Local graph clustering and the closely related seed set expansion problem are primitives on graphs that are central to a wide range of analytic and learning tasks such as local clustering, community detection, nodes ranking and feature inference. Prior work on local graph clustering mostly falls into two categories with numerical and combinatorial roots respectively. In this work, we draw inspiration from both fields and propose a family of convex optimization formulations based on the idea of diffusion with p-norm network flow for $p\in (1,\infty)$. In the context of local clustering, we characterize the optimal solutions for these optimization problems and show their usefulness in finding low conductance cuts around input seed set. In particular, we achieve quadratic approximation of conductance in the case of $p=2$ similar to the Cheeger-type bounds of spectral methods, constant factor approximation when $p\rightarrow\infty$ similar to max-flow based methods, and a smooth transition for general $p$ values in between. Thus, our optimization formulation can be viewed as bridging the numerical and combinatorial approaches, and we can achieve the best of both worlds in terms of speed and noise robustness. We show that the proposed problem can be solved in strongly local running time for $p\ge 2$ and conduct empirical evaluations on both synthetic and real-world graphs to illustrate our approach compares favorably with existing methods.
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）
We suggest using the max-norm as a convex surrogate constraint for clustering. We show how this yields a better exact cluster recovery guarantee than previously suggested nuclear-norm relaxation, and study the effectiveness of our method, and other related convex relaxations, compared to other clustering approaches.
Cornell University（康奈尔大学）；University of Waterloo（滑铁卢大学）
Sum-of-norms clustering is a clustering formulation based on convex optimization that automatically induces hierarchy. Multiple algorithms have been proposed to solve the optimization problem: subgradient descent by Hocking et al., ADMM and ADA by Chi and Lange, stochastic incremental algorithm by Panahi et al. and semismooth Newton-CG augmented Lagrangian method by Sun et al. All algorithms yield approximate solutions, even though an exact solution is demanded to determine the correct cluster assignment. The purpose of this paper is to close the gap between the output from existing algorithms…
Sum-of-norms clustering is a method for assigning $n$ points in $\mathbb{R}^d$ to $K$ clusters, $1\le K\le n$, using convex optimization. Recently, Panahi et al.\ proved that sum-of-norms clustering is guaranteed to recover a mixture of Gaussians under the restriction that the number of samples is not too large. The purpose of this note is to lift this restriction, i.e., show that sum-of-norms clustering with equal weights can recover a mixture of Gaussians even as the number of samples tends to infinity. Our proof relies on an interesting characterization of clusters computed by sum-of-norms …
Clustering algorithms partition a dataset into groups of similar points. The primary contribution of this article is the Multiscale Spatially-Regularized Diffusion Learning (M-SRDL) clustering algorithm, which uses spatially-regularized diffusion distances to efficiently and accurately learn multiple scales of latent structure in hyperspectral images. The M-SRDL clustering algorithm extracts clusterings at many scales from a hyperspectral image and outputs these clusterings' variation of information-barycenter as an exemplar for all underlying cluster structure. We show that incorporating spat…
An unsupervised learning algorithm to cluster hyperspectral image (HSI) data is proposed that exploits spatially-regularized random walks. Markov diffusions are defined on the space of HSI spectra with transitions constrained to near spatial neighbors. The explicit incorporation of spatial regularity into the diffusion construction leads to smoother random processes that are more adapted for unsupervised machine learning than those based on spectra alone. The regularized diffusion process is subsequently used to embed the high-dimensional HSI into a lower dimensional space through diffusion di…
California State University, Fullerton（加州州立大学富勒顿分校）；USC Information Sciences Institute（南加州大学信息科学研究所）；Claremont Graduate University（克莱蒙特研究生大学）；HP Labs（惠普实验室）
Spectral clustering is widely used to partition graphs into distinct modules or communities. Existing methods for spectral clustering use the eigenvalues and eigenvectors of the graph Laplacian, an operator that is closely associated with random walks on graphs. We propose a new spectral partitioning method that exploits the properties of epidemic diffusion. An epidemic is a dynamic process that, unlike the random walk, simultaneously transitions to all the neighbors of a given node. We show that the replicator, an operator describing epidemic diffusion, is equivalent to the symmetric normaliz…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Nanjing University of Science and Technology（南京理工大学）
Cluster analysis, which focuses on the grouping and categorization of similar elements, is widely used in various fields of research. A novel and fast clustering algorithm, fission clustering algorithm, is proposed in recent year. In this article, we propose a robust fission clustering (RFC) algorithm and a self-adaptive noise identification method. The RFC and the self-adaptive noise identification method are combine to propose a self-adaptive robust fission clustering (SARFC) algorithm. Several frequently-used datasets were applied to test the performance of the proposed clustering approach …
Brunel University（布鲁内尔大学）；University of Helsinki（赫尔辛基大学）；Aalto University（阿尔托大学）
Functional magnetic resonance imaging (fMRI) produces data about activity inside the brain, from which spatial maps can be extracted by independent component analysis (ICA). In datasets, there are n spatial maps that contain p voxels. The number of voxels is very high compared to the number of analyzed spatial maps. Clustering of the spatial maps is usually based on correlation matrices. This usually works well, although such a similarity matrix inherently can explain only a certain amount of the total variance contained in the high-dimensional data where n is relatively small but p is large. …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We introduce the {\it diffusion $K$-means} clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion $K$-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. The diffusion $K$-means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given the number of clusters, we propose a po…
This paper proposes and analyzes a novel clustering algorithm that combines graph-based diffusion geometry with techniques based on density and mode estimation. The proposed method is suitable for data generated from mixtures of distributions with densities that are both multimodal and have nonlinear shapes. A crucial aspect of this algorithm is the use of time of a data-adapted diffusion process as a scale parameter that is different from the local spatial scale parameter used in many clustering algorithms. We prove estimates for the behavior of diffusion distances with respect to this time p…
University of Michigan – Ann Arbor（密歇根大学安娜堡分校）；University of Missouri – Rolla（密苏里大学罗拉分校）；GE Global Research（通用电气全球研究中心）
In this paper, we describe an algorithm FARDiff (Fuzzy Adaptive Resonance Dif- fusion) which combines Diffusion Maps and Fuzzy Adaptive Resonance Theory to do clustering on high dimensional data. We describe some applications of this method and some problems for future research.
University of Southampton（南安普顿大学）；Peking University（北京大学）；DAMO Academy, Alibaba Group（阿里巴巴达摩院）
Finding a suitable density function is essential for density-based clustering algorithms such as DBSCAN and DPC. A naive density corresponding to the indicator function of a unit $d$-dimensional Euclidean ball is commonly used in these algorithms. Such density suffers from capturing local features in complex datasets. To tackle this issue, we propose a new kernel diffusion density function, which is adaptive to data of varying local distributional characteristics and smoothness. Furthermore, we develop a surrogate that can be efficiently computed in linear time and space and prove that it is a…
Clustering algorithms partition a dataset into groups of similar points. The clustering problem is very general, and different partitions of the same dataset could be considered correct and useful. To fully understand such data, it must be considered at a variety of scales, ranging from coarse to fine. We introduce the Multiscale Environment for Learning by Diffusion (MELD) data model, which is a family of clusterings parameterized by nonlinear diffusion on the dataset. We show that the MELD data model precisely captures latent multiscale structure in data and facilitates its analysis. To effi…
This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation of Laplacian. Then, we cover the cuts of graph and spectral clustering which applies clustering in a subspace of data. Different optimization variants of Laplacian eigenmap and its out-of-sample extension are explained. Thereafter, we introduce the locality preserving projection and its kernel variant as linear special cases of Laplacian eigenmap. Versions of…
Université de Caen（卡昂大学）；ENSICAEN（卡昂国立高等工程师学院）；CNRS（法国国家科学研究中心）；Université de Sherbrooke（舍布鲁克大学）
The understanding of neurodegenerative diseases undoubtedly passes through the study of human brain white matter fiber tracts. To date, diffusion magnetic resonance imaging (dMRI) is the unique technique to obtain information about the neural architecture of the human brain, thus permitting the study of white matter connections and their integrity. However, a remaining challenge of the dMRI community is to better characterize complex fiber crossing configurations, where diffusion tensor imaging (DTI) is limited but high angular resolution diffusion imaging (HARDI) now brings solutions. This pa…
In this work we build a unifying framework to interpolate between density-driven and geometry-based algorithms for data clustering, and specifically, to connect the mean shift algorithm with spectral clustering at discrete and continuum levels. We seek this connection through the introduction of Fokker-Planck equations on data graphs. Besides introducing new forms of mean shift algorithms on graphs, we provide new theoretical insights on the behavior of the family of diffusion maps in the large sample limit as well as provide new connections between diffusion maps and mean shift dynamics on a …
School of Information Science and Engineering, Lanzhou University（兰州大学信息科学与工程学院）
Although many graph-based clustering methods attempt to model the stationary diffusion state in their objectives, their performance limits to using a predefined graph. We argue that the estimation of the stationary diffusion state can be achieved by gradient descent over neural networks. We specifically design the Stationary Diffusion State Neural Estimation (SDSNE) to exploit multiview structural graph information for co-supervised learning. We explore how to design a graph neural network specially for unsupervised multiview learning and integrate multiple graphs into a unified consensus grap…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
We study the convex relaxation of clustering and hamming embedding, focusing on the asymmetric case (co-clustering and asymmetric hamming embedding), understanding their relationship to LSH as studied by (Charikar 2002) and to the max-norm ball, and the differences between their symmetric and asymmetric versions.
Tufts University（塔夫茨大学）；University of Cambridge（剑桥大学）；City University of Hong Kong（香港城市大学）；Wake Forest University（维克森林大学）
Ash dieback (Hymenoscyphus fraxineus) is an introduced fungal disease that is causing the widespread death of ash trees across Europe. Remote sensing hyperspectral images encode rich structure that has been exploited for the detection of dieback disease in ash trees using supervised machine learning techniques. However, to understand the state of forest health at landscape-scale, accurate unsupervised approaches are needed. This article investigates the use of the unsupervised Diffusion and VCA-Assisted Image Segmentation (D-VIS) clustering algorithm for the detection of ash dieback disease in…
City University of Hong Kong（香港城市大学）；Hong Kong Centre for Cerebro-Cardiovascular Health Engineering（香港脑心血管健康工程中心）；Tufts University（塔夫茨大学）；Wake Forest University（维克森林大学）
Hyperspectral images, which store a hundred or more spectral bands of reflectance, have become an important data source in natural and social sciences. Hyperspectral images are often generated in large quantities at a relatively coarse spatial resolution. As such, unsupervised machine learning algorithms incorporating known structure in hyperspectral imagery are needed to analyze these images automatically. This work introduces the Spatial-Spectral Image Reconstruction and Clustering with Diffusion Geometry (DSIRC) algorithm for partitioning highly mixed hyperspectral images. DSIRC reduces mea…
Tufts University（塔夫茨大学）；City University of Hong Kong（香港城市大学）；Conservation Research Institute, University of Cambridge（剑桥大学保护研究所）；University of Cambridge（剑桥大学）；Wake Forest University（维克森林大学）
Hyperspectral images taken from aircraft or satellites contain information from hundreds of spectral bands, within which lie latent lower-dimensional structures that can be exploited for classifying vegetation and other materials. A disadvantage of working with hyperspectral images is that, due to an inherent trade-off between spectral and spatial resolution, they have a relatively coarse spatial scale, meaning that single pixels may correspond to spatial regions containing multiple materials. This article introduces the Diffusion and Volume maximization-based Image Clustering (D-VIC) algorith…
Clustering is an effective technique in data mining to group a set of objects in terms of some attributes. Among various clustering approaches, the family of K-Means algorithms gains popularity due to simplicity and efficiency. However, most of existing K-Means based clustering algorithms cannot deal with outliers well and are difficult to efficiently solve the problem embedded the $L_0$-norm constraint. To address the above issues and improve the performance of clustering significantly, we propose a novel clustering algorithm, named REFCMFS, which develops a $L_{2,1}$-norm robust loss as the …
Vironova AB（Vironova公司）；KTH Royal Institute of Technology（瑞典皇家理工学院）
In this paper we propose an adaptive approach for clustering and visualization of data by an orthogonalization process. Starting with the data points being represented by a Markov process using the diffusion map framework, the method adaptively increase the orthogonality of the clusters by applying a feedback mechanism inspired by the Gromov-Wasserstein distance. This mechanism iteratively increases the spectral gap and refines the orthogonality of the data to achieve a clustering with high specificity. By using the diffusion map framework and representing the relation between data points usin…
In this study, we propose a new spectral clustering framework that can auto-tune the parameters of the clustering algorithm in the context of speaker diarization. The proposed framework uses normalized maximum eigengap (NME) values to estimate the number of clusters and the parameters for the threshold of the elements of each row in an affinity matrix during spectral clustering, without the use of parameter tuning on the development set. Even through this hands-off approach, we achieve a comparable or better performance across various evaluation sets than the results found using traditional cl…
In this paper, we present a local information theoretic approach to explicitly learn probabilistic clustering of a discrete random variable. Our formulation yields a convex maximization problem for which it is NP-hard to find the global optimum. In order to algorithmically solve this optimization problem, we propose two relaxations that are solved via gradient ascent and alternating maximization. Experiments on the MSR Sentence Completion Challenge, MovieLens 100K, and Reuters21578 datasets demonstrate that our approach is competitive with existing techniques and worthy of further investigatio…
Ben-Gurion University（本-古里安大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We introduce the $(p,q)$-Fair Clustering problem. In this problem, we are given a set of points $P$ and a collection of different weight functions $W$. We would like to find a clustering which minimizes the $\ell_q$-norm of the vector over $W$ of the $\ell_p$-norms of the weighted distances of points in $P$ from the centers. This generalizes various clustering problems, including Socially Fair $k$-Median and $k$-Means, and is closely connected to other problems such as Densest $k$-Subgraph and Min $k$-Union. We utilize convex programming techniques to approximate the $(p,q)$-Fair Clustering pr…
Sum-of-norms clustering does not separate nearby balls
Courant Institute of Mathematical Sciences（柯朗数学科学研究所）；New York University（纽约大学）；Duke University（杜克大学）；Ecole Normale Supérieure de Lyon（里昂高等师范学院）；CNRS（法国国家科学研究中心）
Federated learning is a model for privacy without revealing private data by transfer models instead of personal and private data from local client devices. While, in the global model, it's crucial to recognize each local data is normal. This paper suggests one method to separate normal locals and abnormal locals by Euclidean similarity clustering of vectors extracted by inputting dummy data in local models. In a federated classification model, this method divided locals into normal and abnormal.
A low-rank non-convex norm method for multiview graph clustering
The UM6P Vanguard Center, Mohammed VI Polytechnic University, Green City, Morocco（摩洛哥绿色城穆莱·易斯二世理工大学UM6P先锋中心）；Université du Littoral Cote d’Opale, LMPA（滨海阿尔卑斯大学LMPA）
Max Planck Institute for Computer Science（马克斯·普朗克计算机科学研究所）；Saarland University（萨尔大学）
An important form of prior information in clustering comes in form of cannot-link and must-link constraints. We present a generalization of the popular spectral clustering technique which integrates such constraints. Motivated by the recently proposed $1$-spectral clustering for the unconstrained problem, our method is based on a tight relaxation of the constrained normalized cut into a continuous optimization problem. Opposite to all other methods which have been suggested for constrained spectral clustering, we can always guarantee to satisfy all constraints. Moreover, our soft formulation a…
NTT Advanced Technology Corporation（NTT先进技术公司）；The University of Electro-Communications（电气通信大学）
Sparse convex clustering is to cluster observations and conduct variable selection simultaneously in the framework of convex clustering. Although a weighted $L_1$ norm is usually employed for the regularization term in sparse convex clustering, its use increases the dependence on the data and reduces the estimation accuracy if the sample size is not sufficient. To tackle these problems, this paper proposes a Bayesian sparse convex clustering method based on the ideas of Bayesian lasso and global-local shrinkage priors. We introduce Gibbs sampling algorithms for our method using scale mixtures …
The ordered weighted $\ell_1$ (OWL) norm is a newly developed generalization of the Octogonal Shrinkage and Clustering Algorithm for Regression (OSCAR) norm. This norm has desirable statistical properties and can be used to perform simultaneous clustering and regression. In this paper, we show how to compute the projection of an $n$-dimensional vector onto the OWL norm ball in $O(n\log(n))$ operations. In addition, we illustrate the performance of our algorithm on a synthetic regression test.
This is a survey of the method of graph cuts and its applications to graph clustering of weighted unsigned and signed graphs. I provide a fairly thorough treatment of the method of normalized graph cuts, a deeply original method due to Shi and Malik, including complete proofs. The main thrust of this paper is the method of normalized cuts. I give a detailed account for K = 2 clusters, and also for K > 2 clusters, based on the work of Yu and Shi. I also show how both graph drawing and normalized cut K-clustering can be easily generalized to handle signed graphs, which are weighted graphs in whi…
Robust Fuzzy local k-plane clustering with mixture distance of hinge loss and L1 norm
China Electric Power Research Institute（中国电力科学研究院）；Key Laboratory of Measurement and Test of High Voltage and Heavy Current, State Administration for Market Regulation（高压及大电流测量测试关键实验室，国家市场监督管理总局）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学学院）；National Center for Applied Mathematics in Hubei, Wuhan University（湖北省应用数学中心，武汉大学）；Hubei Key Laboratory of Computational Science, Wuhan University（湖北省计算科学重点实验室，武汉大学）；Wuhan Institute for Math and AI, Wuhan University（武汉数学与人工智能研究院，武汉大学）
