半监督k近邻分类方法

加权KNN(k-nearest neighbor)方法,仅利用了k个最近邻训练样本所提供的类别信息,而没考虑测试样本的贡献,因而常会导致一些误判。针对这个缺陷,提出了半监督KNN分类方法。该方法对序列样本和非序列样本,均能够较好地执行分类。在分类决策时,还考虑了c个最近邻测试样本的贡献,从而提高了分类的正确性。在Cohn-Kanade人脸库上,序列图像的识别率提高了5.95%,在CMU-AMP人脸库上,非序列图像的识别率提高了7.98%。实验结果表明,该方法执行效率高,分类效果好。     

The MinMax k-Means clustering algorithm

Applying k-Means to minimize the sum of the intra-cluster variances is the most popular clustering approach. However, after a bad initialization, poor local optima can be easily obtained. To tackle the initialization problem of k-Means, we propose the MinMax k-Means algorithm, a method that assigns weights to the clusters relative to their variance and optimizes a weighted version of the k-Means objective. Weights are learned together with the cluster assignments, through an iterative procedure. The proposed weighting scheme limits the emergence of large variance clusters and allows high quality solutions to be systematically uncovered, irrespective of the initialization. Experiments verify the effectiveness of our approach and its robustness over bad initializations, as it compares favorably to both k-Means and other methods from the literature that consider the k-Means initialization problem.     

The global k-means clustering algorithm

We present the global k-means algorithm which is an incremental approach to clustering that dynamically adds one cluster center at a time through a deterministic global search procedure consisting of N (with N being the size of the data set) executions of the k-means algorithm from suitable initial positions. We also propose modifications of the method to reduce the computational load without significantly affecting solution quality. The proposed clustering methods are tested on well-known data sets and they compare favorably to the k-means algorithm with random restarts.     

Improved k-nearest neighbor classification

k-nearest neighbor (k-NN) classification is a well-known decision rule that is widely used in pattern classification. However, the traditional implementation of this method is computationally expensive. In this paper we develop two effective techniques, namely, template condensing and preprocessing, to significantly speed up k-NN classification while maintaining the level of accuracy. Our template condensing technique aims at “sparsifying” dense homogeneous clusters of prototypes of any single class. This is implemented by iteratively eliminating patterns which exhibit high attractive capacities. Our preprocessing technique filters a large portion of prototypes which are unlikely to match against the unknown pattern. This again accelerates the classification procedure considerably, especially in cases where the dimensionality of the feature space is high. One of our case studies shows that the incorporation of these two techniques to k-NN rule achieves a seven-fold speed-up without sacrificing accuracy.     

Information cut for clustering using a gradient descent approach

We introduce a new graph cut for clustering which we call the Information Cut. It is derived using Parzen windowing to estimate an information theoretic distance measure between probability density functions. We propose to optimize the Information Cut using a gradient descent-based approach. Our algorithm has several advantages compared to many other graph-based methods in terms of determining an appropriate affinity measure, computational complexity, memory requirements and coping with different data scales. We show that our method may produce clustering and image segmentation results comparable or better than the state-of-the art graph-based methods.     

A k-populations algorithm for clustering categorical data

In this paper, the conventional k-modes-type algorithms for clustering categorical data are extended by representing the clusters of categorical data with k-populations instead of the hard-type centroids used in the conventional algorithms. Use of a population-based centroid representation makes it possible to preserve the uncertainty inherent in data sets as long as possible before actual decisions are made. The k-populations algorithm was found to give markedly better clustering results through various experiments.     

Rough-fuzzy weighted k-nearest leader classifier for large data sets

A leaders set which is derived using the leaders clustering method can be used in place of a large training set to reduce the computational burden of a classifier. Recently, a fast and efficient leader-based classifier called weighted k-nearest leader-based classifier is shown by us to be an efficient and faster classifier. But, there exist some uncertainty while calculating the relative importance (weight) of the prototypes. This paper proposes a generalization over the earlier proposed k-nearest leader-based classifier where a novel soft computing approach is used to resolve the uncertainty. Combined principles of rough set theory and fuzzy set theory are used to analyze the proposed method. The proposed method called rough-fuzzy weighted k-nearest leader classifier (RF-wk-NLC) uses a two level hierarchy of prototypes along with their relative importance. RF-wk-NLC is shown by using some standard data sets to have improved performance and is compared with the earlier related methods.     

Probably correct k-nearest neighbor search in high dimensions

A novel approach for k-nearest neighbor (k-NN) searching with Euclidean metric is described. It is well known that many sophisticated algorithms cannot beat the brute-force algorithm when the dimensionality is high. In this study, a probably correct approach, in which the correct set of k-nearest neighbors is obtained in high probability, is proposed for greatly reducing the searching time. We exploit the marginal distribution of the k th nearest neighbors in low dimensions, which is estimated from the stored data (an empirical percentile approach). We analyze the basic nature of the marginal distribution and show the advantage of the implemented algorithm, which is a probabilistic variant of the partial distance searching. Its query time is sublinear in data size n, that is, O(mnδ) with δ=o(1) in n and δ≤1, for any fixed dimension m.     

Model-based prediction error uncertainty estimation for k-nn method

The k-nearest neighbour estimation method is one of the main tools used in multi-source forest inventories. It is a powerful non-parametric method for which estimates are easy to compute and relatively accurate. One downside of this method is that it lacks an uncertainty measure for predicted values and for areas of an arbitrary size. We present a method to estimate the prediction uncertainty based on the variogram model which derives the necessary formula for the k-nn method. A data application is illustrated for multi-source forest inventory data, and the results are compared at pixel level to the conventional RMSE method. We find that the variogram model-based method which is analytic, is competitive with the RMSE method.     

Fast global k-means clustering using cluster membership and inequality

In this paper, we present a fast global k-means clustering algorithm by making use of the cluster membership and geometrical information of a data point. This algorithm is referred to as MFGKM. The algorithm uses a set of inequalities developed in this paper to determine a starting point for the jth cluster center of global k-means clustering. Adopting multiple cluster center selection (MCS) for MFGKM, we also develop another clustering algorithm called MFGKM+MCS. MCS determines more than one starting point for each step of cluster split; while the available fast and modified global k-means clustering algorithms select one starting point for each cluster split. Our proposed method MFGKM can obtain the least distortion; while MFGKM+MCS may give the least computing time. Compared to the modified global k-means clustering algorithm, our method MFGKM can reduce the computing time and number of distance calculations by a factor of 3.78-5.55 and 21.13-31.41, respectively, with the average distortion reduction of 5,487 for the Statlog data set. Compared to the fast global k-means clustering algorithm, our method MFGKM+MCS can reduce the computing time by a factor of 5.78-8.70 with the average reduction of distortion of 30,564 using the same data set. The performances of our proposed methods are more remarkable when a data set with higher dimension is divided into more clusters.     

Fast exact k nearest neighbors search using an orthogonal search tree

The problem of k nearest neighbors (kNN) is to find the nearest k neighbors for a query point from a given data set. In this paper, a novel fast kNN search method using an orthogonal search tree is proposed. The proposed method creates an orthogonal search tree for a data set using an orthonormal basis evaluated from the data set. To find the kNN for a query point from the data set, projection values of the query point onto orthogonal vectors in the orthonormal basis and a node elimination inequality are applied for pruning unlikely nodes. For a node, which cannot be deleted, a point elimination inequality is further used to reject impossible data points. Experimental results show that the proposed method has good performance on finding kNN for query points and always requires less computation time than available kNN search algorithms, especially for a data set with a big number of data points or a large standard deviation.     

Improvement of the k-means clustering filtering algorithm

In this paper, we present a modified filtering algorithm (MFA) by making use of center variations to speed up clustering process. Our method first divides clusters into static and active groups. We use the information of cluster displacements to reject unlikely cluster centers for all nodes in the kd-tree. We reduce the computational complexity of filtering algorithm (FA) through finding candidates for each node mainly from the set of active cluster centers. Two conditions for determining the set of candidate cluster centers for each node from active clusters are developed. Our approach is different from the major available algorithm, which passes no information from one stage of iteration to the next. Theoretical analysis shows that our method can reduce the computational complexity, in terms of the number of distance calculations, of FA at each stage of iteration by a factor of FC/AC, where FC and AC are the numbers of total clusters and active clusters, respectively. Compared with the FA, our algorithm can effectively reduce the computing time and number of distance calculations. It is noted that our proposed algorithm can generate the same clusters as that produced by hard k-means clustering. The superiority of our method is more remarkable when a larger data set with higher dimension is used.     

Design-based approach to k-nearest neighbours technique for coupling field and remotely sensed data in forest surveys

The statistical properties of the k-NN estimators are investigated in a design-based framework, avoiding any assumption about the population under study. The issue of coupling remotely sensed digital imagery with data arising from forest inventories conducted using probabilistic sampling schemes is considered. General results are obtained for the k-NN estimator at the pixel level. When averages (or totals) of forest attributes for the whole study area or sub-areas are of interest, the use of the empirical difference estimator is proposed. The estimator is shown to be approximately unbiased with a variance admitting unbiased or conservative estimators. The performance of the empirical difference estimator is evaluated by an extensive simulation study performed on several populations whose dimensions and covariate values are taken from a real case study. Samples are selected from the populations by means of simple random sampling without replacement. Comparisons with the generalized regression estimator and Horvitz-Thompson estimators are also performed. An application to a local forest inventory on a test area of central Italy is considered.     

A time-efficient pattern reduction algorithm for k-means clustering

This paper presents an efficient algorithm, called pattern reduction (PR), for reducing the computation time of k-means and k-means-based clustering algorithms. The proposed algorithm works by compressing and removing at each iteration patterns that are unlikely to change their membership thereafter. Not only is the proposed algorithm simple and easy to implement, but it can also be applied to many other iterative clustering algorithms such as kernel-based and population-based clustering algorithms. Our experiments—from 2 to 1000 dimensions and 150 to 10,000,000 patterns—indicate that with a small loss of quality, the proposed algorithm can significantly reduce the computation time of all state-of-the-art clustering algorithms evaluated in this paper, especially for large and high-dimensional data sets.     

A direct boosting algorithm for the k-nearest neighbor classifier via local warping of the distance metric

Though the k-nearest neighbor (k-NN) pattern classifier is an effective learning algorithm, it can result in large model sizes. To compensate, a number of variant algorithms have been developed that condense the model size of the k-NN classifier at the expense of accuracy. To increase the accuracy of these condensed models, we present a direct boosting algorithm for the k-NN classifier that creates an ensemble of models with locally modified distance weighting. An empirical study conducted on 10 standard databases from the UCI repository shows that this new Boosted k-NN algorithm has increased generalization accuracy in the majority of the datasets and never performs worse than standard k-NN.     

Upper signed k-domination in a general graph

Let k be a positive integer, and let G=(V,E) be a graph with minimum degree at least k−1. A function f:V→{−1,1} is said to be a signed k-dominating function (SkDF) if _∑u∈N[v]f(u)?k for every v∈V. An SkDF f of a graph G is minimal if there exists no SkDF g such that g≠f and g(v)?f(v) for every v∈V. The maximum of the values of _∑v∈Vf(v), taken over all minimal SkDFs f, is called the upper signed k-domination numberΓkS(G). In this paper, we present a sharp upper bound on this number for a general graph.     

Improving the performance of k-means for color quantization

Color quantization is an important operation with many applications in graphics and image processing. Most quantization methods are essentially based on data clustering algorithms. However, despite its popularity as a general purpose clustering algorithm, k-means has not received much respect in the color quantization literature because of its high computational requirements and sensitivity to initialization. In this paper, we investigate the performance of k-means as a color quantizer. We implement fast and exact variants of k-means with several initialization schemes and then compare the resulting quantizers to some of the most popular quantizers in the literature. Experiments on a diverse set of images demonstrate that an efficient implementation of k-means with an appropriate initialization strategy can in fact serve as a very effective color quantizer.     

Equally contributory privacy-preserving k-means clustering over vertically partitioned data

In recent years, there have been numerous attempts to extend the k-means clustering protocol for single database to a distributed multiple database setting and meanwhile keep privacy of each data site. Current solutions for (whether two or more) multiparty k-means clustering, built on one or more secure two-party computation algorithms, are not equally contributory, in other words, each party does not equally contribute to k-means clustering. This may lead a perfidious attack where a party who learns the outcome prior to other parties tells a lie of the outcome to other parties. In this paper, we present an equally contributory multiparty k-means clustering protocol for vertically partitioned data, in which each party equally contributes to k-means clustering. Our protocol is built on ElGamal's encryption scheme, Jakobsson and Juels's plaintext equivalence test protocol, and mix networks, and protects privacy in terms that each iteration of k-means clustering can be performed without revealing the intermediate values.     

A Fan-type result on k-ordered graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. In this paper, we show that if G is a ⌊3k/2⌋-connected graph of order n?100k, and d(u)+d(v)?n for any two vertices u and v with d(u,v)=2, then G is k-ordered hamiltonian. Our result implies the theorem of G. Chen et al. [Ars Combin. 70 (2004) 245-255] [1], which requires the degree sum condition for all pairs of non-adjacent vertices, not just those distance 2 apart.     

Almost k-wise independence versus k-wise independence

We say that a distribution over {0,1}ⁿ is (ε,k)-wise independent if its restriction to every k coordinates results in a distribution that is ε-close to the uniform distribution. A natural question regarding (ε,k)-wise independent distributions is how close they are to some k-wise independent distribution. We show that there exist (ε,k)-wise independent distributions whose statistical distance is at least n^O(k)·ε from any k-wise independent distribution. In addition, we show that for any (ε,k)-wise independent distribution there exists some k-wise independent distribution, whose statistical distance is n^O(k)·ε.