Takanori Maehara

Computing personalized PageRank quickly by exploiting graph structures

We propose a new scalable algorithm that can compute Personalized PageRank (PPR) very quickly. The Power method is a state-of-the-art algorithm for computing exact PPR; however, it requires many iterations. Thus reducing the number of iterations is the main challenge. We achieve this by exploiting graph structures of web graphs and social networks. The convergence of our algorithm is very fast. In fact, it requires up to 7.5 times fewer iterations than the Power method and is up to five times faster in actual computation time. To the best of our knowledge, this is the first time to use graph structures explicitly to solve PPR quickly. Our contributions can be summarized as follows. 1. We provide an algorithm for computing a tree decomposition, which is more efficient and scalable than any previous algorithm. 2. Using the above algorithm, we can obtain a core-tree decomposition of any web graph and social network. This allows us to decompose a web graph and a social network into (1) the core, which behaves like an expander graph, and (2) a small tree-width graph, which behaves like a tree in an algorithmic sense. 3. We apply a direct method to the small tree-width graph to construct an LU decomposition. 4. Building on the LU decomposition and using it as pre-conditoner, we apply GMRES method (a state-of-the-art advanced iterative method) to compute PPR for whole web graphs and social networks.

Efficient PageRank Tracking in Evolving Networks

Real-world networks, such as the World Wide Web and online social networks, are very large and are evolving rapidly. Thus tracking personalized PageRank in such evolving networks is an important challenge in network analysis and graph mining. In this paper, we propose an efficient online algorithm for tracking personalized PageRank in an evolving network. The proposed algorithm tracks personalized PageRank accurately (i.e., within a given accuracy ε > 0). Moreover it can update the personalized PageRank scores in amortized O(1/ε) iterations for each graph modification. In addition, when m edges are randomly and sequentially inserted, the total number of iterations is expected to be O(log m/ε). We evaluated our algorithm in real-world networks. In average case, for each edge insertion and deletion, our algorithm updated the personalized PageRank in 3us in a web graph with 105M vertices and 3.7B edges, and 20ms in a social network with 42M vertices and 1.5B edges. By comparing existing state-of-the-arts algorithms, our algorithm is 2--290 times faster with an equal accuracy.

Algorithm for Error-Controlled Simultaneous Block-Diagonalization of Matrices

An algorithm is given for the problem of finding the finest simultaneous block-diagonalization of a given set of square matrices. This problem has been studied independently in the area of semidefinite programming and independent component analysis. The proposed algorithm considers the commutant algebra of the matrix *-algebra generated by the given matrices. It is simpler than other existing methods, and has the capability of controlling numerical errors. Some numerical examples are presented to demonstrate its merits.

Scalable SimRank join algorithm

Similarity join finds all pairs of objects (i, j) with similarity score s(i, j) greater than some specified threshold θ. This is a fundamental query problem in the database research community, and is used in many practical applications, such as duplicate detection, merge/purge, record linkage, object matching, and reference conciliation. In this paper, we propose a scalable approximation algorithm with an arbitrary accuracy for the similarity join problem with the SimRank similarity measure. The algorithm consists of two phases: filter and verification. The filter phase enumerates similar pair candidates, and the similarity of each candidate is then assessed in the verification phase. The scalability of the proposed algorithm is experimentally verified for large real networks. The complexity depends only on the number of similar pairs, but does not depend on all pairs O(n2). The proposed algorithm scales up to the network of 5M vertices and 70M edges. By comparing the state-of-the-art algorithms, it is about 10 times faster and it requires about 10 times smaller memory.

Risk averse submodular utility maximization

In this study, we investigate risk averse solutions to stochastic submodular utility functions. We formulate the problem as a discrete optimization problem of conditional value-at-risk, and prove hardness results for this problem.

Reply trees in Twitter: data analysis and branching process models

Structure of networks constructed from mentioning relationships between posts in online media may be valuable for understanding how information and opinions spread in these media. We crawled Twitter to collect tweets and replies to construct a large number of so-called reply trees, each of which was rooted at a tweet and joined by replies. Consistent with the previous literature, we found that the empirical trees were characterized by some long path-like reply trees, large star-like trees, and long irregular trees, although their frequencies were not high. We tested several branching process models to explain the empirical frequency of these types of reply trees as well as more basic quantities such as the distributions of the size and depth of the reply tree. Based on our modeling results, we suggest that the in-degree of the tweet that initiates a reply tree (i.e., the number of times that the tweet is directly mentioned by other reply posts) may play an important role in forming the global shape of the reply tree.

A framework of discrete DC programming by discrete convex analysis

A theoretical framework of difference of discrete convex functions (discrete DC functions) and optimization problems for discrete DC functions is established. Standard results in continuous DC theory are exported to the discrete DC theory with the use of discrete convex analysis. A discrete DC algorithm, which is a discrete analogue of the continuous DC algorithm (Concave–Convex procedure in machine learning) is proposed. The algorithm contains the submodular-supermodular procedure as a special case. Exploiting the polyhedral structure of discrete convex functions, the algorithms tailored to specific types of discrete DC functions are proposed.

Second order subspace analysis and simple decompositions

The recovery of the mixture of an N-dimensional signal generated by N independent processes is a well studied problem (see e.g. [1,10]) and robust algorithms that solve this problem by Joint Diagonalization exist. While there is a lot of empirical evidence suggesting that these algorithms are also capable of solving the case where the source signals have block structure (apart from a final permutation recovery step), this claim could not be shown yet - even more, it previously was not known if this model separable at all. We present a precise definition of the subspace model, introducing the notion of simple components, show that the decomposition into simple components is unique and present an algorithm handling the decomposition task.

Jointly learning word embeddings using a corpus and a knowledge base

Methods for representing the meaning of words in vector spaces purely using the information distributed in text corpora have proved to be very valuable in various text mining and natural language processing (NLP) tasks. However, these methods still disregard the valuable semantic relational structure between words in co-occurring contexts. These beneficial semantic relational structures are contained in manually-created knowledge bases (KBs) such as ontologies and semantic lexicons, where the meanings of words are represented by defining the various relationships that exist among those words. We combine the knowledge in both a corpus and a KB to learn better word embeddings. Specifically, we propose a joint word representation learning method that uses the knowledge in the KBs, and simultaneously predicts the co-occurrences of two words in a corpus context. In particular, we use the corpus to define our objective function subject to the relational constrains derived from the KB. We further utilise the corpus co-occurrence statistics to propose two novel approaches, Nearest Neighbour Expansion (NNE) and Hedged Nearest Neighbour Expansion (HNE), that dynamically expand the KB and therefore derive more constraints that guide the optimisation process. Our experimental results over a wide-range of benchmark tasks demonstrate that the proposed method statistically significantly improves the accuracy of the word embeddings learnt. It outperforms a corpus-only baseline and reports an improvement of a number of previously proposed methods that incorporate corpora and KBs in both semantic similarity prediction and word analogy detection tasks.

Continuous relaxation for discrete DC programming

Discrete DC programming with convex extensible functions is studied. A natural approach for this problem is a continuous relaxation that extends the problem to a continuous domain and applies the algorithm in continuous DC programming. By employing a special form of continuous relaxation, which is named “lin-vex extension,” the produced optimal solution of the extended continuous relaxation coincides with the solution of the original discrete problem. The proposed method is demonstrated for the degree-concentrated spanning tree problem, the unfair flow problem, and the correlated knapsack problem.