Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of "components." Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and/or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data.Our main algorithmic results are two randomized algorithms which take as input an m × n matrix A and a rank parameter k. In our first algorithm, C is chosen, and we let A ′ = CC + A, where C + is the Moore-Penrose generalized inverse of C. In our second algorithm C, U , R are chosen, and we let A ′ = CU R. (C and R are matrices that consist of actual columns and rows, respectively, of A, and U is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least 1 − δwhere A k is the "best" rank-k approximation provided by truncating the singular value decomposition (SVD) of A, and where X F is the Frobenius norm of the matrix X. The number of columns of C and rows of R is a low-degree polynomial in k, 1/ǫ, and log(1/δ). Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants of these matrix decompositions over the last ten years. However, our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist. Both of our algorithms are simple and they take time of the order needed to approximately compute the top k singular vectors of A.The technical crux of our analysis is a novel, intuitive sampling method we introduce in this paper called "subspace sampling." In subspace sampling, the sampling probabilities depend on the Euclidean norms of the rows of the top singular vectors. This allows us to obtain provable relative-error guarantees by deconvoluting "subspace" information and "sizeof-A" information in the input matrix. This technique is likely to be useful for other matrix approximation and data analysis problems. * A preliminary version of this paper appeared in manuscript and technical report format as "Polynomial Time Algorithm for Column-Row-Based Relative-Error Low-Rank Matrix Approximation" [27,28]. Preliminary versions of parts of this paper have also appeared as conference proceedings [29,30,31]. IntroductionLarge m × n matrices are common in applications since the data often consist of m objects, each of which is described by n features. Examples of object-feature pairs include: documents and words contained in those documents; genomes and environmental conditions under which gene responses are measured; stocks and their associated temporal resolution; hyperspectral images and frequency resolution; and ...

Least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution. In a typical setting, one lets n be the number of constraints and d be the number of variables, with n d. Then, existing exact methods find a solution vector in O(nd 2 ) time. We present two randomized algorithms that provide accurate relative-error approximations to the optimal value and the solution vector of a least squares approximation problem more rapidly than existing exact algorithms. Both of our algorithms preprocess the data with the Randomized Hadamard transform. One then uniformly randomly samples constraints and solves the smaller problem on those constraints, and the other performs a sparse random projection and solves the smaller problem on those projected coordinates. In both cases, solving the smaller problem provides relative-error approximations, and, if n is sufficiently larger than d, the approximate solution can be computed in O(nd ln d) time.

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Recent advances in personalized recommendation have sparked great interest in the exploitation of rich structured information provided by knowledge graphs. Unlike most existing approaches that only focus on leveraging knowledge graphs for more accurate recommendation, we perform explicit reasoning with knowledge for decision making so that the recommendations are generated and supported by an interpretable causal inference procedure. To this end, we propose a method called Policy-Guided Path Reasoning (PGPR), which couples recommendation and interpretability by providing actual paths in a knowledge graph. Our contributions include four aspects. We first highlight the significance of incorporating knowledge graphs into recommendation to formally define and interpret the reasoning process. Second, we propose a reinforcement learning (RL) approach featuring an innovative soft reward strategy, user-conditional action pruning and a multi-hop scoring function. Third, we design a policy-guided graph search algorithm to efficiently and effectively sample reasoning paths for recommendation. Finally, we extensively evaluate our method on several large-scale real-world benchmark datasets, obtaining favorable results compared with state-of-the-art methods.

We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of 1 − 1 e ≃ 0.632, a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the 1 − 1 e bound. Our main result is a 0.67-approximation online algorithm for stochastic bipartite matching, breaking this 1− 1 e barrier. Furthermore, we show that no online algorithm can produce a 1−ǫ approximation for an arbitrarily small ǫ for this problem.Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-)matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution.At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.