Recently a deterministic method, frequent directions (FD) is proposed to solve the high dimensional low rank approximation problem. It works well in practice, but experiences high computational cost. In this paper, we establish a fast frequent directions algorithm for the low rank approximation problem, which implants a randomized algorithm, sparse subspace embedding (SpEmb) in FD. This new algorithm makes use of FD's natural block structure and sends more information through SpEmb to each block in FD. We prove that our new algorithm produces a good low rank approximation with a sketch of size linear on the rank approximated. Its effectiveness and efficiency are demonstrated by the experimental results on both synthetic and real world datasets, as well as applications in network analysis.
Since its introduction by Clarkson and Woodruff @cite_28 , SpEmb has received great attention due to its efficient time complexity. It has been extensively used as a randomization technique to solve numerical linear algebraic problems, here we look at its application on low rank approximation.
@cite_28
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An "oblivious subspace embedding (OSE)" given some parameters eps,d is a distribution D over matrices B in R^ m x n such that for any linear subspace W in R^n with dim(W) = d it holds that Pr_ B D (forall x in W ||B x||_2 in (1 + - eps)||x||_2) > 2 3 We show an OSE exists with m = O(d^2 eps^2) and where every B in the support of D has exactly s=1 non-zero entries per column. This improves previously best known bound in [Clarkson-Woodruff, arXiv:1207.6365]. Our quadratic dependence on d is optimal for any OSE with s=1 [Nelson-Nguyen, 2012]. We also give two OSE's, which we call Oblivious Sparse Norm-Approximating Projections (OSNAPs), that both allow the parameter settings m = O(d eps^2) and s = polylog(d) eps, or m = O(d^ 1+gamma eps^2) and s=O(1 eps) for any constant gamma>0. This m is nearly optimal since m >= d is required simply to no non-zero vector of W lands in the kernel of B. These are the first constructions with m=o(d^2) to have s=o(d). In fact, our OSNAPs are nothing more than the sparse Johnson-Lindenstrauss matrices of [Kane-Nelson, SODA 2012]. Our analyses all yield OSE's that are sampled using either O(1)-wise or O(log d)-wise independent hash functions, which provides some efficiency advantages over previous work for turnstile streaming applications. Our main result is essentially a Bai-Yin type theorem in random matrix theory and is likely to be of independent interest: i.e. we show that for any U in R^ n x d with orthonormal columns and random sparse B, all singular values of BU lie in [1-eps, 1+eps] with good probability. Plugging OSNAPs into known algorithms for numerical linear algebra problems such as approximate least squares regression, low rank approximation, and approximating leverage scores implies faster algorithms for all these problems. Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis, and this work was performed by individuals from many different research communities. This monograph will provide a detailed overview of recent work on the theory of randomized matrix algorithms as well as the application of those ideas to the solution of practical problems in large-scale data analysis. An emphasis will be placed on a few simple core ideas that underlie not only recent theoretical advances but also the usefulness of these tools in large-scale data applications. Crucial in this context is the connection with the concept of statistical leverage. This concept has long been used in statistical regression diagnostics to identify outliers; and it has recently proved crucial in the development of improved worst-case matrix algorithms that are also amenable to high-quality numerical implementation and that are useful to domain scientists. Randomized methods solve problems such as the linear least-squares problem and the low-rank matrix approximation problem by constructing and operating on a randomized sketch of the input matrix. Depending on the specifics of the situation, when compared with the best previously-existing deterministic algorithms, the resulting randomized algorithms have worst-case running time that is asymptotically faster; their numerical implementations are faster in terms of clock-time; or they can be implemented in parallel computing environments where existing numerical algorithms fail to run at all. Numerous examples illustrating these observations will be described in detail. In this paper we consider sparsity on a tensor level, as given by the n-rank of a tensor. In an important sparse-vector approximation problem (compressed sensing) and the low-rank matrix recovery problem, using a convex relaxation technique proved to be a valuable solution strategy. Here, we will adapt these techniques to the tensor setting. We use the n-rank of a tensor as a sparsity measure and consider the low-n-rank tensor recovery problem, i.e. the problem of finding the tensor of the lowest n-rank that fulfills some linear constraints. We introduce a tractable convex relaxation of the n-rank and propose efficient algorithms to solve the low-n-rank tensor recovery problem numerically. The algorithms are based on the Douglas–Rachford splitting technique and its dual variant, the alternating direction method of multipliers. Censor- [PODC’15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in the past few years. This paper develops further algebraic techniques for designing algorithms in this model. We present deterministic and randomized algorithms, in the congested clique model, for efficiently computing multiple independent instances of matrix products, computing the determinant, the rank and the inverse of a matrix, and solving systems of linear equations. As applications of these techniques, we obtain more efficient algorithms for the computation, again in the congested clique model, of the all-pairs shortest paths and the diameter in directed and undirected graphs with small weights, improving over Censor-’s work. We also obtain algorithms for several other graph-theoretic problems such as computing the number of edges in a maximum matching and the Gallai-Edmonds decomposition of a simple graph, and computing a minimum vertex cover of a bipartite graph.