k nearest neighbor (kNN) method is a popular classification method in data mining and statistics because of its simple implementation and significant classification performance. However, it is impractical for traditional kNN methods to assign a fixed k value (even though set by experts) to all test samples. Previous solutions assign different k values to different test samples by the cross validation method but are usually time-consuming. This paper proposes a kTree method to learn different optimal k values for different test/new samples, by involving a training stage in the kNN classification. Specifically, in the training stage, kTree method first learns optimal k values for all training samples by a new sparse reconstruction model, and then constructs a decision tree (namely, kTree) using training samples and the learned optimal k values. In the test stage, the kTree fast outputs the optimal k value for each test sample, and then, the kNN classification can be conducted using the learned optimal k value and all training samples. As a result, the proposed kTree method has a similar running cost but higher classification accuracy, compared with traditional kNN methods, which assign a fixed k value to all test samples. Moreover, the proposed kTree method needs less running cost but achieves similar classification accuracy, compared with the newly kNN methods, which assign different k values to different test samples. This paper further proposes an improvement version of kTree method (namely, k*Tree method) to speed its test stage by extra storing the information of the training samples in the leaf nodes of kTree, such as the training samples located in the leaf nodes, their kNNs, and the nearest neighbor of these kNNs. We call the resulting decision tree as k*Tree, which enables to conduct kNN classification using a subset of the training samples in the leaf nodes rather than all training samples used in the newly kNN methods. This actually reduces running cost of test stage. Finally, the experimental results on 20 real data sets showed that our proposed methods (ie., kTree and k*Tree) are much more efficient than the compared methods in terms of classification tasks.

The K Nearest Neighbor (kNN) method has widely been used in the applications of data mining and machine learning due to its simple implementation and distinguished performance. However, setting all test data with the same k value in the previous kNN methods has been proven to make these methods impractical in real applications. This article proposes to learn a correlation matrix to reconstruct test data points by training data to assign different k values to different test data points, referred to as the Correlation Matrix kNN (CM-kNN for short) classification. Specifically, the least-squares loss function is employed to minimize the reconstruction error to reconstruct each test data point by all training data points. Then, a graph Laplacian regularizer is advocated to preserve the local structure of the data in the reconstruction process. Moreover, an ℓ ₁ -norm regularizer and an ℓ _{2, 1} -norm regularizer are applied to learn different k values for different test data and to result in low sparsity to remove the redundant/noisy feature from the reconstruction process, respectively. Besides for classification tasks, the kNN methods (including our proposed CM-kNN method) are further utilized to regression and missing data imputation. We conducted sets of experiments for illustrating the efficiency, and experimental results showed that the proposed method was more accurate and efficient than existing kNN methods in data-mining applications, such as classification, regression, and missing data imputation.

Graph-based subspace clustering methods have exhibited promising performance. However, they still suffer some of these drawbacks: they encounter the expensive time overhead, they fail to explore the explicit clusters, and cannot generalize to unseen data points. In this work, we propose a scalable graph learning framework, seeking to address the above three challenges simultaneously. Specifically, it is based on the ideas of anchor points and bipartite graph. Rather than building an $n\times n$ graph, where $n$ is the number of samples, we construct a bipartite graph to depict the relationship between samples and anchor points. Meanwhile, a connectivity constraint is employed to ensure that the connected components indicate clusters directly. We further establish the connection between our method and the $K$ -means clustering. Moreover, a model to process multiview data is also proposed, which is linearly scaled with respect to $n$ . Extensive experiments demonstrate the efficiency and effectiveness of our approach with respect to many state-of-the-art clustering methods.

In image analysis, the images are often represented by multiple visual features (also known as multiview features), that aim to better interpret them for achieving remarkable performance of the learning. Since the processes of feature extraction on each view are separated, the multiple visual features of images may include overlap, noise, and redundancy. Thus, learning with all the derived views of the data could decrease the effectiveness. To address this, this paper simultaneously conducts a hierarchical feature selection and a multiview multilabel (MVML) learning for multiview image classification, via embedding a proposed a new block-row regularizer into the MVML framework. The block-row regularizer concatenating a Frobenius norm (F-norm) regularizer and an l _2,1 -norm regularizer is designed to conduct a hierarchical feature selection, in which the F-norm regularizer is used to conduct a high-level feature selection for selecting the informative views (i.e., discarding the uninformative views) and the 12,1-norm regularizer is then used to conduct a low-level feature selection on the informative views. The rationale of the use of a block-row regularizer is to avoid the issue of the over-fitting (via the block-row regularizer), to remove redundant views and to preserve the natural group structures of data (via the F-norm regularizer), and to remove noisy features (the 12,1-norm regularizer), respectively. We further devise a computationally efficient algorithm to optimize the derived objective function and also theoretically prove the convergence of the proposed optimization method. Finally, the results on real image datasets show that the proposed method outperforms two baseline algorithms and three state-of-the-art algorithms in terms of classification performance.

In this paper, we propose a new unsupervised spectral feature selection model by embedding a graph regularizer into the framework of joint sparse regression for preserving the local structures of data. To do this, we first extract the bases of training data by previous dictionary learning methods and, then, map original data into the basis space to generate their new representations, by proposing a novel joint graph sparse coding (JGSC) model. In JGSC, we first formulate its objective function by simultaneously taking subspace learning and joint sparse regression into account, then, design a new optimization solution to solve the resulting objective function, and further prove the convergence of the proposed solution. Furthermore, we extend JGSC to a robust JGSC (RJGSC) via replacing the least square loss function with a robust loss function, for achieving the same goals and also avoiding the impact of outliers. Finally, experimental results on real data sets showed that both JGSC and RJGSC outperformed the state-of-the-art algorithms in terms of k -nearest neighbor classification performance.

Missing data imputation is a key issue in learning from incomplete data. Various techniques have been developed with great successes on dealing with missing values in data sets with homogeneous attributes (their independent attributes are all either continuous or discrete). This paper studies a new setting of missing data imputation, i.e., imputing missing data in data sets with heterogeneous attributes (their independent attributes are of different types), referred to as imputing mixed-attribute data sets. Although many real applications are in this setting, there is no estimator designed for imputing mixed-attribute data sets. This paper first proposes two consistent estimators for discrete and continuous missing target values, respectively. And then, a mixture-kernel-based iterative estimator is advocated to impute mixed-attribute data sets. The proposed method is evaluated with extensive experiments compared with some typical algorithms, and the result demonstrates that the proposed approach is better than these existing imputation methods in terms of classification accuracy and root mean square error (RMSE) at different missing ratios.