We consider the problem of learning from a similarity matrix (such as spectral clustering and lowd imensional embedding), when computing pairwise similarities are costly, and only a limited number of entries can be observed. We provide a theoretical analysis using standard notions of graph approximation, significantly generalizing previous results (which focused on spectral clustering with two clusters). We also propose a new algorithmic approach based on adaptive sampling, which experimentally matches or improves on previous methods, while being considerably more general and computationally cheaper.
Matrix rank minimization problem is in general NP-hard. The nuclear norm is used to substitute the rank function in many recent studies. Nevertheless, the nuclear norm approximation adds all singular values together and the approximation error may depend heavily on the magnitudes of singular values. This might restrict its capability in dealing with many practical problems. In this paper, an arctangent function is used as a tighter approximation to the rank function. We use it on the challenging subspace clustering problem. For this nonconvex minimization problem, we develop an effective optim…
INRAE（法国国家农业食品与环境研究院）；Institut Agro（法国国家农业、食品与环境学院）；Univ Montpellier（蒙彼利埃大学）；Université Paris-Saclay（巴黎萨克雷大学）；Institut für Mathematik, Universität Potsdam（波茨坦大学数学研究所）
The cost of computing the spectrum of Laplacian matrices hinders the application of spectral clustering to large data sets. While approximations recover computational tractability, they can potentially affect clustering performance. This paper proposes a practical approach to learn spectral clustering based on adaptive stochastic gradient optimization. Crucially, the proposed approach recovers the exact spectrum of Laplacian matrices in the limit of the iterations, and the cost of each iteration is linear in the number of samples. Extensive experimental validation on data sets with up to half …
Spectral clustering techniques are valuable tools in signal processing and machine learning for partitioning complex data sets. The effectiveness of spectral clustering stems from constructing a non-linear embedding based on creating a similarity graph and computing the spectral decomposition of the Laplacian matrix. However, spectral clustering methods fail to scale to large data sets because of high computational cost and memory usage. A popular approach for addressing these problems utilizes the Nystrom method, an efficient sampling-based algorithm for computing low-rank approximations to l…
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…
Opportunistic spectrum access is one of the emerging techniques for maximizing throughput in congested bands and is enabled by predicting idle slots in spectrum. We propose a kernel-based reinforcement learning approach coupled with a novel budget-constrained sparsification technique that efficiently captures the environment to find the best channel access actions. This approach allows learning and planning over the intrinsic state-action space and extends well to large state spaces. We apply our methods to evaluate coexistence of a reinforcement learning-based radio with a multi-channel adver…
In this paper, we study correlation clustering under fairness constraints. Fair variants of $k$-median and $k$-center clustering have been studied recently, and approximation algorithms using a notion called fairlet decomposition have been proposed. We obtain approximation algorithms for fair correlation clustering under several important types of fairness constraints. Our results hinge on obtaining a fairlet decomposition for correlation clustering by introducing a novel combinatorial optimization problem. We define a fairlet decomposition with cost similar to the $k$-median cost and this all…
Motivated by the needs of estimating the proximity clustering with partial distance measurements from vantage points or landmarks for remote networked systems, we show that the proximity clustering problem can be effectively formulated as the Nyström approximation problem, which solves the kernel K-means clustering problem in the complex space. We implement the Nyström approximation based on a landmark based Nonnegative Matrix Factorization (NMF) process. Evaluation results show that the proposed method finds nearly optimal clustering quality on both synthetic and real-world data sets as we va…
We study the task of differentially private clustering. For several basic clustering problems, including Euclidean DensestBall, 1-Cluster, k-means, and k-median, we give efficient differentially private algorithms that achieve essentially the same approximation ratios as those that can be obtained by any non-private algorithm, while incurring only small additive errors. This improves upon existing efficient algorithms that only achieve some large constant approximation factors. Our results also imply an improved algorithm for the Sample and Aggregate privacy framework. Furthermore, we show tha…
Hierarchical clustering is a widely used approach for clustering datasets at multiple levels of granularity. Despite its popularity, existing algorithms such as hierarchical agglomerative clustering (HAC) are limited to the offline setting, and thus require the entire dataset to be available. This prohibits their use on large datasets commonly encountered in modern learning applications. In this paper, we consider hierarchical clustering in the online setting, where points arrive one at a time. We propose two algorithms that seek to optimize the Moseley and Wang (MW) revenue function, a varian…
Southeast University（东南大学）；City University of Hong Kong（香港城市大学）
This paper explores the problem of clustering ensemble, which aims to combine multiple base clusterings to produce better performance than that of the individual one. The existing clustering ensemble methods generally construct a co-association matrix, which indicates the pairwise similarity between samples, as the weighted linear combination of the connective matrices from different base clusterings, and the resulting co-association matrix is then adopted as the input of an off-the-shelf clustering algorithm, e.g., spectral clustering. However, the co-association matrix may be dominated by po…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）
We present an $(e^{O(p)} \frac{\log \ell}{\log\log\ell})$-approximation algorithm for socially fair clustering with the $\ell_p$-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of $\ell$ groups. The goal is to find a $k$-medians, $k$-means, or, more generally, $\ell_p$-clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of $k$ centers $C$ so as to minimize the maximum over all groups $j$ of $\sum_{u \text{ in group }j} d(u,C)^p$. The socially fair clustering problem was independen…
Parameter estimation for model-based clustering using a finite mixture of normal inverse Gaussian (NIG) distributions is achieved through variational Bayes approximations. Univariate NIG mixtures and multivariate NIG mixtures are considered. The use of variational Bayes approximations here is a substantial departure from the traditional EM approach and alleviates some of the associated computational complexities and uncertainties. Our variational algorithm is applied to simulated and real data. The paper concludes with discussion and suggestions for future work.
Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；University Paris-Saclay（巴黎萨克雷大学）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）
In this work, we revisit fast dimension reduction approaches, as with random projections and random sampling. Our goal is to summarize the data to decrease computational costs and memory footprint of subsequent analysis. Such dimension reduction can be very efficient when the signals of interest have a strong structure, such as with images. We focus on this setting and investigate feature clustering schemes for data reductions that capture this structure. An impediment to fast dimension reduction is that good clustering comes with large algorithmic costs. We address it by contributing a linear…
Spectral Clustering: An empirical study of Approximation Algorithms and its Application to the Attrition Problem
University of California, Los Angeles（加州大学洛杉矶分校）；Instituto Tecnológico Autónomo de México（墨西哥自治技术研究所）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）；IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）；Claremont McKenna College（克莱蒙特·麦肯纳学院）
Qualcomm Institute, University of California, San Diego（加州大学圣地亚哥分校高通研究所）；Ben Gurion University（本·古里安大学）
We study k-median clustering under the sequential no-substitution setting. In this setting, a data stream is sequentially observed, and some of the points are selected by the algorithm as cluster centers. However, a point can be selected as a center only immediately after it is observed, before observing the next point. In addition, a selected center cannot be substituted later. We give the first algorithm for this setting that obtains a constant approximation factor on the optimal risk under a random arrival order, an exponential improvement over previous work. This is also the first constant…
Series Expansion of Probability of Correct Selection for Improved Finite Budget Allocation in Ranking and Selection
PKU-Wuhan Institute for Artificial Intelligence（北大-武汉人工智能研究院）；Guanghua School of Management, Peking University（北京大学光华管理学院）；Xiangjiang Laboratory（湘江实验室）；Inria（法国国家信息与自动化研究所）；Univ Rennes（雷恩大学）；CNRS（法国国家科学研究中心）；IRISA（法国信息与随机系统研究所）
In the context of clustering, we assume a generative model where each cluster is the result of sampling points in the neighborhood of an embedded smooth surface; the sample may be contaminated with outliers, which are modeled as points sampled in space away from the clusters. We consider a prototype for a higher-order spectral clustering method based on the residual from a local linear approximation. We obtain theoretical guarantees for this algorithm and show that, in terms of both separation and robustness to outliers, it outperforms the standard spectral clustering algorithm (based on pairw…
Efficient algorithms for $k$-means clustering frequently converge to suboptimal partitions, and given a partition, it is difficult to detect $k$-means optimality. In this paper, we develop an a posteriori certifier of approximate optimality for $k$-means clustering. The certifier is a sub-linear Monte Carlo algorithm based on Peng and Wei's semidefinite relaxation of $k$-means. In particular, solving the relaxation for small random samples of the dataset produces a high-confidence lower bound on the $k$-means objective, and being sub-linear, our algorithm is faster than $k$-means++ when the nu…
University of California San Diego（加利福尼亚大学圣地亚哥分校）；Tel Aviv University（特拉维夫大学）
Clustering is a popular form of unsupervised learning for geometric data. Unfortunately, many clustering algorithms lead to cluster assignments that are hard to explain, partially because they depend on all the features of the data in a complicated way. To improve interpretability, we consider using a small decision tree to partition a data set into clusters, so that clusters can be characterized in a straightforward manner. We study this problem from a theoretical viewpoint, measuring cluster quality by the $k$-means and $k$-medians objectives: Must there exist a tree-induced clustering whose…
Given a similarity graph between items, correlation clustering (CC) groups similar items together and dissimilar ones apart. One of the most popular CC algorithms is KwikCluster: an algorithm that serially clusters neighborhoods of vertices, and obtains a 3-approximation ratio. Unfortunately, KwikCluster in practice requires a large number of clustering rounds, a potential bottleneck for large graphs. We present C4 and ClusterWild!, two algorithms for parallel correlation clustering that run in a polylogarithmic number of rounds and achieve nearly linear speedups, provably. C4 uses concurrency…
