Vertex-Context Sampling for Weighted Network Embedding

In recent years, network embedding methods have garnered increasing attention because of their effectiveness in various information retrieval tasks. The goal is to learn low-dimensional representations of vertexes in an information network and simultaneously capture and preserve the network structure. Critical to the performance of a network embedding method is how the edges/vertexes of the network is sampled for the learning process. Many existing methods adopt a uniform sampling method to reduce learning complexity, but when the network is non-uniform (i.e. a weighted network) such uniform sampling incurs information loss. The goal of this paper is to present a generalized vertex sampling framework that works seamlessly with most existing network embedding methods to support weighted instead of uniform vertex/edge sampling. For efficiency, we propose a delicate sequential vertex-to-context graph data structure, such that sampling a training pair for learning takes only constant time. For scalability and memory efficiency, we design the graph data structure in a way that keeps space consumption low without requiring additional space. In addition to implementing existing network embedding methods, the proposed framework can be used to implement extensions that feature high-order proximity modeling and weighted relation modeling. Experiments conducted on three datasets, including a commercial large-scale one, verify the effectiveness and efficiency of the proposed weighted network embedding methods on a variety of tasks, including word similarity search, multi-label classification, and item recommendation.Comment: 10 page

Reducing Randomness via Irrational Numbers

We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms by Blum and Kannan and can speed up their checking algorithm for sorting programs on a large range of inputs

Collaborative Similarity Embedding for Recommender Systems

We present collaborative similarity embedding (CSE), a unified framework that exploits comprehensive collaborative relations available in a user-item bipartite graph for representation learning and recommendation. In the proposed framework, we differentiate two types of proximity relations: direct proximity and k-th order neighborhood proximity. While learning from the former exploits direct user-item associations observable from the graph, learning from the latter makes use of implicit associations such as user-user similarities and item-item similarities, which can provide valuable information especially when the graph is sparse. Moreover, for improving scalability and flexibility, we propose a sampling technique that is specifically designed to capture the two types of proximity relations. Extensive experiments on eight benchmark datasets show that CSE yields significantly better performance than state-of-the-art recommendation methods.Comment: The shorten version is accepted by WWW'1

Optimal Bid Sequences for Multiple-Object Auctions with Unequal Budgets

In a multiple-object auction, every bidder tries to win as many objects as possible with a bidding algorithm. This paper studies position-randomized auctions, which form a special class of multiple-object auctions where a bidding algorithm consists of an initial bid sequence and an algorithm for randomly permuting the sequence. We are especially concerned with situations where some bidders know the bidding algorithms of others. For the case of only two bidders, we give an optimal bidding algorithm for the disadvantaged bidder. Our result generalizes previous work by allowing the bidders to have unequal budgets. One might naturally anticipate that the optimal expected numbers of objects won by the bidders would be proportional to their budgets. Surprisingly, this is not true. Our new algorithm runs in optimal O(n) time in a straightforward manner. The case with more than two bidders is open.Comment: A preliminary version appeared in In D. T. Lee and S. H. Teng, editors, Lecture Notes in Computer Science 1969: Proceedings of the 11th Annual International Symposium on Algorithms and Computation, pages 84--95, New York, NY, 2000. Springer-Verla

Common-Face Embeddings of Planar Graphs

Given a planar graph G and a sequence C_1,...,C_q, where each C_i is a family of vertex subsets of G, we wish to find a plane embedding of G, if any exists, such that for each i in {1,...,q}, there is a face F_i in the embedding whose boundary contains at least one vertex from each set in C_i. This problem has applications to the recovery of topological information from geographical data and the design of constrained layouts in VLSI. Let I be the input size, i.e., the total number of vertices and edges in G and the families C_i, counting multiplicity. We show that this problem is NP-complete in general. We also show that it is solvable in O(I log I) time for the special case where for each input family C_i, each set in C_i induces a connected subgraph of the input graph G. Note that the classical problem of simply finding a planar embedding is a further special case of this case with q=0. Therefore, the processing of the additional constraints C_1,...,C_q only incurs a logarithmic factor of overhead.Comment: A preliminary version appeared in the Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 1999, pp. 195-20

Task-space coordinated tracking of multiple heterogeneous manipulators via controller-estimator approaches

This paper studies the task-space coordinated tracking of a time-varying leader for multiple heterogeneous manipulators (MHMs), containing redundant manipulators and nonredundant ones. Different from the traditional coordinated control, distributed controller-estimator algorithms (DCEA), which consist of local algorithms and networked algorithms, are developed for MHMs with parametric uncertainties and input disturbances. By invoking differential inclusions, nonsmooth analysis, and input-to-state stability, some conditions (including sufficient conditions, necessary and sufficient conditions) on the asymptotic stability of the task-space tracking errors and the subtask errors are developed. Simulation results are given to show the effectiveness of the presented DCEA.Comment: 17 pages, 7 figures, Journal of the Franklin Institut

Wrapped Loss Function for Regularizing Nonconforming Residual Distributions

Multi-output is essential in machine learning that it might suffer from nonconforming residual distributions, i.e., the multi-output residual distributions are not conforming to the expected distribution. In this paper, we propose "Wrapped Loss Function" to wrap the original loss function to alleviate the problem. This wrapped loss function acts just like the original loss function that its gradient can be used for backpropagation optimization. Empirical evaluations show wrapped loss function has advanced properties of faster convergence, better accuracy, and improving imbalanced data.Comment: 10 page

Sharp Inequalities between Harmonic, Seiffert, Quadratic and Contraharmonic Means

In this paper, we present the greatest values $\alpha$, $\lambda$ and $p$, and the least values $\beta$, $\mu$ and $q$ such that the double inequalities $\alpha D(a,b)+(1-\alpha)H(a,b)<T(a,b)<\beta D(a,b)+(1-\beta) H(a,b)$, $\lambda D(a,b)+(1-\lambda)H(a,b)<C(a,b)<\mu D(a,b)+(1-\mu) H(a,b)$ and $p D(a,b)+(1-p)H(a,b)0$ with $a\neq b$, where $H(a,b)=2ab/(a+b)$, $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$, $Q(a,b)=\sqrt{(a^2+b^2)/2}$, $C(a,b)=(a^2+b^2)/(a+b)$ and $D(a,b)=(a^3+b^3)/(a^2+b^2)$ are the harmonic, Seiffert, quadratic, first contraharmonic and second contraharmonic means of $a$ and $b$, respectively.Comment: 11 page

On the Ramsey Numbers for Bipartite Multigraphs

A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced subgraph with respect to color $c$ is complete.) In this paper, we investigate a variant of the Ramsey problem for the class of complete bipartite multigraphs. (By a multigraph we mean a graph in which multiple edges, but no loops, are allowed.) Unlike the conventional m-coloring scheme in Ramsey theory which imposes a constraint (i.e., $m$) on the total number of colors allowed in a graph, we introduce a relaxed version called m-local coloring which only requires that, for every vertex $v$, the number of colors associated with $v$'s incident edges is bounded by $m$. Note that the number of colors found in a graph under $m$-local coloring may exceed m. We prove that given any $n \times n$ complete bipartite multigraph $G$, every shuffle-preserved $m$-local coloring displays a monochromatic copy of $K_{p,p}$ provided that $2(p-1)(m-1) < n$. Moreover, the above bound is tight when (i) $m=2$, or (ii) $n=2^k$ and $m=3\cdot 2^{k-2}$ for every integer $k\geq 2$. As for the lower bound of $p$, we show that the existence of a monochromatic $K_{p,p}$ is not guaranteed if $p> \lceil \frac{n}{m} \rceil$. Finally, we give a generalization for $k$-partite graphs and a method applicable to general graphs. Many conclusions found in $m$-local coloring can be inferred to similar results of $m$-coloring.Comment: 10 pages, 3 figure

Service Overlay Forest Embedding for Software-Defined Cloud Networks

Network Function Virtualization (NFV) on Software-Defined Networks (SDN) can effectively optimize the allocation of Virtual Network Functions (VNFs) and the routing of network flows simultaneously. Nevertheless, most previous studies on NFV focus on unicast service chains and thereby are not scalable to support a large number of destinations in multicast. On the other hand, the allocation of VNFs has not been supported in the current SDN multicast routing algorithms. In this paper, therefore, we make the first attempt to tackle a new challenging problem for finding a service forest with multiple service trees, where each tree contains multiple VNFs required by each destination. Specifically, we formulate a new optimization, named Service Overlay Forest (SOF), to minimize the total cost of all allocated VNFs and all multicast trees in the forest. We design a new $3\rho_{ST}$-approximation algorithm to solve the problem, where $\rho_{ST}$ denotes the best approximation ratio of the Steiner Tree problem, and the distributed implementation of the algorithm is also presented. Simulation results on real networks for data centers manifest that the proposed algorithm outperforms the existing ones by over 25%. Moreover, the implementation of an experimental SDN with HP OpenFlow switches indicates that SOF can significantly improve the QoE of the Youtube service.Comment: Technical Repor