Abstract Spectral clustering algorithms typically require a priori selection of input parameters such as the number of clusters, a scaling parameter for the affinity measure, or ranges of these values for parameter tuning. Despite efforts for automating the process of spectral clustering, the task of grouping data in multi-scale and higher dimensional spaces is yet to be explored. This study presents a spectral clustering heuristic algorithm that obviates the need for any input by estimating the parameters from the data itself. Specifically, it introduces the heuristic of iterative eigengap search with (1) global scaling and (2) local scaling. These approaches estimate the scaling parameter and implement iterative eigengap quantification along a search tree to reveal dissimilarities at different scales of a feature space and identify clusters. The performance of these approaches has been tested on various real-world datasets of power variation with multi-scale nature and gene expression. Our findings show that iterative eigengap search with a PCA-based global scaling scheme can discover different patterns with an accuracy of higher than 90 in most cases without asking for a priori input information.
Spectral clustering has gained popularity due to their ease of implementation and efficiency in clustering @cite_42 @cite_33 . Therefore, in recent decades, several clustering algorithms have been proposed and used for different applications. The focus in these algorithms has been on the application of the similarity matrix spectrum for dimensionality reduction and feature space transformation to introduce convexity. One of the well-known algorithms in this field is the one proposed by Ng, Jordan, and Weiss (referred to as NJW) @cite_2 . In addition to the efforts in the formalization of spectral clustering algorithms, a number of studies have focused on expanding the algorithms into instances, which are capable of self-tuning or automated identification of natural partitions (or groups) in the data. Natural in this context refers to the clusters (or groups) that represent the actual physical separation in the data.
@cite_42 @cite_33 @cite_2
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Based on spectral graph theory, spectral clustering is an optimal graph partition problem. It has been proven that the spectral clustering is equivalent to nonnegative matrix factorization (NMF) under certain conditions. Based on the equivalence, some spectral clustering methods are proposed, but the global discriminative information of the dataset is neglected. In this paper, based on the equivalence between spectral clustering and NMF, we simultaneously maximize the between-class scatter matrix and minimize the within-class scatter matrix to enhance the discriminating power. We integrate the geometrical structure and discriminative structure in a joint framework. With a global discriminative regularization term added into the nonnegative matrix factorization framework, we propose two novel spectral clustering methods, named global discriminative-based nonnegative and spectral clustering (GDBNSC-Ncut and GDBNSC-Rcut) These new spectral clustering algorithms can preserve both the global geometrical structure and global discriminative structure. The intrinsic geometrical information of the dataset is detected, and clustering quality is improved with enhanced discriminating power. In addition, the proposed algorithms also have very good abilities of handling out-of-sample data. Experimental results on real word data demonstrate that the proposed algorithms outperform some state-of-the-art methods with good clustering qualities. We maximize the between-class scatter matrix.Meanwhile, we minimize the within-class scatter matrix.We integrate the geometrical structure and discriminative structure in a framework.A global discriminative regularization term added into the objective functions.The new algorithms can preserve the global geometrical and discriminative structure. The technique of spectral clustering is widely used to segment a range of data from graphs to images. Our work marks a natural progression of spectral clustering from the original passive unsupervised formulation to our active semi-supervised formulation. We follow the widely used area of constrained clustering and allow supervision in the form of pair wise relations between two nodes: Must-Link and Cannot-Link. Unlike most previous constrained clustering work, our constraints are specified incrementally by querying an oracle (domain expert). Since in practice, each query comes with a cost, our goal is to maximally improve the result with as few queries as possible. The advantages of our approach include: 1) it is principled by querying the constraints which maximally reduce the expected error, 2) it can incorporate both hard and soft constraints which are prevalent in practice. We empirically show that our method significantly outperforms the baseline approach, namely constrained spectral clustering with randomly selected constraints, on UCI benchmark data sets. Spectral clustering aims to partition a data set into several groups by using the Laplacian of the graph such that data points in the same group are similar while data points in different groups are dissimilar to each other. Spectral clustering is very simple to implement and has many advantages over the traditional clustering algorithms such as k-means. Non-negative matrix factorization (NMF) factorizes a non-negative data matrix into a product of two non-negative (lower rank) matrices so as to achieve dimension reduction and part-based data representation. In this work, we proved that the spectral clustering under some conditions is equivalent to NMF. Unlike the previous work, we formulate the spectral clustering as a factorization of data matrix (or scaled data matrix) rather than the symmetrical factorization of the symmetrical pairwise similarity matrix as the previous study did. Under the NMF framework, where regularization can be easily incorporated into the spectral clustering, we propose several non-negative and sparse spectral clustering algorithms. Empirical studies on real world data show much better clustering accuracy of the proposed algorithms than some state-of-the-art methods such as ratio cut and normalized cut spectral clustering and non-negative Laplacian embedding. HighlightsWe proved that the spectral clustering is equivalent to NMF.Under the NMF framework, we propose nonnegative sparse spectral clustering model.We propose several algorithms to solve the proposed models.Experiment results on real world data show much better performance. Spectral clustering is a popular and successful approach for partitioning the nodes of a graph into clusters for which the ratio of outside connections compared to the volume (sum of degrees) is small. In order to partition into k clusters, one first computes an approximation of the bottom k eigenvectors of the (normalized) Laplacian of G, uses it to embed the vertices of G into k-dimensional Euclidean space R^k, and then partitions the resulting points via a k-means clustering algorithm. It is an important task for theory to explain the success of spectral clustering. (COLT, 2015) made an important step in this direction. They showed that spectral clustering provably works if the gap between the (k+1)-th and the k-th eigenvalue of the normalized Laplacian is sufficiently large. They proved a structural and an algorithmic result. The algorithmic result needs a considerably stronger gap assumption and does not analyze the standard spectral clustering paradigm; it replaces spectral embedding by heat kernel embedding and k-means clustering by locality sensitive hashing. We extend their work in two directions. Structurally, we improve the quality guarantee for spectral clustering by a factor of k and simultaneously weaken the gap assumption. Algorithmically, we show that the standard paradigm for spectral clustering works. Moreover, it even works with the same gap assumption as required for the structural result.