The Laplacian matrix of a graph can be used in many areas of mathematical research and has a physical interpretation in various theories. However, there are a few open issues in the Laplacian graph construction: (i) Selecting the appropriate scale of analysis, (ii) Selecting the appropriate number of neighbors, (iii) Handling multiscale data, and, (iv) Dealing with noise and outliers. In this paper, we propose that the affinity between pairs of samples could be computed using sparse representation with proper constraints. This parameter free setting automatically produces the Laplacian graph, leads to significant reduction in computation cost and robustness to the outliers and noise. We further provide an efficient algorithm to solve the difficult optimization problem based on improvement of existing algorithms. To demonstrate our motivation, we conduct spectral clustering experiments with benchmark methods. Empirical experiments on 9 data sets demonstrate the effectiveness of our method.

Multi-task clustering and multi-view clustering have severally found wide applications and received much attention in recent years. Nevertheless, there are many clustering problems that involve both multi-task clustering and multi-view clustering, i.e., the tasks are closely related and each task can be analyzed from multiple views. In this paper, for non-negative data (e.g., documents), we introduce a multi-task multi-view clustering (MTMVC) framework which integrates within-view-task clustering, multi-view relationship learning and multi-task relationship learning. We then propose a specific algorithm to optimize the MTMVC framework. Experimental results show the superiority of the proposed algorithm over either multi-task clustering algorithms or multi-view clustering algorithms for multi-task clustering of multi-view data.

As an important machine learning topic, dimensionality reduction has been widely studied and utilized in various kinds of areas. A multitude of dimensionality reduction methods have been developed, among which unsupervised dimensionality reduction is more desirable when obtaining label information requires onerous work. However, most previous unsupervised dimensionality reduction methods call for an affinity graph constructed beforehand, with which the following dimensionality reduction steps can be then performed. Separation of graph construction and dimensionality reduction leads the dimensionality reduction process highly dependent on quality of the input graph. In this paper, we propose a novel graph embedding method for unsupervised dimensionality reduction. We simultaneously conduct dimensionality reduction along with graph construction by assigning adaptive and optimal neighbors according to the projected local distances. Our method doesn’t need an affinity graph constructed in advance, but instead learns the graph concurrently with dimensionality reduction. Thus, the learned graph is optimal for dimensionality reduction. Meanwhile, our learned graph has an explicit block diagonal structure, from which the clustering results could be directly revealed without any postprocessing steps. Extensive empirical results on dimensionality reduction as well as clustering are presented to corroborate the performance of our method.

We propose a Multi-task Multi-dimensional Hawkes Process (MMHP) for modeling event sequences where there exist multiple triggering patterns within sequences and structures across sequences.MMHP is able to model the dynamics of multiple sequences jointly by imposing structural constraints and thus systematically uncover clustering structure among sequences.We propose an effective and robust optimization algorithm to learn MMHP models, which takes advantage of alternating direction method of multipliers (ADMM), majorization minimization and Euler-Lagrange equations.Our experimental results demonstrate that MMHP performs well on both synthetic and real data

We address an ensemble clustering problem, where reliable clusters are locally embedded in given multiple partitions. We propose a new nonnegative matrix factorization (NMF)-based method, in which locally reliable clusters are explicitly considered by using instance-wise weights over clusters. Our method factorizes the input cluster assignment matrix into two matrices H and W, which are optimized by iteratively 1) updating H and W while keeping the weight matrix constant and 2) updating the weight matrix while keeping H and W constant, alternatively. The weights in the second step were updated by solving a convex problem, which makes our algorithm significantly faster than existing NMF-based ensemble clustering methods. We empirically proved that our method outperformed a lot of cutting-edge ensemble clustering methods by using a variety of datasets.

Transfer learning has attracted a lot of attention in the past decade. One crucial research issue in transfer learning is how to find a good representation for instances of different domains such that the divergence between domains can be reduced with the new representation. Recently, deep learning has been proposed to learn more robust or higher-level features for transfer learning. However, to the best of our knowledge, most of the previous approaches neither minimize the difference between domains explicitly nor encode label information in learning the representation. In this paper, we propose a supervised representation learning method based on deep autoencoders for transfer learning. The proposed deep autoencoder consists of two encoding layers: an embedding layer and a label encoding layer. In the embedding layer, the distance in distributions of the embedded instances between the source and target domains is minimized in terms of KL-Divergence. In the label encoding layer, label information of the source domain is encoded using a softmax regression model. Extensive experiments conducted on three real-world image datasets demonstrate the effectiveness of our proposed method compared with several state-of-the-art baseline methods.

Multi-view outlier detection is a challenging problem due to the inconsistent behaviors and complicated distributions of samples across different views. The existing approaches are designed to identify the outlier exhibiting inconsistent characteristics across different views. However, due to the inevitable system errors caused by data-captured sensors or others, there always exists another type of outlier, which consistently behaves abnormally in individual view. Unfortunately, this kind of outlier is neglected by all the existing multi-view outlier detection methods, consequently their outlier detection performances are dramatically harmed.In this paper, we propose a novel Dual-regularized Multi-view Outlier Detection method (DMOD) to detect both kinds of anomalies simultaneously. By representing the multi-view data with latent coefficients and sample-specific errors, we characterize each kind of outlier explicitly. Moreover, an outlier measurement criterion is well-designed to quantify the inconsistency. To solve the proposed non-smooth model, a novel optimization algorithm is proposed in an iterative manner. We evaluate our method on five datasets with different outlier settings. The consistent superior results to other state-of-the-art methods demonstrate the effectiveness of our approach.

It is also useful for scientific discovery like in neuroscience, like correlation of X, Y only test for (univariate) to see if a stimulus X (say an image) is independent linear independence, natural alternatives like of the brain activity Y (say fMRI) in a relevant part of mutual information of X, Y are hard to estimate the brain. Since detecting nonlinear correlations is much easier due to a serious curse of dimensionality. A recent than estimating a nonparametric regression function (of approach, avoiding both issues, estimates norms of Y onto X), it can be done at smaller sample sizes, with further an operator in Reproducing Kernel Hilbert Spaces samples collected for estimation only if an effect is detected (RKHSs). Our main contribution is strong empirical by the hypothesis test. For such situations, correlation evidence that by employing shrunk operators only tests for univariate linear independence, while other when the sample size is small, one can attain an improvement statistics like mutual information that do characterize multivariate in power at low false positive rates. We independence are hard to estimate from data, suffering analyze the effects of Stein shrinkage on a popular from a serious curse of dimensionality. A recent popular test statistic called HSIC (Hilbert-Schmidt Independence approach for this problem (and a related two-sample testing Criterion). Our observations provide insights problem) involve the use of quantities defined in reproducing into two recently proposed shrinkage estimators, kernel Hilbert spaces (RKHSs) - see [Gretton et al., 2006; SCOSE and FCOSE - we prove that SCOSE Harchaoui et al., 2007; Gretton et al., 2005b; 2005a].

The goal of subspace segmentation is to partition a set of data drawn from a union of subspace into their underlying subspaces. The performance of  spectral clustering based approaches heavily depends on learned data affinity matrices, which are usually constructed either directly from the raw data or from their computed representations. In this paper, we propose a novel method to simultaneously learn the representations of data and the affinity matrix of representation in a unified optimization framework. A novel Augmented Lagrangian Multiplier based algorithm is designed to effectively and efficiently seek the optimal solution of the problem. The experimental results on both synthetic and real data demonstrate the efficacy of the proposed method and its superior performance over the state-of-the-art alternatives.

In this paper, we propose a stratified topic model (STM). Instead of directly modeling and inferring flat topics or hierarchically structured topics, we use the stratified relationships in topic hierarchies to regularize the flat topics. The topic structures are captured by a hierarchical clustering method and play as constraints during the learning process. We propose two theoretically sound and practical inference methods to solve the model. Experimental results with two real world data sets and various evaluation metrics demonstrate the effectiveness of the proposed model.