Efficient Frequent Directions Algorithm for Sparse Matrices

Ghashami, Mina; Liberty, Edo; Phillips, Jeff M.

doi:10.1145/2939672.2939800

Cited by 24 publications

(26 citation statements)

References 45 publications

(58 reference statements)

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“…Fiber subset selection, also called tensor cross approximation (TCA), finds a small subset of fibers which approximates the entire data tensor. For the matrix case, this problem is known as the Column/Row Subset Selection or CUR Problem which has been thoroughly investigated and for which there exist several algorithms with almost matching lower bounds [64,82,140].…”

Section: Tensor Sketching Using Tucker Modelmentioning

confidence: 99%

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Cichocki

Lee

Oseledets

et al. 2016

FNT in Machine Learning

373

342

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Modern applications in engineering and data science are increasingly based on multidimensional data of exceedingly high volume, variety, and structural richness. However, standard machine learning algorithms typically scale exponentially with data volume and complexity of cross-modal couplings - the so called curse of dimensionality - which is prohibitive to the analysis of large-scale, multi-modal and multi-relational datasets. Given that such data are often efficiently represented as multiway arrays or tensors, it is therefore timely and valuable for the multidisciplinary machine learning and data analytic communities to review low-rank tensor decompositions and tensor networks as emerging tools for dimensionality reduction and large scale optimization problems. Our particular emphasis is on elucidating that, by virtue of the underlying low-rank approximations, tensor networks have the ability to alleviate the curse of dimensionality in a number of applied areas. In Part 1 of this monograph we provide innovative solutions to low-rank tensor network decompositions and easy to interpret graphical representations of the mathematical operations on tensor networks. Such a conceptual insight allows for seamless migration of ideas from the flat-view matrices to tensor network operations and vice versa, and provides a platform for further developments, practical applications, and non-Euclidean extensions. It also permits the introduction of various tensor network operations without an explicit notion of mathematical expressions, which may be beneficial for many research communities that do not directly rely on multilinear algebra. Our focus is on the Tucker and tensor train (TT) decompositions and their extensions, and on demonstrating the ability of tensor networks to provide linearly or even super-linearly (e.g., logarithmically) scalable solutions, as illustrated in detail in Part 2 of this monograph

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Section: Tensor Sketching Using Tucker Modelmentioning

confidence: 99%

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Cichocki

Lee

Oseledets

et al. 2016

FNT in Machine Learning

373

342

View full text Add to dashboard Cite

show abstract

“…Graph sketches [Ahn et al 2012;Liberty 2013;Ghashami et al 2016], or data synopses obtained by applying linear projections, are also relevant. Graph sketching can be viewed as linear dimensionality reduction, where the linearity of sketches makes them applicable to the analysis of streaming graphs with node and edge additions and deletions and distributed settings, such as MapReduce [Dean and Ghemawat 2004].…”

Section: Simplification-based Methodsmentioning

confidence: 99%

Graph Summarization Methods and Applications

et al. 2018

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While advances in computing resources have made processing enormous amounts of data possible, human ability to identify patterns in such data has not scaled accordingly. Efficient computational methods for condensing and simplifying data are thus becoming vital for extracting actionable insights. In particular, while data summarization techniques have been studied extensively, only recently has summarizing interconnected data, or graphs, become popular. This survey is a structured, comprehensive overview of the state-of-the-art methods for summarizing graph data. We first broach the motivation behind, and the challenges of, graph summarization. We then categorize summarization approaches by the type of graphs taken as input and further organize each category by core methodology. Finally, we discuss applications of summarization on real-world graphs and conclude by describing some open problems in the field.

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“…In summary, these techniques estimate the properties of the original graph, estimate relative frequencies of its substructures and then create a small sample subgraph that resembles the original graph. Also, there are techniques [10,22] that use linear dimensionality reduction on the complex graph to generate simplified graph sketches or data synopses. Grouping-based methods.…”

Section: Related Workmentioning

confidence: 99%

Utility-driven graph summarization

Kumar

Efstathopoulos

2018

Proc. VLDB Endow.

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A lot of the large datasets analyzed today represent graphs. In many real-world applications, summarizing large graphs is beneficial (or necessary) so as to reduce a graph's size and, thus, achieve a number of benefits, including but not limited to 1) significant speed-up for graph algorithms, 2) graph storage space reduction, 3) faster network transmission, 4) improved data privacy, 5) more effective graph visualization, etc. During the summarization process, potentially useful information is removed from the graph (nodes and edges are removed or transformed). Consequently, one important problem with graph summarization is that, although it reduces the size of the input graph, it also adversely affects and reduces its utility. The key question that we pose in this paper is, can we summarize and compress a graph while ensuring that its utility or usefulness does not drop below a certain user-specified utility threshold? We explore this question and propose a novel iterative utilitydriven graph summarization approach. During iterative summarization, we incrementally keep track of the utility of the graph summary. This enables a user to query a graph summary that is conditioned on a user-specified utility value. We present both exhaustive and scalable approaches for implementing our proposed solution. Our experimental results on real-world graph datasets show the effectiveness of our proposed approach. Finally, through multiple real-world applications we demonstrate the practicality of our notion of utility of the computed graph summary.

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Efficient Frequent Directions Algorithm for Sparse Matrices

Cited by 24 publications

References 45 publications

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Graph Summarization Methods and Applications

Utility-driven graph summarization

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