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PU Learning for Matrix Completion
 Inderjit S. Dhillon Dept of Computer Science UT Austin LowRank Bilinear Prediction
, 2015
"... In this paper, we consider the matrix completion problem when the observations are onebit measurements of some underlying matrix M, and in particular the observed samples consist only of ones and no zeros. This problem is motivated by modern applications such as recommender systems and social net ..."
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In this paper, we consider the matrix completion problem when the observations are onebit measurements of some underlying matrix M, and in particular the observed samples consist only of ones and no zeros. This problem is motivated by modern applications such as recommender systems and social networks where only “likes ” or “friendships ” are observed. The problem is an instance of PU (positiveunlabeled) learning, i.e. learning from only positive and unlabeled examples that has been studied in the context of binary classification. Under the assumption thatM has bounded nuclear norm, we provide recovery guarantees for two different observation models: 1) M parameterizes a distribution that generates a binary matrix, 2) M is thresholded to obtain a binary matrix. For the first case, we propose a “shifted matrix completion ” method that recovers M using only a subset of indices corresponding to ones; for the second case, we propose a “biased matrix completion ” method that recovers the (thresholded) binary matrix. Both methods yield strong error bounds — if M ∈ Rn×n, the error is bounded as O
Recommending Tumblr Blogs to Follow with Inductive Matrix Completion Donghyuk Shin
"... In microblogging sites, recommending blogs (users) to follow is one of the core tasks for enhancing user experience. In this paper, we propose a novel inductive matrix completion based blog recommendation method to effectively utilize multiple rich sources of evidence such as the social network and ..."
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In microblogging sites, recommending blogs (users) to follow is one of the core tasks for enhancing user experience. In this paper, we propose a novel inductive matrix completion based blog recommendation method to effectively utilize multiple rich sources of evidence such as the social network and the content as well as the activity data from users and blogs. Experiments on a largescale realworld dataset from Tumblr show the effectiveness of the proposed blog recommendation method.
MultiScale Spectral Decomposition of Massive Graphs
"... Computing the k dominant eigenvalues and eigenvectors of massive graphs is a key operation in numerous machine learning applications; however, popular solvers suffer from slow convergence, especially when k is reasonably large. In this paper, we propose and analyze a novel multiscale spectral decom ..."
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Computing the k dominant eigenvalues and eigenvectors of massive graphs is a key operation in numerous machine learning applications; however, popular solvers suffer from slow convergence, especially when k is reasonably large. In this paper, we propose and analyze a novel multiscale spectral decomposition method (MSEIGS), which first clusters the graph into smaller clusters whose spectral decomposition can be computed efficiently and independently. We show theoretically as well as empirically that the union of all cluster’s subspaces has significant overlap with the dominant subspace of the original graph, provided that the graph is clustered appropriately. Thus, eigenvectors of the clusters serve as good initializations to a block Lanczos algorithm that is used to compute spectral decomposition of the original graph. We further use hierarchical clustering to speed up the computation and adopt a fast early termination strategy to compute quality approximations. Our method outperforms widely used solvers in terms of convergence speed and approximation quality. Furthermore, our method is naturally parallelizable and exhibits significant speedups in sharedmemory parallel settings. For example, on a graph with more than 82 million nodes and 3.6 billion edges, MSEIGS takes less than 3 hours on a singlecore machine while Randomized SVD takes more than 6 hours, to obtain a similar approximation of the top50 eigenvectors. Using 16 cores, we can reduce this time to less than 40 minutes. 1