The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph p-Laplacian. Unlike the existing two-node graph Laplacians, this generalization makes it possible to analyze hypergraphs, where the edges are allowed to connect any number of nodes. Moreover, we propose a semi-supervised learning method based on the proposed hypergraph p-Laplacian, and formalize them as the analogue to the Dirichlet problem, which often appears in physics. We further explore theoretical connections to normalized hypergraph cut on a hypergraph, and propose normalized cut corresponding to hypergraph p-Laplacian. The proposed p-Laplacian is shown to outperform standard hypergraph Laplacians in the experiment on a hypergraph semi-supervised learning and normalized cut setting.

Recent advances in deep domain adaptation reveal that adversarial learning can be embedded into deep networks to learn transferable features that reduce distribution discrepancy between the source and target domains. Existing domain adversarial adaptation methods based on single domain discriminator only align the source and target data distributions without exploiting the complex multimode structures. In this paper, we present a multi-adversarial domain adaptation (MADA) approach, which captures multimode structures to enable fine-grained alignment of different data distributions based on multiple domain discriminators. The adaptation can be achieved by stochastic gradient descent with the gradients computed by back-propagation in linear-time. Empirical evidence demonstrates that the proposed model outperforms state of the art methods on standard domain adaptation datasets.

In this paper, we propose a unified framework and an algorithm for the problem of group recommendation where a fixed number of items or alternatives can be recommended to a group of users. The problem of group recommendation arises naturally in many real world contexts, and is closely related to the budgeted social choice problem studied in economics. We frame the group recommendation problem as choosing a subgraph with the largest group consensus score in a completely connected graph defined over the item affinity matrix. We propose a fast greedy algorithm with strong theoretical guarantees, and show that the proposed algorithm compares favorably to the state-of-the-art group recommendation algorithms according to commonly used relevance and coverage performance measures on benchmark dataset.

Several methods of normalizing convolution kernels have been proposed in the literature to train convolutional neural networks (CNNs), and have shown some success. However, our understanding of these methods has lagged behind their success in application; there are a lot of open questions, such as why a certain type of kernel normalization is effective and what type of normalization should be employed for each (e.g., higher or lower) layer of a CNN. As the first step towards answering these questions, we propose a framework that enables us to use a variety of kernel normalization methods at any layer of a CNN. A naive integration of kernel normalization with a general optimization method, such as SGD, often entails instability while updating parameters. Thus, existing methods employ ad-hoc procedures to empirically assure convergence. In this study, we pose estimation of convolution kernels under normalization constraints as constraint-free optimization on kernel submanifolds that are identified by the employed constraints. Note that naive application of the established optimization methods for matrix manifolds to the aforementioned problems is not feasible because of the hierarchical nature of CNNs. To this end, we propose an algorithm for optimization on kernel manifolds in CNNs by appropriate scaling of the space of kernels based on structure of CNNs and statistics of data. We theoretically prove that the proposed algorithm has assurance of almost sure convergence to a solution at single minimum. Our experimental results show that the proposed method can successfully train popular CNN models using several different types of kernel normalization methods. Moreover, they show that the proposed method improves classification performance of baseline CNNs, and provides state-of-the-art performance for major image classification benchmarks.

We study the problem of within network classification, where given a partially labeled network, we infer the labels of the remaining nodes based on the link structure. Conventional loss functions penalize a node based on a function of its predicted label and target label. Such loss functions under-perform while learning on a network having overlapping classes. In relational setting, even though the ground truth is not known for the unlabeled nodes, some evidence is present in the form of labeling acquired by the nodes in their neighborhood. We propose a structural loss function for learning in networks based on the hypothesis that loss is induced when a node fails to acquire a label that is consistent with the labels of the majority of the nodes in its neighborhood. We further combine this with a novel semantic regularizer, which we call homophily regularizer, to capture the smooth transition of discriminatory power and behavior of semantically similar nodes. The proposed structural loss along with the regularizer permits relaxation labeling. Through extensive comparative study on different real-world datasets, we found that our method improves over the state-of-the-art approaches.

Just as semantic hashing can accelerate information retrieval, binary valued embeddings can significantly reduce latency in the retrieval of graphical data. We introduce a simple but effective model for learning such binary vectors for nodes in a graph. By imagining the embeddings as independent coin flips of varying bias, continuous optimization techniques can be applied to the approximate expected loss. Embeddings optimized in this fashion consistently outperform the quantization of both spectral graph embeddings and various learned real-valued embeddings, on both ranking and pre-ranking tasks for a variety of datasets.

Learning representation from relative similarity comparisons, often called ordinal embedding, gains rising attention in recent years. Most of the existing methods are batch methods designed mainly based on the convex optimization, say, the projected gradient descent method. However, they are generally time-consuming due to that the singular value decomposition (SVD) is commonly adopted during the update, especially when the data size is very large. To overcome this challenge, we propose a stochastic algorithm called SVRG-SBB, which has the following features: (a) SVD-free via dropping convexity, with good scalability by the use of stochastic algorithm, i.e., stochastic variance reduced gradient (SVRG), and (b) adaptive step size choice via introducing a new stabilized Barzilai-Borwein (SBB) method as the original version for convex problems might fail for the considered stochastic non-convex optimization problem. Moreover, we show that the proposed algorithm converges to a stationary point at a rate O (1/ T ) in our setting, where T is the number of total iterations. Numerous simulations and real-world data experiments are conducted to show the effectiveness of the proposed algorithm via comparing with the state-of-the-art methods, particularly, much lower computational cost with good prediction performance.

Multi-view clustering has attracted intensive attention due to the effectiveness of exploiting multiple views of data. However, most existing multi-view clustering methods only aim to explore the consistency or enhance the diversity of different views. In this paper, we propose a novel multi-view subspace clustering method (CSMSC), where consistency and specificity are jointly exploited for subspace representation learning. We formulate the multi-view self-representation property using a shared consistent representation and a set of specific representations, which better fits the real-world datasets. Specifically, consistency models the common properties among all views, while specificity captures the inherent difference in each view. In addition, to optimize the non-convex problem, we introduce a convex relaxation and develop an alternating optimization algorithm to recover the corresponding data representations. Experimental evaluations on four benchmark datasets demonstrate that the proposed approach achieves better performance over several state-of-the-arts.

Spectral Clustering (SC) is a widely used data clustering method which first learns a low-dimensional embedding U of data by computing the eigenvectors of the normalized Laplacian matrix, and then performs k-means on U T to get the final clustering result. The Sparse Spectral Clustering (SSC) method extends SC with a sparse regularization on UU T by using the block diagonal structure prior of UU T in the ideal case. However, encouraging UU T to be sparse leads to a heavily nonconvex problem which is challenging to solve and the work (Lu, Yan, and Lin 2016) proposes a convex relaxation in the pursuit of this aim indirectly. However, the convex relaxation generally leads to a loose approximation and the quality of the solution is not clear. This work instead considers to solve the nonconvex formulation of SSC which directly encourages UU T to be sparse. We propose an efficient Alternating Direction Method of Multipliers (ADMM) to solve the nonconvex SSC and provide the convergence guarantee. In particular, we prove that the sequences generated by ADMM always exist a limit point and any limit point is a stationary point. Our analysis does not impose any assumptions on the iterates and thus is practical. Our proposed ADMM for nonconvex problems allows the stepsize to be increasing but upper bounded, and this makes it very efficient in practice. Experimental analysis on several real data sets verifies the effectiveness of our method.

Sparse representation based classification (SRC) has gained great success in image recognition. Motivated by the fact that kernel trick can capture the nonlinear similarity of features, which may help improve the separability and margin between nearby data points, we propose Euler SRC for image classification, which is essentially the SRC with Euler sparse representation. To be specific, it first maps the images into the complex space by Euler representation, which has a negligible effect for outliers and illumination, and then performs complex SRC with Euler representation. The major advantage of our method is that Euler representation is explicit with no increase of the image space dimensionality, thereby enabling this technique to be easily deployed in real applications. To solve Euler SRC, we present an efficient algorithm, which is fast and has good convergence. Extensive experimental results illustrate that Euler SRC outperforms traditional SRC and achieves better performance for image classification.