# Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates

@article{Zhu2015SpectralSA,
title={Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates},
author={Zeyuan Allen Zhu and Zhenyu A. Liao and Lorenzo Orecchia},
journal={Proceedings of the forty-seventh annual ACM symposium on Theory of Computing},
year={2015}
}
• Published 14 June 2015
• Computer Science
• Proceedings of the forty-seventh annual ACM symposium on Theory of Computing
In this paper, we provide a novel construction of the linear-sized spectral sparsifiers of Batson, Spielman and Srivastava [11]. While previous constructions required Ω(n4) running time [11, 45], our sparsification routine can be implemented in almost-quadratic running time O(n2+ε). The fundamental conceptual novelty of our work is the leveraging of a strong connection between sparsification and a regret minimization problem over density matrices. This connection was known to provide an…
101 Citations

## Tables from this paper

### Dynamic Streaming Spectral Sparsification in Nearly Linear Time and Space

• Computer Science, Mathematics
ArXiv
• 2019
This work provides both the first efficient $\ell_2$-sparse recovery algorithm for graphs and new primitives for manipulating the effective resistance embedding of a graph, both of which it is hoped have further applications.

### Speeding Up Sparsification using Inner Product Search Data Structures

• Computer Science
ArXiv
• 2022
The heart of the work is the design of a variety of different inner product search data structures that have efficient initialization, query and update time, compatible to dimensionality reduction and robust against adaptive adversary.

### An SDP-based algorithm for linear-sized spectral sparsification

• Mathematics, Computer Science
STOC
• 2017
An algorithm is presented that outputs a (1+ε)-spectral sparsifier of G with O(n/ε2) edges in Ο(m/εO(1)) time, based on a new potential function which is much easier to compute yet has similar guarantees as the potential functions used in previous references.

### Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

• Computer Science, Mathematics
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
• 2015
This work presents the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs, using a novel combination of two techniques used in literature for constructing spectralSparsification: Random sampling by effective resistance and adaptive constructions based on barrier functions.

### Flows in almost linear time via adaptive preconditioning

• Computer Science
STOC
• 2019
This work gives an alternate approach for approximating undirected max-flow, and the first almost-linear time approximations of discretizations of total variation minimization objectives.

### Online convex optimization: algorithms, learning, and duality

• Computer Science
• 2019
Taking a bird’s-eyes view of the connections shown throughout the text, forming a “genealogy” of OCO algorithms is formed, and some possible path for future research is discussed.

### Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

We present an almost-linear time algorithm for constructing a spectral sparsifier with the number of edges linear in its number of vertices. This improves all previous constructions of linear-sized

### Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

We present an almost-linear time algorithm for constructing a spectral sparsifier with the number of edges linear in its number of vertices. This improves all previous constructions of linear-sized

### Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

We present an almost-linear time algorithm for constructing a spectral sparsifier with the number of edges linear in its number of vertices. This improves all previous constructions of linear-sized

### Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

We present an almost-linear time algorithm for constructing a spectral sparsifier with the number of edges linear in its number of vertices. This improves all previous constructions of linear-sized

## References

SHOWING 1-10 OF 51 REFERENCES

### Sparse Sums of Positive Semidefinite Matrices

• Computer Science, Mathematics
TALG
• 2015
This article considers a more general task of approximating sums of symmetric, positive semidefinite matrices of arbitrary rank and presents two deterministic, polynomial time algorithms for solving this problem.

### A Matrix Hyperbolic Cosine Algorithm and Applications

This paper generalizes Spencer's hyperbolic cosine algorithm to the matrix-valued setting, and gives an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrixSparsification, which implies an improved deterministic algorithm for spectral graph sparsify of dense graphs.

### Twice-ramanujan sparsifiers

• Mathematics, Computer Science
STOC '09
• 2009
It is proved that every graph has a spectral sparsifier with a number of edges linear in its number of vertices, and an elementary deterministic polynomial time algorithm is given for constructing H, which approximates G spectrally at least as well as a Ramanujan expander with dn/2 edges approximates the complete graph.

### Graph sparsification by effective resistances

• Mathematics, Computer Science
SIAM J. Comput.
• 2011
A key ingredient in the algorithm is a subroutine of independent interest: a nearly-linear time algorithm that builds a data structure from which the authors can query the approximate effective resistance between any two vertices in a graph in O(log n) time.

### Near-Optimal Algorithms for Online Matrix Prediction

• Computer Science
COLT
• 2012
This paper isolates a property of matrices, which is called $(\beta,\tau)$-decomposability, and derives an efficient online learning algorithm that enjoys a regret bound of $\tilde{O}(\sqrt{\beta\, \tau\,T})$ for all problems in which the comparison class is composed of $(\ beta,\Tau)$- decomposable matrices.

### Vertex Sparsifiers and Abstract Rounding Algorithms

• Computer Science
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
• 2010
It is shown that any rounding algorithm which also works for the $0$-extension relaxation can be used to construct good vertex-sparsifiers for which the optimization problem is easy, and that for many natural optimization problems, the integrality gap of the linear program is always at most $O(\log k)$ times the integralality gap restricted to trees.

### An Efficient Algorithm for Unweighted Spectral Graph Sparsification

• Computer Science, Mathematics
ArXiv
• 2014
The algorithm can efficiently compute unweighted graph sparsifiers for weighted graphs, leading to sparsified graphs that retain the weights of the original graphs.

### Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut

• Jonah Sherman
• Computer Science
2009 50th Annual IEEE Symposium on Foundations of Computer Science
• 2009
The core of the algorithm is a stronger, algorithmic version of Arora et al.'s structure theorem, where it is shown that matching-chaining argument at the heart of their proof can be viewed as an algorithm that finds good augmenting paths in certain geometric multicommodity flow networks.

### Beating the adaptive bandit with high probability

• Computer Science, Mathematics
2009 Information Theory and Applications Workshop
• 2009
We provide a principled way of proving Õ(√T) high-probability guarantees for partial-information (bandit) problems over arbitrary convex decision sets. First, we prove a regret guarantee for the