Solving real world problems with data science concepts is so exciting and it yields so fun. A famous dialogue you could listen from the data science people. It could be true if we add it's so challenging at the end of the dialogue. The foremost challenge starts from categorising the problem itself. The first level of categorising could be whether supervised or unsupervised learning.

Multilayer graphs are commonly used for representing different relations between entities and handling heterogeneous data processing tasks. New challenges arise in multilayer graph clustering for assigning clusters to a common multilayer node set and for combining information from each layer. This paper presents a theoretical framework for multilayer spectral graph clustering of the nodes via convex layer aggregation. Under a novel multilayer signal plus noise model, we provide a phase transition analysis that establishes the existence of a critical value on the noise level that permits reliable cluster separation. The analysis also specifies analytical upper and lower bounds on the critical value, where the bounds become exact when the clusters have identical sizes. Numerical experiments on synthetic multilayer graphs are conducted to validate the phase transition analysis and study the effect of layer weights and noise levels on clustering reliability.

Hybrid clustering combines partitional and hierarchical clustering for computational effectiveness and versatility in cluster shape. In such clustering, a dissimilarity measure plays a crucial role in the hierarchical merging. The dissimilarity measure has great impact on the final clustering, and data-independent properties are needed to choose the right dissimilarity measure for the problem at hand. Properties for distance-based dissimilarity measures have been studied for decades, but properties for density-based dissimilarity measures have so far received little attention. Here, we propose six data-independent properties to evaluate density-based dissimilarity measures associated with hybrid clustering, regarding equality, orthogonality, symmetry, outlier and noise observations, and light-tailed models for heavy-tailed clusters. The significance of the properties is investigated, and we study some well-known dissimilarity measures based on Shannon entropy, misclassification rate, Bhattacharyya distance and Kullback-Leibler divergence with respect to the proposed properties. As none of them satisfy all the proposed properties, we introduce a new dissimilarity measure based on the Kullback-Leibler information and show that it satisfies all proposed properties. The effect of the proposed properties is also illustrated on several real and simulated data sets.

The goal of graph clustering is to group the nodes into clusters of high similarity. Applications of graph clustering, also known as community detection [1, 2], include but are not limited to graph signal processing [3-11], multivariate data clustering [12-14], image segmentation [15, 16], and network vulnerability assessment [17]. Spectral clustering [12-14] is a popular method for graph clustering, which we refer to as spectral graph clustering (SGC). It works by transforming the graph adjacency matrix into a graph Laplacian matrix [18], computing its eigendecomposition, and performing K-means clustering [19] on the eigenvectors to partition the nodes into clusters. Although heuristic methods have been proposed to automatically select the number of clusters [12,13,20], rigorous theoretical justifications on the selection of the number of eigenvectors for clustering are still lacking and little is known about the capabilities and limitations of spectral clustering on graphs. Based on a recent development of clustering reliability analysis for SGC under the random interconnection model (RIM) [21], we propose a novel automated model order selection (AMOS) algorithm for SGC. AMOS works by incrementally increasing the number of clusters, estimating the quality of identified clusters, and providing a series of clustering reliability tests.

Cluster analysis methods are used to identify homogeneous subgroups in a data set. In biomedical applications, one frequently applies cluster analysis in order to identify biologically interesting subgroups. In particular, one may wish to identify subgroups that are associated with a particular outcome of interest. Conventional clustering methods generally do not identify such subgroups, particularly when there are a large number of high-variance features in the data set. Conventional methods may identify clusters associated with these high-variance features when one wishes to obtain secondary clusters that are more interesting biologically or more strongly associated with a particular outcome of interest. A modification of sparse clustering can be used to identify such secondary clusters or clusters associated with an outcome of interest. This method correctly identifies such clusters of interest in several simulation scenarios. The method is also applied to a large prospective cohort study of temporomandibular disorders and a leukemia microarray data set.

The prescribed viewpoint appears to be good enough to accommodate most of the existing spectral graph techniques as a consequence of just one single formalism and algorithm. Keywords and phrases: Nonparametric spectral graph analysis; Graph correlation density field (GraField); Empirical and smoothed spectral graph analysis; High-dimensional discrete data smoothing.

We analyze a compression scheme for large data sets that randomly keeps a small percentage of the components of each data sample. The benefit is that the output is a sparse matrix and therefore subsequent processing, such as PCA or K-means, is significantly faster, especially in a distributed-data setting. Furthermore, the sampling is single-pass and applicable to streaming data. The sampling mechanism is a variant of previous methods proposed in the literature combined with a randomized preconditioning to smooth the data. We provide guarantees for PCA in terms of the covariance matrix, and guarantees for K-means in terms of the error in the center estimators at a given step. We present numerical evidence to show both that our bounds are nearly tight and that our algorithms provide a real benefit when applied to standard test data sets, as well as providing certain benefits over related sampling approaches.

I recently came across this question on Cross Validated, and I thought it offered a great opportunity to use R and ggplot2 to explore, in depth, the assumptions underlying the k-means algorithm. The question, and my response, follow. K-means is a widely used method in cluster analysis. In my understanding, this method does NOT require ANY assumptions, i.e., give me a data set and a pre-specified number of clusters, k, then I just apply this algorithm which minimize the SSE, the within cluster square error. So k-means, it is essentially an optimization problem.

Symmetric Nonnegative Matrix Factorization (SNMF) models arise naturally as simple reformulations of many standard clustering algorithms including the popular spectral clustering method. Recent work has demonstrated that an elementary instance of SNMF provides superior clustering quality compared to many classic clustering algorithms on a variety of synthetic and real world data sets. In this work, we present novel reformulations of this instance of SNMF based on the notion of variable splitting and produce two fast and effective algorithms for its optimization using i) the provably convergent Accelerated Proximal Gradient (APG) procedure and ii) a heuristic version of the Alternating Direction Method of Multipliers (ADMM) framework. Our two algorithms present an interesting tradeoff between computational speed and mathematical convergence guarantee: while the former method is provably convergent it is considerably slower than the latter approach, for which we also provide significant but less stringent mathematical proof regarding its convergence. Through extensive experiments we show not only that the efficacy of these approaches is equal to that of the state of the art SNMF algorithm, but also that the latter of our algorithms is extremely fast being one to two orders of magnitude faster in terms of total computation time than the state of the art approach, outperforming even spectral clustering in terms of computation time on large data sets.

The two most extended density-based approaches to clustering are surely mixture model clustering and modal clustering. In the mixture model approach, the density is represented as a mixture and clusters are associated to the different mixture components. In modal clustering, clusters are understood as regions of high density separated from each other by zones of lower density, so that they are closely related to certain regions around the density modes. If the true density is indeed in the assumed class of mixture densities, then mixture model clustering allows to scrutinize more subtle situations than modal clustering. However, when mixture modeling is used in a nonparametric way, taking advantage of the denseness of the sieve of mixture densities to approximate any density, then the correspondence between clusters and mixture components may become questionable. In this paper we introduce two methods to adopt a modal clustering point of view after a mixture model fit. Numerous examples are provided to illustrate that mixture modeling can also be used for clustering in a nonparametric sense, as long as clusters are understood as the domains of attraction of the density modes.