Multiple Kernel Clustering Algorithms Research Articles

Local Sample-Weighted Multiple Kernel Clustering With Consensus Discriminative Graph.

Multiple kernel clustering (MKC) is committed to achieving optimal information fusion from a set of base kernels. Constructing precise and local kernel matrices is proven to be of vital significance in applications since the unreliable distant-distance similarity estimation would degrade clustering performance. Although existing localized MKC algorithms exhibit improved performance compared with globally designed competitors, most of them widely adopt the KNN mechanism to localize kernel matrix by accounting for τ -nearest neighbors. However, such a coarse manner follows an unreasonable strategy that the ranking importance of different neighbors is equal, which is impractical in applications. To alleviate such problems, this article proposes a novel local sample-weighted MKC (LSWMKC) model. We first construct a consensus discriminative affinity graph in kernel space, revealing the latent local structures. Furthermore, an optimal neighborhood kernel for the learned affinity graph is output with naturally sparse property and clear block diagonal structure. Moreover, LSWMKC implicitly optimizes adaptive weights on different neighbors with corresponding samples. Experimental results demonstrate that our LSWMKC possesses better local manifold representation and outperforms existing kernel or graph-based clustering algorithms. The source code of LSWMKC can be publicly accessed from https://github.com/liliangnudt/LSWMKC.

On the Consistency and Large-Scale Extension of Multiple Kernel Clustering.

Existing multiple kernel clustering (MKC) algorithms have two ubiquitous problems. From the theoretical perspective, most MKC algorithms lack sufficient theoretical analysis, especially the consistency of learned parameters, such as the kernel weights. From the practical perspective, the high complexity makes MKC unable to handle large-scale datasets. This paper tries to address the above two issues. We first make a consistency analysis of an influential MKC method named Simple Multiple Kernel k-Means (SimpleMKKM). Specifically, suppose that ∧γn are the kernel weights learned by SimpleMKKM from the training samples. We also define the expected version of SimpleMKKM and denote its solution as γ*. We establish an upper bound of ||∧γn-γ*||∞ in the order of ~O(1/√n), where n is the sample number. Based on this result, we also derive its excess clustering risk calculated by a standard clustering loss function. For the large-scale extension, we replace the eigen decomposition of SimpleMKKM with singular value decomposition (SVD). Consequently, the complexity can be decreased to O(n) such that SimpleMKKM can be implemented on large-scale datasets. We then deduce several theoretical results to verify the approximation ability of the proposed SVD-based method. The results of comprehensive experiments demonstrate the superiority of the proposed method. The code is publicly available at https://github.com/weixuan-liang/SVD-based-SimpleMKKM.

Regularized Simple Multiple Kernel k -Means With Kernel Average Alignment.

Multiple kernel clustering (MKC) aims to learn an optimal kernel to better serve for clustering from several precomputed basic kernels. Most MKC algorithms adhere to a common assumption that an optimal kernel is linearly combined by basic kernels. Based on a min-max framework, a newly proposed MKC method termed simple multiple kernel k -means (SimpleMKKM) can acquire a high-quality unified kernel. Although SimpleMKKM has achieved promising clustering performance, we observe that it cannot benefit from any prior knowledge. This would cause the learned partition matrix may seriously deviate from the expected one, especially in clustering tasks where the ground truth is absent during the learning course. To tackle this issue, we propose a novel algorithm termed regularized simple multiple kernel k -means with kernel average alignment (R-SMKKM-KAA). According to the experimental results of existing MKC algorithms, the average partition is a strong baseline to reflect true clustering. To gain knowledge from the average partition, we add the average alignment as a regularization term to prevent the learned unified partition from being far from the average partition. After that, we have designed an efficient solving algorithm to optimize the new resulting problem. In this way, both the incorporated prior knowledge and the combination of basic kernels are helpful to learn better unified partition. Consequently, the clustering performance can be significantly improved. Extensive experiments on nine common datasets have sufficiently demonstrated the effectiveness of incorporation of prior knowledge into SimpleMKKM.

Late Fusion Multiple Kernel Clustering With Proxy Graph Refinement.

Multiple kernel clustering (MKC) optimally utilizes a group of pre-specified base kernels to improve clustering performance. Among existing MKC algorithms, the recently proposed late fusion MKC methods demonstrate promising clustering performance in various applications and enjoy considerable computational acceleration. However, we observe that the kernel partition learning and late fusion processes are separated from each other in the existing mechanism, which may lead to suboptimal solutions and adversely affect the clustering performance. In this article, we propose a novel late fusion multiple kernel clustering with proxy graph refinement (LFMKC-PGR) framework to address these issues. First, we theoretically revisit the connection between late fusion kernel base partition and traditional spectral embedding. Based on this observation, we construct a proxy self-expressive graph from kernel base partitions. The proxy graph in return refines the individual kernel partitions and also captures partition relations in graph structure rather than simple linear transformation. We also provide theoretical connections and considerations between the proposed framework and the multiple kernel subspace clustering. An alternate algorithm with proved convergence is then developed to solve the resultant optimization problem. After that, extensive experiments are conducted on 12 multi-kernel benchmark datasets, and the results demonstrate the effectiveness of our proposed algorithm. The code of the proposed algorithm is publicly available at https://github.com/wangsiwei2010/graphlatefusion_MKC.

Simple multiple kernel k-means with kernel weight regularization

Multiple kernel clustering (MKC) aims to determine the optimal kernel from several pre-computed basic kernels. Most of MKC algorithms follow a common assumption that the optimal kernel is linearly combined by basic kernels. Based on a min–max framework, a newly proposed MKC method termed simple multiple kernel k-means can acquire a high-quality unified kernel. Although it has achieved promising clustering performance, we have observed that it cannot benefit from any regularization or prior knowledge, thus the learned kernel weight coefficients may be seriously sparse or over-selected. To tackle this issue, we have developed a new algorithm termed simple multiple kernel k-means with kernel weight regularization (SMKKM-KWR), where we introduce average coefficients to avoid too sparse kernel weights. Specially, we add the average kernel coefficients as a regularization term to prevent the learned weight coefficients being far from the average values. After that, an efficient optimization strategy is proposed to solve the new resultant problem. By this way, the unified partition can learn clustering structure from fusion information of all the kernel views, with the goal to reach better clustering performance. Extensive experiments on nine benchmark datasets and four large-scale datasets have demonstrated the effectiveness and efficiency of the proposed algorithm.

Spare simple MKKM with semi‐infinite linear program optimization

Multiple kernel clustering (MKC) optimally combines a group of predefined kernel matrices to improve clustering performance. Although demonstrating promising performance in various applications, most of existing approaches adopt the min–min formulation, which could be sensitive to perturbation with adversarial samples. Moreover, existing MKC algorithms often involve several hypermeters preventing them into further real applications. To address these issues, we propose a parameter-free effective sparse simple multiple kernel k-means algorithm with max–min optimization formulation in this paper. To be specific, we propose to optimize the widely used unsupervised kernel alignment criterion by minimizing the kernel coefficient and maximizing the clustering partition matrix. Unlike traditional min–min formulation, the max–min kernel alignment is robust to adversarial sample perturbation and free of hyper-parameters. An optimization method based on semi-infinite linear program is designed to solve the complicated optimization problem. Extensive experiments on six multiple kernel benchmark data sets demonstrate the effectiveness of the proposed method.

Optimal Neighborhood Multiple Kernel Clustering with Adaptive Local Kernels

Multiple kernel clustering (MKC) algorithm aims to group data into different categories by optimally integrating information from a group of pre-specified kernels. Though demonstrating superiorities in various applications, we observe that existing MKC algorithms usually do not sufficiently consider the local density around individual data samples and excessively limit the representation capacity of the learned optimal kernel, leading to unsatisfying performance. In this paper, we propose an algorithm, called optimal neighborhood MKC with adaptive local kernels (ON-ALK), to address the two issues. In specific, we construct adaptive local kernels to sufficiently consider the local density around individual data samples, where different numbers of neighbors are discriminatingly selected on each sample. Further, the proposed ON-ALK algorithm boosts the representation of the learned optimal kernel via relaxing it into the neighborhood area of weighted combination of the pre-specified kernels. To solve the resultant optimization problem, a three-step iterative algorithm is designed and theoretically proven to be convergent. After that, we also study the generalization bound of the proposed algorithm. Extensive experiments have been conducted to evaluate the clustering performance. As indicated, the algorithm significantly outperforms state-of-the-art methods in recent literatures on six challenging benchmark datasets, verifying its advantages and effectiveness.

Multiple Kernel Clustering With Neighbor-Kernel Subspace Segmentation.

Multiple kernel clustering (MKC) has been intensively studied during the last few decades. Even though they demonstrate promising clustering performance in various applications, existing MKC algorithms do not sufficiently consider the intrinsic neighborhood structure among base kernels, which could adversely affect the clustering performance. In this paper, we propose a simple yet effective neighbor-kernel-based MKC algorithm to address this issue. Specifically, we first define a neighbor kernel, which can be utilized to preserve the block diagonal structure and strengthen the robustness against noise and outliers among base kernels. After that, we linearly combine these base neighbor kernels to extract a consensus affinity matrix through an exact-rank-constrained subspace segmentation. The naturally possessed block diagonal structure of neighbor kernels better serves the subsequent subspace segmentation, and in turn, the extracted shared structure is further refined through subspace segmentation based on the combined neighbor kernels. In this manner, the above two learning processes can be seamlessly coupled and negotiate with each other to achieve better clustering. Furthermore, we carefully design an efficient iterative optimization algorithm with proven convergence to address the resultant optimization problem. As a by-product, we reveal an interesting insight into the exact-rank constraint in ridge regression by careful theoretical analysis: it back-projects the solution of the unconstrained counterpart to its principal components. Comprehensive experiments have been conducted on several benchmark data sets, and the results demonstrate the effectiveness of the proposed algorithm.

Multiple Kernel Clustering With Global and Local Structure Alignment

Multiple kernel clustering (MKC) based on global structure alignment (GSA) has unified many existing MKC algorithms, and shown outstanding clustering performance. However, we observe that most of existing GSA-based MKC algorithms only maximally align global structure of data with an ideal similarity matrix, while ignoring the local geometrical structure hidden in data, which is regarded to be important in improving the clustering performance. To address this issue, we propose a global and local structure alignment framework for MKC (GLSAMKC) which well considers both the alignment between the global structure and local structure of data with the same ideal similarity matrix. To illustrate the effectiveness of the proposed framework, we instantiate two specific GLSAMKC-based algorithms by exploiting the local structure with local linear embedding and locality preserving projection, respectively. A two-step alternate iterative and convergent optimization algorithm is developed to implement the resultant optimization problem. Extensive experimental results on five benchmark data sets demonstrate the superiority of proposed algorithms compared with the many state-of-the-art MKC algorithms, indicating the effectiveness of the proposed framework.

Multiple kernel clustering with corrupted kernels

Multiple kernel clustering (MKC) algorithms usually learn an optimal kernel from a group of pre-specified base kernels to improve the clustering performance. However, we observe that existing MKC algorithms do not well handle the situation that kernels are corrupted with noise and outliers. In this paper, we first propose a novel method to learn an optimal consensus kernel from a group of pre-specified kernel matrices, each of which can be decomposed into the optimal consensus kernel matrix and a sparse error matrix. Further, we propose a scheme to address the problem of considerable corrupted kernels, where each given kernel is adaptively adjusted according to its corresponding error matrix. The inexact augmented Lagrange multiplier scheme is developed for solving the corresponding optimization problem, where the optimal consensus kernel and the localized weight variables are jointly optimized. Extensive experiments well demonstrate the effectiveness and robustness of the proposed algorithm.

Multiple Kernel k-Means with Incomplete Kernels.

Multiple kernel clustering (MKC) algorithms optimally combine a group of pre-specified base kernel matrices to improve clustering performance. However, existing MKC algorithms cannot efficiently address the situation where some rows and columns of base kernel matrices are absent. This paper proposes two simple yet effective algorithms to address this issue. Different from existing approaches where incomplete kernel matrices are first imputed and a standard MKC algorithm is applied to the imputed kernel matrices, our first algorithm integrates imputation and clustering into a unified learning procedure. Specifically, we perform multiple kernel clustering directly with the presence of incomplete kernel matrices, which are treated as auxiliary variables to be jointly optimized. Our algorithm does not require that there be at least one complete base kernel matrix over all the samples. Also, it adaptively imputes incomplete kernel matrices and combines them to best serve clustering. Moreover, we further improve this algorithm by encouraging these incomplete kernel matrices to mutually complete each other. The three-step iterative algorithm is designed to solve the resultant optimization problems. After that, we theoretically study the generalization bound of the proposed algorithms. Extensive experiments are conducted on 13 benchmark data sets to compare the proposed algorithms with existing imputation-based methods. Our algorithms consistently achieve superior performance and the improvement becomes more significant with increasing missing ratio, verifying the effectiveness and advantages of the proposed joint imputation and clustering.

Multiple Kernel k-Means with Incomplete Kernels

Multiple kernel clustering (MKC) algorithms optimally combine a group of pre-specified base kernels to improve clustering performance. However, existing MKC algorithms cannot efficiently address the situation where some rows and columns of base kernels are absent. This paper proposes a simple while effective algorithm to address this issue. Different from existing approaches where incomplete kernels are firstly imputed and a standard MKC algorithm is applied to the imputed kernels, our algorithm integrates imputation and clustering into a unified learning procedure. Specifically, we perform multiple kernel clustering directly with the presence of incomplete kernels, which are treated as auxiliary variables to be jointly optimized. Our algorithm does not require that there be at least one complete base kernel over all the samples. Also, it adaptively imputes incomplete kernels and combines them to best serve clustering. A three-step iterative algorithm with proved convergence is designed to solve the resultant optimization problem. Extensive experiments are conducted on four benchmark data sets to compare the proposed algorithm with existing imputation-based methods. Our algorithm consistently achieves superior performance and the improvement becomes more significant with increasing missing ratio, verifying the effectiveness and advantages of the proposed joint imputation and clustering.

Multiple kernel clustering based on centered kernel alignment

Multiple kernel clustering (MKC), which performs kernel-based data fusion for data clustering, is an emerging topic. It aims at solving clustering problems with multiple cues. Most MKC methods usually extend existing clustering methods with a multiple kernel learning (MKL) setting. In this paper, we propose a novel MKC method that is different from those popular approaches. Centered kernel alignment—an effective kernel evaluation measure—is employed in order to unify the two tasks of clustering and MKL into a single optimization framework. To solve the formulated optimization problem, an efficient two-step iterative algorithm is developed. Experiments on several UCI datasets and face image datasets validate the effectiveness and efficiency of our MKC algorithm.