Accelerated sparse Kernel Spectral Clustering for large scale data clustering problems

An improved version of the sparse multiway kernel spectral clustering (KSC) is presented in this brief. The original algorithm is derived from weighted kernel principal component (KPCA) analysis formulated within the primal-dual least-squares support vector machine (LS-SVM) framework. Sparsity is achieved then by the combination of the incomplete Cholesky decomposition (ICD) based low rank approximation of the kernel matrix with the so called reduced set method. The original ICD based sparse KSC algorithm was reported to be computationally far too demanding, especially when applied on large scale data clustering problems that actually it was designed for, which has prevented to gain more than simply theoretical relevance so far. This is altered by the modifications reported in this brief that drastically improve the computational characteristics. Solving the alternative, symmetrized version of the computationally most demanding core eigenvalue problem eliminates the necessity of forming and SVD of large matrices during the model construction. This results in solving clustering problems now within seconds that were reported to require hours without altering the results. Furthermore, sparsity is also improved significantly, leading to more compact model representation, increasing further not only the computational efficiency but also the descriptive power. These transform the original, only theoretically relevant ICD based sparse KSC algorithm applicable for large scale practical clustering problems. Theoretical results and improvements are demonstrated by computational experiments on carefully selected synthetic data as well as on real life problems such as image segmentation.

Kernel Spectral Clustering and applications

In this chapter we review the main literature related to kernel spectral clustering (KSC), an approach to clustering cast within a kernel-based optimization setting. KSC represents a least-squares support vector machine based formulation of spectral clustering described by a weighted kernel PCA objective. Just as in the classifier case, the binary clustering model is expressed by a hyperplane in a high dimensional space induced by a kernel. In addition, the multi-way clustering can be obtained by combining a set of binary decision functions via an Error Correcting Output Codes (ECOC) encoding scheme. Because of its model-based nature, the KSC method encompasses three main steps: training, validation, testing. In the validation stage model selection is performed to obtain tuning parameters, like the number of clusters present in the data. This is a major advantage compared to classical spectral clustering where the determination of the clustering parameters is unclear and relies on heuristics. Once a KSC model is trained on a small subset of the entire data, it is able to generalize well to unseen test points. Beyond the basic formulation, sparse KSC algorithms based on the Incomplete Cholesky Decomposition (ICD) and $L_0$, $L_1, L_0 + L_1$, Group Lasso regularization are reviewed. In that respect, we show how it is possible to handle large scale data. Also, two possible ways to perform hierarchical clustering and a soft clustering method are presented. Finally, real-world applications such as image segmentation, power load time-series clustering, document clustering and big data learning are considered.

Clustering evolving data using kernel-based methods

In this thesis, we propose several modelling strategies to tackle evolving data in different contexts. In the framework of static clustering, we start by introducing a soft kernel spectral clustering (SKSC) algorithm, which can better deal with overlapping clusters with respect to kernel spectral clustering (KSC) and provides more interpretable outcomes. Afterwards, a whole strategy based upon KSC for community detection of static networks is proposed, where the extraction of a high quality training sub-graph, the choice of the kernel function, the model selection and the applicability to large-scale data are key aspects. This paves the way for the development of a novel clustering algorithm for the analysis of evolving networks called kernel spectral clustering with memory effect (MKSC), where the temporal smoothness between clustering results in successive time steps is incorporated at the level of the primal optimization problem, by properly modifying the KSC formulation. Later on, an application of KSC to fault detection of an industrial machine is presented. Here, a smart pre-processing of the data by means of a proper windowing operation is necessary to catch the ongoing degradation process affecting the machine. In this way, in a genuinely unsupervised manner, it is possible to raise an early warning when necessary, in an online fashion. Finally, we propose a new algorithm called incremental kernel spectral clustering (IKSC) for online learning of non-stationary data. This ambitious challenge is faced by taking advantage of the out-of-sample property of kernel spectral clustering (KSC) to adapt the initial model, in order to tackle merging, splitting or drifting of clusters across time. Real-world applications considered in this thesis include image segmentation, time-series clustering, community detection of static and evolving networks.