Kernel-based K-means clustering has gained popularity due to its simplicity and the power of its implicit non-linear representation of the data. A dominant concern is the memory requirement since memory scales as the square of the number of data points. We provide a new analysis of a class of approximate kernel methods that have more modest memory requirements, and propose a specific one-pass randomized kernel approximation followed by standard K-means on the transformed data. The analysis and experiments suggest the method is accurate, while requiring drastically less memory than standard kernel K-means and significantly less memory than Nystrom based approximations.
Kernel methods are popular in clustering due to their generality and discriminating power. However, we show that many kernel clustering criteria have density biases theoretically explaining some practically significant artifacts empirically observed in the past. For example, we provide conditions and formally prove the density mode isolation bias in kernel K-means for a common class of kernels. We call it Breiman's bias due to its similarity to the histogram mode isolation previously discovered by Breiman in decision tree learning with Gini impurity. We also extend our analysis to other popula…
In this paper, we consider the nonparametric least square regression in a Reproducing Kernel Hilbert Space (RKHS). We propose a new randomized algorithm that has optimal generalization error bounds with respect to the square loss, closing a long-standing gap between upper and lower bounds. Moreover, we show that our algorithm has faster finite-time and asymptotic rates on problems where the Bayes risk with respect to the square loss is small. We state our results using standard tools from the theory of least square regression in RKHSs, namely, the decay of the eigenvalues of the associated int…
In many artificial intelligence and computer vision systems, the same object can be observed at distinct viewpoints or by diverse sensors, which raises the challenges for recognizing objects from different, even heterogeneous views. Multi-view discriminant analysis (MvDA) is an effective multi-view subspace learning method, which finds a discriminant common subspace by jointly learning multiple view-specific linear projections for object recognition from multiple views, in a non-pairwise way. In this paper, we propose the kernel version of multi-view discriminant analysis, called kernel multi-…
Multiple kernel methods based on k-means aims to integrate a group of kernels to improve the performance of kernel k-means clustering. However, we observe that most existing multiple kernel k-means methods exploit the nonlinear relationship within kernels, whereas the local manifold structure among multiple kernel space is not sufficiently considered. In this paper, we adopt the manifold adaptive kernel, instead of the original kernel, to integrate the local manifold structure of kernels. Thus, the induced multiple manifold adaptive kernels not only reflect the nonlinear relationship but also …
Dimensionality reduction is an important step in processing the hyperspectral images (HSI) to overcome the curse of dimensionality problem. Linear dimensionality reduction methods such as Independent component analysis (ICA) and Linear discriminant analysis (LDA) are commonly employed to reduce the dimensionality of HSI. These methods fail to capture non-linear dependency in the HSI data, as data lies in the nonlinear manifold. To handle this, nonlinear transformation techniques based on kernel methods were introduced for dimensionality reduction of HSI. However, the kernel methods involve cub…
National University of Defense Technology（国防科技大学）；Edinburgh University（爱丁堡大学）；Hefei University of Technology（合肥工业大学）
We propose a simple yet effective multiple kernel clustering algorithm, termed simple multiple kernel k-means (SimpleMKKM). It extends the widely used supervised kernel alignment criterion to multi-kernel clustering. Our criterion is given by an intractable minimization-maximization problem in the kernel coefficient and clustering partition matrix. To optimize it, we re-formulate the problem as a smooth minimization one, which can be solved efficiently using a reduced gradient descent algorithm. We theoretically analyze the performance of SimpleMKKM in terms of its clustering generalization er…
International Max Planck Research School for Intelligent Systems（马克斯·普朗克智能系统国际研究学院）；IMPRS-IS（马克斯·普朗克智能系统国际研究学院）；University of Tübingen（图宾根大学）；Technical University of Munich（慕尼黑工业大学）
This paper studies the optimality of kernel methods in high-dimensional data clustering. Recent works have studied the large sample performance of kernel clustering in the high-dimensional regime, where Euclidean distance becomes less informative. However, it is unknown whether popular methods, such as kernel k-means, are optimal in this regime. We consider the problem of high-dimensional Gaussian clustering and show that, with the exponential kernel function, the sufficient conditions for partial recovery of clusters using the NP-hard kernel k-means objective matches the known information-the…
A New Framework for Convex Clustering in Kernel Spaces: Finite Sample Bounds, Consistency and Performance Insights
Indian Statistical Institute, Kolkata（印度统计研究院，加尔各答）；Electronics and Communication Sciences Unit, Indian Statistical Institute（印度统计研究院电子与通信科学单位）；Department of Statistics, University of Oxford（牛津大学统计系）；Department of Statistics, University of Michigan（密歇根大学统计系）
University of Michigan（密歇根大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We present memory-efficient and scalable algorithms for kernel methods used in machine learning. Using hierarchical matrix approximations for the kernel matrix the memory requirements, the number of floating point operations, and the execution time are drastically reduced compared to standard dense linear algebra routines. We consider both the general $\mathcal{H}$ matrix hierarchical format as well as Hierarchically Semi-Separable (HSS) matrices. Furthermore, we investigate the impact of several preprocessing and clustering techniques on the hierarchical matrix compression. Effective clusteri…
University of California at Berkeley（加州大学伯克利分校）；International Computer Science Institute（国际计算机科学研究所）
One approach to improving the running time of kernel-based machine learning methods is to build a small sketch of the input and use it in lieu of the full kernel matrix in the machine learning task of interest. Here, we describe a version of this approach that comes with running time guarantees as well as improved guarantees on its statistical performance. By extending the notion of \emph{statistical leverage scores} to the setting of kernel ridge regression, our main statistical result is to identify an importance sampling distribution that reduces the size of the sketch (i.e., the required n…
Discriminative Similarity for Clustering and Semi-Supervised Learning
Snap Research（Snap 研究院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Microsoft Research（微软研究院）；National University of Singapore（新加坡国立大学）；Beckman Institute, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校贝克曼研究所）
University of Electronic Science and Technology of China（电子科技大学）；JD AI Research（京东人工智能研究院）
Multiple kernel learning (MKL) method is generally believed to perform better than single kernel method. However, some empirical studies show that this is not always true: the combination of multiple kernels may even yield an even worse performance than using a single kernel. There are two possible reasons for the failure: (i) most existing MKL methods assume that the optimal kernel is a linear combination of base kernels, which may not hold true; and (ii) some kernel weights are inappropriately assigned due to noises and carelessly designed algorithms. In this paper, we propose a novel MKL fr…
University of California Santa Barbara（加州大学圣巴巴拉分校）
A new method for hierarchical clustering is presented. It combines treelets, a particular multiscale decomposition of data, with a projection on a reproducing kernel Hilbert space. The proposed approach, called kernel treelets (KT), effectively substitutes the correlation coefficient matrix used in treelets with a symmetric, positive semi-definite matrix efficiently constructed from a kernel function. Unlike most clustering methods, which require data sets to be numeric, KT can be applied to more general data and yield a multi-resolution sequence of basis on the data directly in feature space.…
Shanghai Jiao Tong University（上海交通大学）；Zhejiang University（浙江大学）；University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）；Cornell University（康奈尔大学）
Divide-and-conquer is a general strategy to deal with large scale problems. It is typically applied to generate ensemble instances, which potentially limits the problem size it can handle. Additionally, the data are often divided by random sampling which may be suboptimal. To address these concerns, we propose the $DC^2$ algorithm. Instead of ensemble instances, we produce structure-preserving signature pieces to be assembled and conquered. $DC^2$ achieves the efficiency of sampling-based large scale kernel methods while enabling parallel multicore or clustered computation. The data partition …
Center for Modern Statistics, School of Mathematics, Northwest University（西北大学数学学院现代统计中心）
Spectral clustering has been one of the widely used methods for community detection in networks. However, large-scale networks bring computational challenges to the eigenvalue decomposition therein. In this paper, we study the spectral clustering using randomized sketching algorithms from a statistical perspective, where we typically assume the network data are generated from a stochastic block model that is not necessarily of full rank. To do this, we first use the recently developed sketching algorithms to obtain two randomized spectral clustering algorithms, namely, the random projection-ba…
Amirkabir University of Technology（阿米卡比尔理工大学）；Université de Mons（蒙斯大学）；Faculté Polytechnique（理工学院）
Sparse subspace clustering (SSC) is one of the current state-of-the-art methods for partitioning data points into the union of subspaces, with strong theoretical guarantees. However, it is not practical for large data sets as it requires solving a LASSO problem for each data point, where the number of variables in each LASSO problem is the number of data points. To improve the scalability of SSC, we propose to select a few sets of anchor points using a randomized hierarchical clustering method, and, for each set of anchor points, solve the LASSO problems for each data point allowing only ancho…
This paper considers the partially functional linear model (PFLM) where all predictive features consist of a functional covariate and a high dimensional scalar vector. Over an infinite dimensional reproducing kernel Hilbert space, the proposed estimation for PFLM is a least square approach with two mixed regularizations of a function-norm and an $\ell_1$-norm. Our main task in this paper is to establish the minimax rates for PFLM under high dimensional setting, and the optimal minimax rates of estimation is established by using various techniques in empirical process theory for analyzing kerne…
Directed networks are broadly used to represent asymmetric relationships among units. Co-clustering aims to cluster the senders and receivers of directed networks simultaneously. In particular, the well-known spectral clustering algorithm could be modified as the spectral co-clustering to co-cluster directed networks. However, large-scale networks pose great computational challenges to it. In this paper, we leverage sketching techniques and derive two randomized spectral co-clustering algorithms, one \emph{random-projection-based} and the other \emph{random-sampling-based}, to accelerate the c…
Kernel-based clustering algorithm can identify and capture the non-linear structure in datasets, and thereby it can achieve better performance than linear clustering. However, computing and storing the entire kernel matrix occupy so large memory that it is difficult for kernel-based clustering to deal with large-scale datasets. In this paper, we employ incomplete Cholesky factorization to accelerate kernel clustering and save memory space. The key idea of the proposed kernel $k$-means clustering using incomplete Cholesky factorization is that we approximate the entire kernel matrix by the prod…
University of Science and Technology of China（中国科学技术大学）
To cluster data that are not linearly separable in the original feature space, $k$-means clustering was extended to the kernel version. However, the performance of kernel $k$-means clustering largely depends on the choice of kernel function. To mitigate this problem, multiple kernel learning has been introduced into the $k$-means clustering to obtain an optimal kernel combination for clustering. Despite the success of multiple kernel $k$-means clustering in various scenarios, few of the existing work update the combination coefficients based on the diversity of kernels, which leads to the resu…
This paper presents a sequential randomized lowrank matrix factorization approach for incrementally predicting values of an unknown function at test points using the Gaussian Processes framework. It is well-known that in the Gaussian processes framework, the computational bottlenecks are the inversion of the (regularized) kernel matrix and the computation of the hyper-parameters defining the kernel. The main contributions of this paper are two-fold. First, we formalize an approach to compute the inverse of the kernel matrix using randomized matrix factorization algorithms in a streaming scenar…
University of Tübingen（图宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Technical University of Munich（慕尼黑工业大学）
Despite the ubiquity of kernel-based clustering, surprisingly few statistical guarantees exist beyond settings that consider strong structural assumptions on the data generation process. In this work, we take a step towards bridging this gap by studying the statistical performance of kernel-based clustering algorithms under non-parametric mixture models. We provide necessary and sufficient separability conditions under which these algorithms can consistently recover the underlying true clustering. Our analysis provides guarantees for kernel clustering approaches without structural assumptions …
Bilkent University（比尔肯特大学）；DataBoss A. S.（DataBoss股份公司）
We investigate spatio-temporal event analysis using point processes. Inferring the dynamics of event sequences spatiotemporally has many practical applications including crime prediction, social media analysis, and traffic forecasting. In particular, we focus on spatio-temporal Hawkes processes that are commonly used due to their capability to capture excitations between event occurrences. We introduce a novel inference framework based on randomized transformations and gradient descent to learn the process. We replace the spatial kernel calculations by randomized Fourier feature-based transfor…
This article explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is first applied to a sub-matrix of the graph's adjacency matrix associated with a reduced graph sketch constructed using random sampling. Then, the clusters of the full graph are inferred based on the clusters extracted from the sketch using a correlation-based retrieval step. Uniform random node sampling is shown to improve the computational complexity over clu…
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；Inception Institute of Artificial Intelligence（启智人工智能研究院）
In this paper, we study the statistical properties of kernel $k$-means and obtain a nearly optimal excess clustering risk bound, substantially improving the state-of-art bounds in the existing clustering risk analyses. We further analyze the statistical effect of computational approximations of the Nyström kernel $k$-means, and prove that it achieves the same statistical accuracy as the exact kernel $k$-means considering only $Ω(\sqrt{nk})$ Nyström landmark points. To the best of our knowledge, such sharp excess clustering risk bounds for kernel (or approximate kernel) $k$-means have never bee…
Robust, randomized preconditioning for kernel ridge regression
Johns Hopkins University（约翰斯·霍普金斯大学）；University of California Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）；California Institute of Technology（加州理工学院）；University of California San Diego（加州大学圣地亚哥分校）
Stevens Institute of Technology（斯蒂文斯理工学院）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；International Computer Science Institute（国际计算机科学研究所）；University of California at Berkeley（加州大学伯克利分校）
Kernel $k$-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear $k$-means clustering algorithm. However, kernel $k$-means clustering is computationally expensive when the non-linear feature map is high-dimensional and there are many input points. Kernel approximation, e.g., the Nyström method, has been applied in previous works to approximately solve kernel learning problems when both of the above conditions are present. This work analyzes the application of this paradigm to kernel $k$-means clustering, and shows that applying t…
National Key Laboratory for Novel Software Technology, Nanjing University（南京大学计算机软件新技术全国重点实验室）；School of Information Technology, Deakin University（迪肯大学信息技术学院）
Measuring similarity between two objects is the core operation in existing clustering algorithms in grouping similar objects into clusters. This paper introduces a new similarity measure called point-set kernel which computes the similarity between an object and a set of objects. The proposed clustering procedure utilizes this new measure to characterize every cluster grown from a seed object. We show that the new clustering procedure is both effective and efficient that enables it to deal with large scale datasets. In contrast, existing clustering algorithms are either efficient or effective.…
We derive and analyze a generic, recursive algorithm for estimating all splits in a finite cluster tree as well as the corresponding clusters. We further investigate statistical properties of this generic clustering algorithm when it receives level set estimates from a kernel density estimator. In particular, we derive finite sample guarantees, consistency, rates of convergence, and an adaptive data-driven strategy for choosing the kernel bandwidth. For these results we do not need continuity assumptions on the density such as Hölder continuity, but only require intuitive geometric assumptions…
Aalto University（阿尔托大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
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 t…
We propose a deep learning approach for discovering kernels tailored to identifying clusters over sample data. Our neural network produces sample embeddings that are motivated by--and are at least as expressive as--spectral clustering. Our training objective, based on the Hilbert Schmidt Information Criterion, can be optimized via gradient adaptations on the Stiefel manifold, leading to significant acceleration over spectral methods relying on eigendecompositions. Finally, our trained embedding can be directly applied to out-of-sample data. We show experimentally that our approach outperforms …
In machine learning, no data point stands alone. We believe that context is an underappreciated concept in many machine learning methods. We propose Attention-Based Clustering (ABC), a neural architecture based on the attention mechanism, which is designed to learn latent representations that adapt to context within an input set, and which is inherently agnostic to input sizes and number of clusters. By learning a similarity kernel, our method directly combines with any out-of-the-box kernel-based clustering approach. We present competitive results for clustering Omniglot characters and includ…
In this paper, we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in $\mathbb{R}^n$. Our approach focuses on high dimensions and relies solely on the concentration properties of the components in the mixture. We give several results describing of the structure of kernel matrices for a sample drawn from such a mixture. Based on these results, we analyze the ability of kernel PCA to cluster high dimensional mixtures. In particular, we exhibit a specific kernel leading to a simple spectral algorithm for clustering mixtures with possibly common means but dif…
Random features approach has been widely used for kernel approximation in large-scale machine learning. A number of recent studies have explored data-dependent sampling of features, modifying the stochastic oracle from which random features are sampled. While proposed techniques in this realm improve the approximation, their suitability is often verified on a single learning task. In this paper, we propose a task-specific scoring rule for selecting random features, which can be employed for different applications with some adjustments. We restrict our attention to Canonical Correlation Analysi…
We present a study on how to effectively reduce the dimensions of the $k$-means clustering problem, so that provably accurate approximations are obtained. Four algorithms are presented, two \textit{feature selection} and two \textit{feature extraction} based algorithms, all of which are randomized.
University of Electronic Science and Technology of China（电子科技大学）
Constructing the adjacency graph is fundamental to graph-based clustering. Graph learning in kernel space has shown impressive performance on a number of benchmark data sets. However, its performance is largely determined by the chosen kernel matrix. To address this issue, the previous multiple kernel learning algorithm has been applied to learn an optimal kernel from a group of predefined kernels. This approach might be sensitive to noise and limits the representation ability of the consensus kernel. In contrast to existing methods, we propose to learn a low-rank kernel matrix which exploits …
University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）；Linkedin Inc（领英公司）；Indigo Agriculture Inc（靛蓝农业公司）；DiDi Research America（滴滴美国研究院）
Similarity plays a fundamental role in many areas, including data mining, machine learning, statistics and various applied domains. Inspired by the success of ensemble methods and the flexibility of trees, we propose to learn a similarity kernel called rpf-kernel through random projection forests (rpForests). Our theoretical analysis reveals a highly desirable property of rpf-kernel: far-away (dissimilar) points have a low similarity value while nearby (similar) points would have a high similarity}, and the similarities have a native interpretation as the probability of points remaining in the…
