Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Lee, Yin Tat; Sun, He

doi:10.1109/focs.2015.24

Cited by 48 publications

(90 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This problem can be solved in time O(mn 1/3 ) using the multiplicative weights framework as applied in [7,6]. The resulting algorithm for implementing the k-oracle would take time O(m + n 4/3 ), by taking advantage of spectral sparsification algorithms [49,32,31]. Instead, we will come up with a faster algorithm.…”

Section: Implementing An O(log N)-oracle In Nearly Linear Timementioning

confidence: 99%

Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

Cohen

Mądry

Tsipras

2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time O m log κ log 2 (1/ε) where ε is the amount of error we are willing to tolerate. Here, κ represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever κ is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time O(m 3/2 log(1/ε)).In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.

show abstract

Section: Implementing An O(log N)-oracle In Nearly Linear Timementioning

confidence: 99%

Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

Cohen

Mądry

Tsipras

2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

show abstract

“…Note that an -spectral sparsifier is also an -cut sparsifier. Cut sparsification and spectral sparsification of graphs have been studied intensively [1,2,17,21,22,29], and it is known that any graph with n vertices can be spectrally sparsified with O(n/ ) edges [17] or O(n/ 2 ) edges [22], where O(·) hides a polylogarithmic factor.…”

Section: Introductionmentioning

confidence: 99%

Spectral Sparsification of Hypergraphs

Soma¹,

Yoshida²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

For an undirected/directed hypergraph G = (V, E), its Laplacian L G : R V → R V is defined such that its "quadratic form" x L G (x) captures the cut information of G. In particular,In this paper, we present a polynomial-time algorithm that, given an undirected/directed hypergraph G on n vertices, constructs an -spectral sparsifier of G with O(n 3 log n/ 2 ) hyperedges/hyperarcs.The proposed spectral sparsification can be used to improve the time and space complexities of algorithms for solving problems that involve the quadratic form, such as computing the eigenvalues of L G , computing the effective resistance between a pair of vertices in G, semisupervised learning based on L G , and cut problems on G. In addition, our sparsification result implies that any submodular function f : 2 V → R + with f (∅) = f (V ) = 0 can be concisely represented by a directed hypergraph. Accordingly, we show that, for any distribution, we can properly and agnostically learn submodular functions f : 2 V → [0, 1] with f (∅) = f (V ) = 0, with O(n 4 log(n/ )/ 4 ) samples.

show abstract

“…Spectral sparsifiers approximately preserve the eigenvalues of the original matrix, preserve all cuts in the graphs associated with the Laplacian matrices, serve as good preconditioners for solving Laplacian linear systems, and more. Furthermore, spectral sparsifiers preserve the quadratic form of the pseudoinverse and therefore preserve many natural measures of distance between vertices, like effective resistance distances and roundtrip commute times.Given their numerous applications, obtaining faster algorithms for constructing sparser ǫ-spectral sparsiers has been an incredibly active area of research over the past decade [3,4,2,9,10,11,12]. Recently, this work has culminated in the results of [12] which showed how to construct ǫ-spectral sparsifiers with O(n/ǫ 2 )-non-zero entries in nearly linear time.…”

mentioning

confidence: 99%

“…Given their numerous applications, obtaining faster algorithms for constructing sparser ǫ-spectral sparsiers has been an incredibly active area of research over the past decade [3,4,2,9,10,11,12]. Recently, this work has culminated in the results of [12] which showed how to construct ǫ-spectral sparsifiers with O(n/ǫ 2 )-non-zero entries in nearly linear time.…”

mentioning

confidence: 99%

Efficient Õ(n/∊) Spectral Sketches for the Laplacian and its Pseudoinverse

Jambulapati

Sidford

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

In this paper we consider the problem of efficiently computing -sketches for the Laplacian and its pseudoinverse. Given a Laplacian and an error tolerance , we seek to construct a function f such that for any vector x (chosen obliviously from f ), with high probability (1 − )x Ax ≤ f (x) ≤ (1 + )x Ax where A is either the Laplacian or its pseudoinverse. Our goal is to construct such a sketch f efficiently and to store it in the least space possible.We provide nearly-linear time algorithms that, when given a Laplacian matrix L ∈ R n×n and an error tolerance , produceÕ(n/ )-size sketches of both L and its pseudoinverse. Our algorithms improve upon the previous best sketch size of O(n/

show abstract

Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Cited by 48 publications

References 26 publications

Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

Spectral Sparsification of Hypergraphs

Efficient Õ(n/∊) Spectral Sketches for the Laplacian and its Pseudoinverse

Contact Info

Product

Resources

About