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39
Spectral Sparsification of Graphs: Theory and Algorithms
, 2013
"... Graph sparsification is the approximation of an arbitrary graph by a sparse graph. We explain what it means for one graph to be a spectral approximation of another and review the development of algorithms for spectral sparsification. In addition to being an interesting concept, spectral sparsificati ..."
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Graph sparsification is the approximation of an arbitrary graph by a sparse graph. We explain what it means for one graph to be a spectral approximation of another and review the development of algorithms for spectral sparsification. In addition to being an interesting concept, spectral sparsification has been an important tool in the design of nearly lineartime algorithms for solving systems of linear equations in symmetric, diagonally dominant matrices. The fast solution of these linear systems has already led to breakthrough results in combinatorial optimization, including a faster algorithm for finding approximate maximum flows and minimum cuts in an undirected network.
Graph stream algorithms: A survey
, 2013
"... Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was twofold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine ..."
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Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was twofold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine and b) considering graph problems yields new insights into the complexity of stream computation. However, the techniques developed in this area are now finding applications in other areas including data structures for dynamic graphs, approximation algorithms, and distributed and parallel computation. We survey the stateoftheart results; identify general techniques; and highlight some simple algorithms that illustrate basic ideas. 1.
Random Projections, Graph Sparsification, and Differential Privacy
"... Abstract. This paper initiates the study of preserving differential privacy (DP) when the dataset is sparse. We study the problem of constructing efficient sanitizer that preserves DP and guarantees high utility for answering cutqueries on graphs. The main motivation for studying sparse graphs ari ..."
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Abstract. This paper initiates the study of preserving differential privacy (DP) when the dataset is sparse. We study the problem of constructing efficient sanitizer that preserves DP and guarantees high utility for answering cutqueries on graphs. The main motivation for studying sparse graphs arises from the empirical evidences that social networking sites are sparse graphs. We also motivate and advocate the necessity to include the efficiency of sanitizers, in addition to the utility guarantee, if one wishes to have a practical deployment of privacy preserving sanitizers. We show that the technique of Blocki et al. [3] (BBDS) can be adapted to preserve DP for answering cutqueries on sparse graphs, with an asymptotically efficient sanitizer than BBDS. We use this as the base technique to construct an efficient sanitizer for arbitrary graphs. In particular, we use a preconditioning step that preserves the spectral properties (and therefore, size of any cut is preserved), and then apply our basic sanitizer.
NEARLY LINEAR TIME ALGORITHMS FOR PRECONDITIONING AND SOLVING SYMMETRIC, DIAGONALLY DOMINANT LINEAR SYSTEMS
, 2014
"... We present a randomized algorithm that on input a symmetric, weakly diagonally dominant nbyn matrix A with m nonzero entries and an nvector b produces an x ̃ such that ‖x ̃ − A†b‖A ≤ ‖A†b‖A in expected time O(m logc n log(1/)) for some constant c. By applying this algorithm inside the inverse p ..."
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We present a randomized algorithm that on input a symmetric, weakly diagonally dominant nbyn matrix A with m nonzero entries and an nvector b produces an x ̃ such that ‖x ̃ − A†b‖A ≤ ‖A†b‖A in expected time O(m logc n log(1/)) for some constant c. By applying this algorithm inside the inverse power method, we compute approximate Fiedler vectors in a similar amount of time. The algorithm applies subgraph preconditioners in a recursive fashion. These preconditioners improve upon the subgraph preconditioners first introduced by Vaidya in 1990. For any symmetric, weakly diagonally dominant matrix A with nonpositive offdiagonal entries and k ≥ 1, we construct in time O(m logc n) a preconditioner B of A with at most 2(n − 1) +O((m/k) log39 n) nonzero offdiagonal entries such that the finite generalized condition number κf (A,B) is at most k, for some other constant c. In the special case when the nonzero structure of the matrix is planar the corresponding linear system solver runs in expected time O(n log2 n+n logn log logn log(1/)). We hope that our introduction of algorithms of low asymptotic complexity will lead to the development of algorithms that are also fast in practice.
LargeScale Machine Learning for Classification and Search
, 2012
"... With the rapid development of the Internet, nowadays tremendous amounts of data including images and videos, up to millions or billions, can be collected for training machine learning models. Inspired by this trend, this thesis is dedicated to developing largescale machine learning techniques for t ..."
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With the rapid development of the Internet, nowadays tremendous amounts of data including images and videos, up to millions or billions, can be collected for training machine learning models. Inspired by this trend, this thesis is dedicated to developing largescale machine learning techniques for the purpose of making classification and nearest neighbor search practical on gigantic databases. Our first approach is to explore data graphs to aid classification and nearest neighbor search. A graph offers an attractive way of representing data and discovering the essential information such as the neighborhood structure. However, both of the graph construction process and graphbased learning techniques become computationally prohibitive at a large scale. To this end, we present an efficient large graph construction approach and subsequently apply it to develop scalable semisupervised learning and unsupervised hashing algorithms. Our unique contributions on the graphrelated topics include: 1. Large Graph Construction: Conventional neighborhood graphs such as kNN graphs require a quadratic time complexity, which is inadequate for largescale applications mentioned above. To overcome this bottleneck, we present a novel graph construction approach,
Fast spectral clustering via the nyström method
 Berlin Heidelberg
, 2013
"... Abstract. We propose and analyze a fast spectral clustering algorithm with computational complexity linear in the number of data points that is directly applicable to largescale datasets. The algorithm combines two powerful techniques in machine learning: spectral clustering algorithms and Nyström ..."
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Abstract. We propose and analyze a fast spectral clustering algorithm with computational complexity linear in the number of data points that is directly applicable to largescale datasets. The algorithm combines two powerful techniques in machine learning: spectral clustering algorithms and Nyström methods commonly used to obtain good quality low rank approximations of large matrices. The proposed algorithm applies the Nyström approximation to the graph Laplacian to perform clustering. We provide theoretical analysis of the performance of the algorithm and show the error bound it achieves and we discuss the conditions under which the algorithm performance is comparable to spectral clustering with the original graph Laplacian. We also present empirical results.
Spanners and Sparsifiers in Dynamic Streams
"... Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing spaceefficient algorithms for classica ..."
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Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing spaceefficient algorithms for classical problems such as frequency moment estimation and computing heavy hitters, and was very recently shown to be a powerful technique for solving graph problems in dynamic streams [AGM’12]. Ideally, one would like to obtain algorithms that use one or a small constant number of passes over the data and a small amount of space (i.e. sketching dimension) to preserve some useful properties of the input graph presented as a sequence of edge insertions and edge deletions. In this paper, we concentrate on the problem of constructing linear sketches of graphs that (approximately) preserve the spectral information of the graph in a few passes over the stream. We do so by giving the first sketchbased algorithm for constructing multiplicative graph spanners in only two passes over the stream. Our spanners use Õ(n1+1/k) bits of space and have stretch 2 k. While this stretch is larger than the conjectured optimal 2k − 1 for this amount of space, we show for an appropriate k that it implies the first 2pass spectral sparsifier with n 1+o(1) bits of space. Previous constructions of spectral sparsifiers in this model with a constant number of passes would require n 1+c bits of space for a constant c> 0. We also give an algorithm for constructing spanners that provides an additive approximation to the shortest path metric using a single pass over the data stream, also achieving an essentially best possible space/approximation tradeoff. 1.
Sampling GMRFs by Subgraph Correction
 In: NIPS 2012 Workshop: Perturbations, Optimization, and Statistics
, 2012
"... The problem of efficiently drawing samples from a Gaussian Markov random field is studied. We introduce the subgraph correction sampling algorithm, which makes use of any preexisting tractable sampling algorithm for a subgraph by perturbing this algorithm so as to yield asymptotically exact sample ..."
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The problem of efficiently drawing samples from a Gaussian Markov random field is studied. We introduce the subgraph correction sampling algorithm, which makes use of any preexisting tractable sampling algorithm for a subgraph by perturbing this algorithm so as to yield asymptotically exact samples for the intended distribution. The subgraph can have any structure for which efficient sampling algorithms exist: for example, treestructured, with low treewidth, or with a small feedback vertex set. Preliminary experimental results demonstrate that the subgraph correction algorithm yields accurate samples much faster than many traditional sampling methods—such as Gibbs sampling—for many graph topologies. 1