weak partitions" of a data set, tangles provide a framework to aggregate these weak partitions such that they "point in the direction of a cluster". As a result, a cluster is softly characterized by a set of consistent pointers. This mechanism provides a highly flexible way of solving soft clustering problems in a variety of setups, ranging from questionnaires over community detection in graphs to clustering points in metric spaces. Conceptually, tangles have many intriguing properties: (1) Similar to boosting, which combines many weak classifiers to a strong classifier, tangles provide a formal way to combine many weak partitions to obtain few strong clusters. (2) In terms of computational complexity, tangles allow us to use simple, fast algorithms to produce the weak partitions. The complexity of identifying the strong partitions is dominated by the number of weak partitions, not the number of data points, leading to an interesting trade-off between the two. (3) If the weak partitions are interpretable, so are the strong partitions induced by the tangles, resulting in one of the rare algorithms to produce interpretable clusters. (4) The output of tangles is of a hierarchical nature, inducing the notion of a soft dendrogram that can be helpful in data visualization.

Kernel functions are a powerful tool to enhance the $k$-means clustering algorithm via the kernel trick. It is known that the parameters of the chosen kernel function can have a dramatic impact on the result. In supervised settings, these can be tuned via cross-validation, but for clustering this is not straightforward and heuristics are usually employed. In this paper we study the impact of kernel parameters on kernel $k$-means. In particular, we derive a lower bound, tight up to constant factors, below which the parameter of the RBF kernel will render kernel $k$-means meaningless. We argue that grid search can be ineffective for hyperparameter search in this context and propose an alternative algorithm for this purpose. In addition, we offer an efficient implementation based on fast approximate exponentiation with provable quality guarantees. Our experimental results demonstrate the ability of our method to efficiently reveal a rich and useful set of hyperparameter values.

In this paper, we introduce the Dissimilarity Mixture Autoencoder (DMAE), a novel neural network model that uses a dissimilarity function to generalize a family of density estimation and clustering methods. It is formulated in such a way that it internally estimates the parameters of a probability distribution through gradient-based optimization. Also, the proposed model can leverage from deep representation learning due to its straightforward incorporation into deep learning architectures, because, it consists of an encoder-decoder network that computes a probabilistic representation. Experimental evaluation was performed on image and text clustering benchmark datasets showing that the method is competitive in terms of unsupervised classification accuracy and normalized mutual information. The source code to replicate the experiments is publicly available at https://github.com/larajuse/DMAE

BIRCH clustering is a widely known approach for clustering, that has influenced much subsequent research and commercial products. The key contribution of BIRCH is the Clustering Feature tree (CF-Tree), which is a compressed representation of the input data. As new data arrives, the tree is eventually rebuilt to increase the compression. Afterward, the leaves of the tree are used for clustering. Because of the data compression, this method is very scalable. The idea has been adopted for example for k-means, data stream, and density-based clustering. Clustering features used by BIRCH are simple summary statistics that can easily be updated with new data: the number of points, the linear sums, and the sum of squared values. Unfortunately, how the sum of squares is then used in BIRCH is prone to catastrophic cancellation. We introduce a replacement cluster feature that does not have this numeric problem, that is not much more expensive to maintain, and which makes many computations simpler and hence more efficient. These cluster features can also easily be used in other work derived from BIRCH, such as algorithms for streaming data. In the experiments, we demonstrate the numerical problem and compare the performance of the original algorithm compared to the improved cluster features.

Community detection methods attempt to divide a network into groups of nodes that share similar properties, thus revealing its large-scale structure. A major challenge when employing such methods is that they are often degenerate, typically yielding a complex landscape of competing answers. As an attempt to extract understanding from a population of alternative solutions, many methods exist to establish a consensus among them in the form of a single partition "point estimate" that summarizes the whole distribution. Here we show that it is in general not possible to obtain a consistent answer from such point estimates when the underlying distribution is too heterogeneous. As an alternative, we provide a comprehensive set of methods designed to characterize and summarize complex populations of partitions in a manner that captures not only the existing consensus, but also the dissensus between elements of the population. Our approach is able to model mixed populations of partitions where multiple consensuses can coexist, representing different competing hypotheses for the network structure. We also show how our methods can be used to compare pairs of partitions, how they can be generalized to hierarchical divisions, and be used to perform statistical model selection between competing hypotheses.

In this paper, we initiate the study of fair clustering that ensures distributional similarity among similar individuals. In response to improving fairness in machine learning, recent papers have investigated fairness in clustering algorithms and have focused on the paradigm of statistical parity/group fairness. These efforts attempt to minimize bias against some protected groups in the population. However, to the best of our knowledge, the alternative viewpoint of individual fairness, introduced by Dwork et al. (ITCS 2012) in the context of classification, has not been considered for clustering so far. Similar to Dwork et al., we adopt the individual fairness notion which mandates that similar individuals should be treated similarly for clustering problems. We use the notion of $f$-divergence as a measure of statistical similarity that significantly generalizes the ones used by Dwork et al. We introduce a framework for assigning individuals, embedded in a metric space, to probability distributions over a bounded number of cluster centers. The objective is to ensure (a) low cost of clustering in expectation and (b) individuals that are close to each other in a given fairness space are mapped to statistically similar distributions. We provide an algorithm for clustering with $p$-norm objective ($k$-center, $k$-means are special cases) and individual fairness constraints with provable approximation guarantee. We extend this framework to include both group fairness and individual fairness inside the protected groups. Finally, we observe conditions under which individual fairness implies group fairness. We present extensive experimental evidence that justifies the effectiveness of our approach.

Cluster analysis organizes data into sensible groupings and is one of fundamental modes of understanding and learning. The widely used K-means and hierarchical clustering methods can be dramatically suboptimal due to local minima. Recently introduced convex clustering approach formulates clustering as a convex optimization problem and ensures a globally optimal solution. However, the state-of-the-art convex clustering algorithms, based on the alternating direction method of multipliers (ADMM) or the alternating minimization algorithm (AMA), require large computation and memory space, which limits their applications. In this paper, we develop a very efficient smoothing proximal gradient algorithm (Sproga) for convex clustering. Our Sproga is faster than ADMM- or AMA-based convex clustering algorithms by one to two orders of magnitude. The memory space required by Sproga is less than that required by ADMM and AMA by at least one order of magnitude. Computer simulations and real data analysis show that Sproga outperforms several well known clustering algorithms including K-means and hierarchical clustering. The efficiency and superior performance of our algorithm will help convex clustering to find its wide application.

The information contained in hierarchical topology, intrinsic to many networks, is currently underutilised. A novel architecture is explored which exploits this information through a multiscale decomposition. A dendrogram is produced by a Girvan-Newman hierarchical clustering algorithm. It is segmented and fed through graph convolutional layers, allowing the architecture to learn multiple scale latent space representations of the network, from fine to coarse grained. The architecture is tested on a benchmark citation network, demonstrating competitive performance. Given the abundance of hierarchical networks, possible applications include quantum molecular property prediction, protein interface prediction and multiscale computational substrates for partial differential equations.

In machine learning, observation features are measured in a metric space to obtain their distance function for optimization. Given similar features that are statistically sufficient as a population, a statistical distance between two probability distributions can be calculated for more precise learning. Provided the observed features are multi-valued, the statistical distance function is still efficient. However, due to its scalar output, it cannot be applied to represent detailed distances between feature elements. To resolve this problem, this paper extends the traditional statistical distance to a matrix form, called a statistical distance matrix. In experiments, the proposed approach performs well in object recognition tasks and clearly and intuitively represents the dissimilarities between cat and dog images in the CIFAR dataset, even when directly calculated using the image pixels. By using the hierarchical clustering of the statistical distance matrix, the image pixels can be separated into several clusters that are geometrically arranged around a center like a Mandala pattern. The statistical distance matrix with clustering, called the Information Mandala, is beyond ordinary saliency maps and can help to understand the basic principles of the convolution neural network.