This paper gives an attempt to explore the manifold in the label space for multi-label learning. Traditional label space is logical, where no manifold exists. In order to study the label manifold, the label space should be extended to a Euclidean space. However, the label manifold is not explicitly available from the training examples. Fortunately, according to the smoothness assumption that the points close to each other are more likely to share a label, the local topological structure can be shared between the feature manifold and the label manifold. Based on this, we propose a novel method called ML2, i.e., Multi-Label Manifold Learning, to reconstruct and exploit the label manifold. To our best knowledge, it is one of the first attempts to explore the manifold in the label space in multi-label learning. Extensive experiments show that the performance of multi-label learning can be improved significantly with the label manifold.

In this work, we introduce a new generalized nonlinear tensor regression framework called kernel-based multiblock tensor partial least squares (KMTPLS) for predicting a set of dependent tensor blocks from a set of independent tensor blocks through the extraction of a small number of common and discriminative latent components. By considering both common and discriminative features, KMTPLS effectively fuses the information from multiple tensorial data sources and unifies the single and multiblock tensor regression scenarios into one general model. Moreover, in contrast to multilinear model, KMTPLS successfully addresses the nonlinear dependencies between multiple response and predictor tensor blocks by combining kernel machines with joint Tucker decomposition, resulting in a significant performance gain in terms of predictability. An efficient learning algorithm for KMTPLS based on sequentially extracting common and discriminative latent vectors is also presented. Finally, to show the effectiveness and advantages of our approach, we test it on the real-life regression task in computer vision, i.e., reconstruction of human pose from multiview video sequences.

Vanishing Component Analysis (VCA) is a recently proposed prominent work in machine learning. It narrows the gap between tools and computational algebra: the vanishing ideal and its applications to classification problem. In this paper, we will analyze VCA in the kernel view, which is also another important research direction in machine learning. Under a very weak assumption, we provide a different point of view to VCA and make the kernel trick on VCA become possible. We demonstrate that the projection matrix derived by VCA is located in the same space as that of Kernel Principal Component Analysis (KPCA) with a polynomial kernel. Two groups of projections can express each other by linear transformation. Furthermore, we prove that KPCA and VCA have identical discriminative power, provided that the ratio trace criteria is employed as the measurement. We also show that the kernel formulated by the inner products of VCA's projections can be expressed by the KPCA's kernel linearly. Based on the analysis above, we proposed a novel Discriminative Vanishing Component Analysis (DVCA) approach. Experimental results are provided for demonstration.

Spatial centrality, whereby samples closer to the center of a dataset tend to be closer to all other samples, is regarded as one source of hubness. Hubness is well known to degrade k-nearest-neighbor (k-NN) classification. Spatial centrality can be removed by centering, i.e., shifting the origin to the global center of the dataset, in cases where inner product similarity is used. However, when Euclidean distance is used, centering has no effect on spatial centrality because the distance between the samples is the same before and after centering. As described in this paper, we propose a solution for the hubness problem when Euclidean distance is considered. We provide a theoretical explanation to demonstrate how the solution eliminates spatial centrality and reduces hubness. We then present some discussion of the reason the proposed solution works, from a viewpoint of density gradient, which is regarded as the origin of spatial centrality and hubness. We demonstrate that the solution corresponds to flattening the density gradient. Using real-world datasets, we demonstrate that the proposed method improves k-NN classification performance and outperforms an existing hub-reduction method.

The Graph-based Convex Clustering (GCC) method has gained increasing attention recently. The GCC method adopts a fused regularizer to learn the cluster centers and obtains a geometric clusterpath by varying the regularization parameter. One major limitation is that solving the GCC model is computationally expensive. In this paper, we develop efficient graph reduction techniques for the GCC model to eliminate edges, each of which corresponds to two data points from the same cluster, without solving the optimization problem in the GCC method, leading to improved computational efficiency. Specifically, two reduction techniques are proposed according to tree-based and cyclic-graph-based convex clustering methods separately. The proposed reduction techniques are appealing since they only need to scan the data once with negligibly additional cost and they are independent of solvers for the GCC method, making them capable of improving the efficiency of any existing solver. Experiments on both synthetic and real-world datasets show that our methods can largely improve the efficiency of the GCC model.

When there are insufficient labeled samples for training a supervised model, we can adopt active learning to select the most informative samples for human labeling, or transfer learning to transfer knowledge from related labeled data source. Combining transfer learning with active learning has attracted much research interest in recent years. Most existing works follow the setting where the class labels in source domain are the same as the ones in target domain. In this paper, we focus on a more challenging cross-class setting where the class labels are totally different in two domains but related to each other in an intermediary attribute space, which is barely investigated before. We propose a novel and effective method that utilizes the attribute representation as the seed parameters to generate the classification models for classes. And we propose a joint learning framework that takes into account the knowledge from the related classes in source domain, and the information in the target domain. Besides, it is simple to perform uncertainty sampling, a fundamental technique for active learning, based on the framework. We conduct experiments on three benchmark datasets and the results demonstrate the efficacy of the proposed method.

Gaussian Conditional Random Fields (GCRF) are atype of structured regression model that incorporatesmultiple predictors and multiple graphs. This isachieved by defining quadratic term feature functions inGaussian canonical form which makes the conditionallog-likelihood function convex and hence allows findingthe optimal parameters by learning from data. In thiswork, the parameter space for the GCRF model is extendedto facilitate joint modelling of positive and negativeinfluences. This is achieved by restricting the modelto a single graph and formulating linear bounds on convexitywith respect to the models parameters. In addition,our formulation for the model using one networkallows calculating gradients much faster than alternativeimplementations. Lastly, we extend the model onestep farther and incorporate a bias term into our linkweight. This bias is solved as part of the convex optimization.Benefits of the proposed model in terms ofimproved accuracy and speed are characterized on severalsynthetic graphs with 2 million links as well as on ahospital admissions prediction task represented as a humandisease-symptom similarity network correspondingto more than 35 million hospitalization records inCalifornia over 9 years.

Matrix concentration inequalities have attracted much attention in diverse applications such as linear algebra, statistical estimation, combinatorial optimization, etc. In this paper, we present new Bernstein concentration inequalities depending only on the first moments of random matrices, whereas previous Bernstein inequalities are heavily relevant to the first and second moments. Based on those results, we analyze the empirical risk minimization in the presence of label noise. We find that many popular losses used in risk minimization can be decomposed into two parts, where the first part won't be affected and only the second part will be affected by noisy labels. We show that the influence of noisy labels on the second part can be reduced by our proposed LICS (Labeled Instance Centroid Smoothing) approach. The effectiveness of the LICS algorithm is justified both theoretically and empirically.

For the AI community, the lasso proposed by Tibshirani is an important regression approach in finding explanatory predictors in high dimensional data. The coordinate descent algorithm is a standard approach to solve the lasso which iteratively updates weights of predictors in a round-robin style until convergence. However, it has high computation cost. This paper proposes Sling, a fast approach to the lasso. It achieves high efficiency by skipping unnecessary updates for the predictors whose weight is zero in the iterations. Sling can obtain high prediction accuracy with fewer predictors than the standard approach. Experiments show that Sling can enhance the efficiency and the effectiveness of the lasso.

Stochastic partition processes for exchangeable graphs produce axis-aligned blocks on a product space. In relational modeling, the resulting blocks uncover the underlying interactions between two sets of entities of the relational data. Although some flexible axis-aligned partition processes, such as the Mondrian process, have been able to capture complex interacting patterns in a hierarchical fashion, they are still in short of capturing dependence between dimensions. To overcome this limitation, we propose the Ostomachion process (OP), which relaxes the cutting direction by allowing for oblique cuts. The partitions generated by an OP are convex polygons that can capture inter-dimensional dependence. The OP also exhibits interesting properties: 1) Along the time line the cutting times can be characterized by a homogeneous Poisson process, and 2) on the partition space the areas of the resulting components comply with a Dirichlet distribution. We can thus control the expected number of cuts and the expected areas of components through hyper-parameters. We adapt the reversible-jump MCMC algorithm for inferring OP partition structures. The experimental results on relational modeling and decision tree classification have validated the merit of the OP.