In this paper we present a novel geometric method for the problem of global pairwise alignment of protein-protein interaction (PPI) networks. A PPI network can be viewed as a node-edge graph and its alignment often needs to solve some generalized version of the subgraph isomorphism problem which is notoriously challenging and NP-hard. All existing research has focused on designing algorithms with good practical performance. In this paper we propose a two-step algorithm for the global pairwise PPI network alignment which consists of a Geometric Step and an MCMF Step. Our algorithm first applies a graph embedding technique that preserves the topological structure of the original PPI networks and maps the problem from graph domain to geometric domain, and computes a rigid transformation for one of the embedded PPI networks so as to minimize its Earth Mover's Distance (EMD) to the other PPI network. It then solves a Min-Cost Max-Flow problem using the (scaled) inverse of sequence similarity scores as edge weight. By using the flow values from the two steps (i.e., EMD and Min-Cost Max-Flow) as the matching scores, we are able to combine the two matching results to obtain the desired alignment. Unlike other popular alignment algorithms which are either greedy or incremental, our algorithm globally optimizes the problem to yield an alignment with better quality.

In this paper, we study a generalization of the classical Voronoi diagram, called the clustering induced Voronoi diagram (CIVD). Different from the traditional model, CIVD takes as its sites the power set $U$ of an input set $P$ of objects. For each subset $C$ of $P$, CIVD uses an influence function $F(C,q)$ to measure the total (or joint) influence of all objects in $C$ on an arbitrary point $q$ in the space $\mathbb{R}^d$ and determines the influence-based Voronoi cell in $\mathbb{R}^d$ for $C$. This generalized model offers a number of new features (e.g., simultaneous clustering and space partition) to the Voronoi diagram which are useful in various new applications. We investigate the general conditions for the influence function which ensure the existence of a small-size (e.g., nearly linear) approximate CIVD for a set $P$ of $n$ points in $\mathbb{R}^d$ for some fixed $d$. To construct CIVD, we first present a stand-alone new technique, called approximate influence (AI) decomposition, for the general CIVD problem. With only $O(n\log n)$ time, the AI decomposition partitions the space $\mathbb{R}^{d}$ into a nearly linear number of cells so that all points in each cell receive their approximate maximum influence from the same (possibly unknown) site (i.e., a subset of $P$). Based on this technique, we develop assignment algorithms to determine a proper site for each cell in the decomposition and form various $(1-\epsilon)$-approximate CIVDs for some small fixed $\epsilon>0$. Particularly, we consider two representative CIVD problems, vector CIVD and density-based CIVD, and show that both of them admit fast assignment algorithms; consequently, their $(1-\epsilon)$-approximate CIVDs can be built in $O(n \log^{\max\{3,d+1\}}n)$ and $O(n \log^{2} n)$ time, respectively.

In this paper we consider the problem of training a Support Vector Machine (SVM) online using a stream of data in random order. We provide a fast online training algorithm for general SVM on very large datasets. Based on the geometric interpretation of SVM known as the polytope distance, our algorithm uses a gradient descent procedure to solve the problem. With high probability our algorithm outputs an (ε; δ)-approximation result in constant time and space, which is independent of the size of the dataset, where (ε; δ)-approximation means that the separating margin of the classifier is almost optimal (with error ≤ ε), and the number of misclassified training points is very small (with error ≤ δ). Experimental results show that our algorithm outperforms most of existing online algorithms, especially in the space requirement aspect, while maintaining high accuracy.

In this paper, we consider the distributed version of Support Vector Machine (SVM) under the coordinator model, where all input data (i.e., points in R^d space) of SVM are arbitrarily distributed among k nodes in some network with a coordinator which can communicate with all nodes. We investigate two variants of this problem, with and without outliers. For distributed SVM without outliers, we prove a lower bound on the communication complexity and give a distributed (1-epsilon)-approximation algorithm to reach this lower bound, where epsilon is a user specified small constant. For distributed SVM with outliers, we present a (1-epsilon)-approximation algorithm to explicitly remove the influence of outliers. Our algorithm is based on a deterministic distributed top t selection algorithm with communication complexity of O(k log (t)) in the coordinator model. Experimental results on benchmark datasets confirm the theoretical guarantees of our algorithms.

In this paper, we study a generalization of the classical Voronoi diagram, called clustering induced Voronoi diagram (CIVD). Different from the traditional model, CIVD takes as its sites the power set U of an input set P of objects. For each subset C of P, CIVD uses an influence function F(C, q) to measure the total (or joint) influence of all objects in C on an arbitrary point q in the space ℝd, and determines the influence-based Voronoi cell in ℝdfor C. This generalized model offers a number of new features (e.g., simultaneous clustering and space partition) to Voronoi diagram which are useful in various new applications. We investigate the general conditions for the influence function which ensure the existence of a small-size (e.g., nearly linear) approximate CIVD for a set P of n points in ℝdfor some fixed d. To construct CIVD, we first present a standalone new technique, called approximate influence (AI) decomposition, for the general CIVD problem. With only O(n log n) time, the AI decomposition partitions the space ℝdinto a nearly linear number of cells so that all points in each cell receive their approximate maximum influence from the same (possibly unknown) site (i.e., a subset of P). Based on this technique, we develop assignment algorithms to determine a proper site for each cell in the decomposition and form various (1-ε)-approximate CIVDs for some small fixed € > 0. Particularly, we consider two representative CIVD problems, vector CIVD and density-based CIVD, and show that both of them admit fast assignment algorithms; consequently, their (1 - €)-approximate CIVDs can be built in O(n logd+1n) and O(n log2n) time, respectively.