In this paper we propose an online approximate k-nn graph building algorithm, which is able to quickly update a k-nn graph using a flow of data points. One very important step of the algorithm consists in using the current distributed graph to search for the neighbors of a new node. Hence we also propose a distributed partitioning method based on balanced k-medoids clustering, that we use to optimize the distributed search process. Finally, we present the improved sequential search procedure that is used inside each partition. We also perform an experimental evaluation of the different algorithms, where we study the influence of the parameters and compare the result of our algorithms to existing state of the art. This experimental evaluation confirms that the fast online k-nn graph building algorithm produces a graph that is highly similar to the graph produced by an offline exhaustive algorithm, while it requires less similarity computations.
However, when it comes to k-means, few balanced versions exist. In @cite_10 , the authors proposed a method that has a complexity @math , which makes it too complex for large graphs. In @cite_6 , the authors proposed the Frequency Sensitive Competitive Learning (FSCL) method, where the distance between a point and a centroid is multiplied by the number of points already assigned to this centroid. Bigger clusters are therefore less likely to win additional points. In @cite_12 , the authors used FSCL with additive bias instead of multiplicative bias. However, both methods offer no guarantee on the final number of points in each partition, and experimental results have shown the resulting partitioning is often largely imbalanced.
@cite_10 @cite_12 @cite_6
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We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Node-weighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as Min-CSPs, and as Submodular Multiway Partition problems. These two generalizations lead to two natural relaxations that we call respectively the Local Distribution LP, and the Lovasz relaxation. The Local Distribution LP is generally stronger than the Lovasz relaxation, but applicable only to Min-CSP with predicates of constant size. The relaxations coincide in some cases such as Multiway Cut where they are both equivalent to the CKR relaxation. We show that the Lovasz relaxation gives a (2--2 k)-approximation for Submodular Multiway Partition with k terminals, improving a recent 2-approximation [2]. We prove that this factor is optimal in two senses: (1) A (2--2 k -- e)-approximation for Submodular Multiway Partition with k terminals would require exponentially many value queries (in the oracle model), or imply NP = RP (for certain explicit submodular functions). (2) For Hypergraph Multiway Cut and Node-weighted Multiway Cut with k terminals, both special cases of Submodular Multiway Partition, we prove that a (2--2 k -- e)-approximation is NP-hard, assuming the Unique Games Conjecture. Both our hardness results are more general: (1) We show that the notion of symmetry gap, previously used for submodular maximization problems [19, 6], also implies hardness results for submodular minimization problems. (2) Assuming the Unique Games Conjecture, we show that the Local Distribution LP gives an optimal approximation for every Min-CSP that includes the Not-Equal predicate. Finally, we connect the two hardness techniques by proving that the integrality gap of the Local Distribution LP coincides with the symmetry gap of the multilinear relaxation (for a related instance). This shows that the appearance of the same hardness threshold for a Min-CSP and the related submodular minimization problem is not a coincidence. Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the k-nearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the lightest possible conditions. In addition, a strong approximation result is obtained and the choice of the optimal set of weights is discussed. In particular, the classical k-nearest neighbor estimate is not optimal in a sense described in the manuscript. The proposed method has been implemented to recover level sets in both simulated and real-life data. In this paper, we present "k-means+ID3", a method to cascade k-means clustering and the ID3 decision tree learning methods for classifying anomalous and normal activities in a computer network, an active electronic circuit, and a mechanical mass-beam system. The k-means clustering method first partitions the training instances into k clusters using Euclidean distance similarity. On each cluster, representing a density region of normal or anomaly instances, we build an ID3 decision tree. The decision tree on each cluster refines the decision boundaries by learning the subgroups within the cluster. To obtain a final decision on classification, the decisions of the k-means and ID3 methods are combined using two rules: 1) the nearest-neighbor rule and 2) the nearest-consensus rule. We perform experiments on three data sets: 1) network anomaly data (NAD), 2) Duffing equation data (DED), and 3) mechanical system data (MSD), which contain measurements from three distinct application domains of computer networks, an electronic circuit implementing a forced Duffing equation, and a mechanical system, respectively. Results show that the detection accuracy of the k-means+ID3 method is as high as 96.24 percent at a false-positive-rate of 0.03 percent on NAD; the total accuracy is as high as 80.01 percent on MSD and 79.9 percent on DED In this paper we describe several new clustering algorithms for nodes in a mobile ad hoc network. We propose to combine two known approaches into a single clustering algorithm which considers connectivity as a primary criterion and lower ID as secondary criterion for selecting cluster heads. The goal is to minimize the number of clusters, which results in dominating sets of smaller sizes (this is important for applications in broadcasting and Bluetooth formation). We also describe algorithms for modifying cluster structure in the presence of topological changes. Next, we generalize the cluster definition so that a cluster contains all nodes that are at a distance of at most k hops from the cluster head. The efficiency of four clustering algorithms (k-lowestID and k-CONID, k=1 and k=2) is tested by measuring the average number of created clusters, the number of border nodes, and the cluster size in random unit graphs. The most interesting experimental result is stability of the ratio of the sum of CHs and border nodes in the set. It was constantly 60-70 for 1-lowestID and 46-56 for 1-ConID, for any value of n (number of nodes) and d (average node degree). Similar conclusions and similar number were obtained for k=2. We also proposed a unified framework for most existing and new clustering algorithms where a properly defined weight at each node is the only difference in the algorithm. Finally, we propose a framework for generating random unit graphs with obstacles.