The K-means algorithm is among the most commonly used data clustering methods. However, the regular K-means can only be applied in the input space and it is applicable when clusters are linearly separable. The kernel K-means, which extends K-means into the kernel space, is able to capture nonlinear structures and identify arbitrarily shaped clusters. However, kernel methods often operate on the kernel matrix of the data, which scale poorly with the size of the matrix or suffer from the high clustering cost due to the repetitive calculations of kernel values. Another issue is that algorithms access the data only through evaluations of $K(x_i, x_j)$, which limits many processes that can be done on data through the clustering task. This paper proposes a method to combine the advantages of the linear and nonlinear approaches by using driven corresponding approximate finite-dimensional feature maps based on spectral analysis. Applying approximate finite-dimensional feature maps were only discussed in the Support Vector Machines (SVM) problems before. We suggest using this method in kernel K-means era as alleviates storing huge kernel matrix in memory, further calculating cluster centers more efficiently and access the data explicitly in feature space. These explicit feature maps enable us to access the data in the feature space explicitly and take advantage of K-means extensions in that space. We demonstrate our Explicit Kernel Minkowski Weighted K-mean (Explicit KMWK-mean) method is able to be more adopted and find best-fitting values in new space by applying additional Minkowski exponent and feature weights parameter. Moreover, it can reduce the impact of concentration on nearest neighbour search by suggesting investigate among other norms instead of Euclidean norm, includes Minkowski norms and fractional norms (as an extension of the Minkowski norms with p<1).
Wellcome Centre for Human Genetics, University of Oxford（牛津大学惠康人类遗传学中心）；University of Oxford（牛津大学）；MINES ParisTech, PSL Research University（巴黎高等矿业学院，巴黎文理研究大学）；Institut Curie, PSL Research University（居里研究所，巴黎文理研究大学）；INSERM（法国国家健康与医学研究院）；CNRS（法国国家科学研究中心）
We propose new positive definite kernels for permutations. First we introduce a weighted version of the Kendall kernel, which allows to weight unequally the contributions of different item pairs in the permutations depending on their ranks. Like the Kendall kernel, we show that the weighted version is invariant to relabeling of items and can be computed efficiently in $O(n \ln(n))$ operations, where $n$ is the number of items in the permutation. Second, we propose a supervised approach to learn the weights by jointly optimizing them with the function estimated by a kernel machine. Third, while…
Interest in generative models has grown tremendously in the past decade. However, their training performance can be adversely affected by contamination, where outliers are encoded in the representation of the model. This results in the generation of noisy data. In this paper, we introduce weighted conjugate feature duality in the framework of Restricted Kernel Machines (RKMs). The RKM formulation allows for an easy integration of methods from classical robust statistics. This formulation is used to fine-tune the latent space of generative RKMs using a weighting function based on the Minimum Co…
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…
Cluster-Adaptive Feature Extraction and its Theoretical Foundation with Minkowski Weighted k-Means
School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe, UK（埃塞克斯大学计算机科学与电子工程学院，英国威文豪斯）；Département d’informatique, Université du Québec à Montréal, C.P. 8888 succ. Centre-Ville, Montreal (QC) H3C 3P8 Canada（魁北克大学蒙特利尔分校计算机科学系，加拿大蒙特利尔（QC）H3C 3P8）；Mila - Quebec AI Institute, Montreal, QC, Canada（魁北克人工智能研究所，加拿大蒙特利尔（QC））
We propose a theoretical framework of multi-way similarity to model real-valued data into hypergraphs for clustering via spectral embedding. For graph cut based spectral clustering, it is common to model real-valued data into graph by modeling pairwise similarities using kernel function. This is because the kernel function has a theoretical connection to the graph cut. For problems where using multi-way similarities are more suitable than pairwise ones, it is natural to model as a hypergraph, which is generalization of a graph. However, although the hypergraph cut is well-studied, there is not…
Kernel functions in support vector machines (SVM) are needed to assess the similarity of input samples in order to classify these samples, for instance. Besides standard kernels such as Gaussian (i.e., radial basis function, RBF) or polynomial kernels, there are also specific kernels tailored to consider structure in the data for similarity assessment. In this article, we will capture structure in data by means of probabilistic mixture density models, for example Gaussian mixtures in the case of real-valued input spaces. From the distance measures that are inherently contained in these models,…
Federal University of Bahia（巴伊亚联邦大学）；Federal Institute of Pernambuco（伯南布哥联邦学院）；University of São Paulo（圣保罗大学）
Improvement of statistical learning models in order to increase efficiency in solving classification or regression problems is still a goal pursued by the scientific community. In this way, the support vector machine model is one of the most successful and powerful algorithms for those tasks. However, its performance depends directly from the choice of the kernel function and their hyperparameters. The traditional choice of them, actually, can be computationally expensive to do the kernel choice and the tuning processes. In this article, it is proposed a novel framework to deal with the kernel…
In this paper, we propose a new homological method to study weighted directed networks. Our model of such networks is a directed graph $Q$ equipped with a weight function $w$ on the set $Q_{1}$ of arrows in $Q$. We require that the range $W$ of our weight function is equipped with an addition or a multiplication, i.e., $W$ is a monoid in the mathematical terminology. When $W$ is equipped with a representation on a vector space $M$, the standard method of homological algebra allows us to define the homology groups $H_{*}(Q,w;M)$. It is known that when $Q$ has no oriented cycles, $H_{n}(Q,w;M)=0…
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；University of Bath（巴斯大学）；TU Chemnitz（开姆尼茨工业大学）
High-dimensional data in the form of tensors are challenging for kernel classification methods. To both reduce the computational complexity and extract informative features, kernels based on low-rank tensor decompositions have been proposed. However, what decisive features of the tensors are exploited by these kernels is often unclear. In this paper we propose a novel kernel that is based on the Tucker decomposition. For this kernel the Tucker factors are computed based on re-weighting of the Tucker matrices with tuneable powers of singular values from the HOSVD decomposition. This provides a …
University of Amsterdam（阿姆斯特丹大学）；Qualcomm AI Research（高通人工智能研究院）
In this work we propose a batch Bayesian optimization method for combinatorial problems on permutations, which is well suited for expensive-to-evaluate objectives. We first introduce LAW, an efficient batch acquisition method based on determinantal point processes using the acquisition weighted kernel. Relying on multiple parallel evaluations, LAW enables accelerated search on combinatorial spaces. We then apply the framework to permutation problems, which have so far received little attention in the Bayesian Optimization literature, despite their practical importance. We call this method LAW2…
Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves performance superior to any other weighted herding method. We demonstrate empirically a rate of convergence faster than O(1/N). Our resu…
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室有限公司）；Tokyo（东京）
The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed-form formula for Gaussian mixture models when parameterized by integer exponents. This result extends to arbitrary mixtures of exponential families with natural parameter spaces being cones: This includes the binomial, the multinomial, the zero-centered Laplacian, the Gaussian a…
The Neural Tangent Kernel (NTK) has recently attracted intense study, as it describes the evolution of an over-parameterized Neural Network (NN) trained by gradient descent. However, it is now well-known that gradient descent is not always a good optimizer for NNs, which can partially explain the unsatisfactory practical performance of the NTK regression estimator. In this paper, we introduce the Weighted Neural Tangent Kernel (WNTK), a generalized and improved tool, which can capture an over-parameterized NN's training dynamics under different optimizers. Theoretically, in the infinite-width …
Kernel weighted importance sampling for off-policy evaluation in contextual bandits
Institute of Child Health, University College London（伦敦大学学院儿童健康研究所）；Strive Health Ltd.（奋进健康有限公司）；National Institute for Health Research（国家卫生研究院）；University College London, Biomedical Research Council（伦敦大学学院生物医学研究委员会）；Great Ormond Street Hospital（大奥蒙德街医院）；McGill University（麦吉尔大学）
We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d. samples from the underlying measure and evaluation of the kernel and that result in a small worst-case error. In addition to our theoretical results and the benefits resulting from convex weights, our experiments indicate that this construction can compete with the optimal bounds in well-known examples.
Joint modeling of language and vision has been drawing increasing interest. A multimodal data representation allowing for bidirectional retrieval of images by sentences and vice versa is a key aspect. In this paper we present three contributions in canonical correlation analysis (CCA) based multimodal retrieval. Firstly, we show that an asymmetric weighting of the canonical weights, while achieving a cross view mapping from the search to the query space, improves the retrieval performance. Secondly, we devise a computationally efficient model selection, crucial to generalization and stability,…
University of Science and Technology Beijing（北京科技大学）；Chinese Academy of Sciences（中国科学院）
Classifier ensemble generally should combine diverse component classifiers. However, it is difficult to give a definitive connection between diversity measure and ensemble accuracy. Given a list of available component classifiers, how to adaptively and diversely ensemble classifiers becomes a big challenge in the literature. In this paper, we argue that diversity, not direct diversity on samples but adaptive diversity with data, is highly correlated to ensemble accuracy, and we propose a novel technology for classifier ensemble, learning to diversify, which learns to adaptively combine classif…
National University of Defense Technology（国防科技大学）；Hunan University（湖南大学）；Supercomputing and Cloud Computing Institute（超级计算与云计算研究院）；State University of New York, New Paltz（纽约州立大学新帕尔茨分校）
Multiple kernel clustering (MKC) is committed to achieving optimal information fusion from a set of base kernels. Constructing precise and local kernel matrices is proved to be of vital significance in applications since the unreliable distant-distance similarity estimation would degrade clustering per-formance. Although existing localized MKC algorithms exhibit improved performance compared to globally-designed competi-tors, most of them widely adopt KNN mechanism to localize kernel matrix by accounting for τ -nearest neighbors. However, such a coarse manner follows an unreasonable strategy t…
Recent technology and equipment advancements provide with us opportunities to better analyze Alzheimer's disease (AD), where we could collect and employ the data from different image and genetic modalities that may potentially enhance the predictive performance. To perform better clustering in AD analysis, in this paper we propose a novel model to leverage data from all different modalities/views, which can learn the weights of each view adaptively. Different from previous vanilla Non-negative Matrix Factorization which assumes data is linearly separable, we propose a simple yet efficient meth…
