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Efficient O(n/ϵ) Spectral Sketches for the Laplacian and its Pseudoinverse [article]

Arun Jambulapati, Aaron Sidford
2018 arXiv   pre-print
In this paper we consider the problem of efficiently computing ϵ-sketches for the Laplacian and its pseudoinverse.  ...  Our algorithms improve upon the previous best sketch size of O(n / ϵ^1.6) for sketching the Laplacian form by Andoni et al (2015) and O(n / ϵ^2) for sketching the Laplacian pseudoinverse by Batson, Spielman  ...  Sketching Laplacian Pseudoinverses. In contrast to spectral sketching the Laplacian, the problem of ǫ-spectral sketching the pseudoinverse of a Laplacian has remained unexplored.  ... 
arXiv:1711.00571v2 fatcat:lo2p67i2szhpxdkb3vmfxzc5nu

Efficient Structured Matrix Recovery and Nearly-Linear Time Algorithms for Solving Inverse Symmetric M-Matrices [article]

Arun Jambulapati, Kirankumar Shiragur, Aaron Sidford
2018 arXiv   pre-print
Key results include: ∙ A nearly linear time algorithm for solving the inverse of symmetric M-matrices, a strict superset of Laplacians and SDD matrices. ∙ An Õ(n^2) time algorithm for solving n × n linear  ...  multiplies with its Laplacian matrix.  ...  Theorem 3. 6 ( 6 Laplacian Recovery and Laplacian Pseudoinverse Solver).  ... 
arXiv:1812.06295v1 fatcat:wepktj5cs5fbzbrdaef33lbirq

Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space

Jack Murtagh, Omer Reingold, Aaron Sidford, Salil Vadhan
2017 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)  
We give a deterministicÕ(log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and  ...  Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC '04; Peng and Spielman, STOC '14) with ideas used to show that UNDIRECTED S-T CONNECTIVITY is in deterministic  ...  Here we will sketch their algorithm and how we obtain a space-efficient analogue of it.  ... 
doi:10.1109/focs.2017.79 dblp:conf/focs/MurtaghRSV17 fatcat:hw3sxqd5wvgnljea3yrdqb54oy

Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space [article]

Jack Murtagh, Omer Reingold, Aaron Sidford, Salil Vadhan
2017 arXiv   pre-print
We give a deterministic Õ( n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape  ...  Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC '04; Peng and Spielman, STOC '14) with ideas used to show that Undirected S-T Connectivity is in deterministic  ...  Here we will sketch their algorithm and how we obtain a space-efficient analogue of it.  ... 
arXiv:1708.04634v1 fatcat:mooju4ooy5ezlpnhajkoyghcy4

High-precision Estimation of Random Walks in Small Space

AmirMahdi Ahmadinejad, Jonathan Kelner, Jack Murtagh, John Peebles, Aaron Sidford, Salil Vadhan
2020 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)  
graphs that is preserved under powering, and giving the first deterministic Õ(log N )-space algorithm for inverting Eulerian Laplacian matrices.  ...  The latter algorithm builds on the work of Murtagh et al. (FOCS '17) that gave a deterministic Õ(log N )-space algorithm for inverting undirected Laplacian matrices, and the work of Cohen et al.  ...  Acknowledgements We thank William Hoza for pointing out an error in an earlier version of Theorem 4.8.  ... 
doi:10.1109/focs46700.2020.00123 fatcat:oydcdvq7cjg4leezgeeljnldce

High-precision Estimation of Random Walks in Small Space [article]

AmirMahdi Ahmadinejad, Jonathan Kelner, Jack Murtagh, John Peebles, Aaron Sidford, Salil Vadhan
2022 arXiv   pre-print
graphs that is preserved under powering, and giving the first deterministic Õ(log N)-space algorithm for inverting Eulerian Laplacian matrices.  ...  The latter algorithm builds on the work of Murtagh et al. (FOCS 17) that gave a deterministic Õ(log N)-space algorithm for inverting undirected Laplacian matrices, and the work of Cohen et al.  ...  We thank William Hoza for pointing out an error in an earlier version of Theorem 4.8 and Ori Sberlo and Dean Doron for finding a bug in an earlier version of the proof of Theorem 5.9.  ... 
arXiv:1912.04524v3 fatcat:peiozxcjrff75c53cvzlx2daju

Single Pass Spectral Sparsification in Dynamic Streams

M. Kapralov, Y. T. Lee, C. N. Musco, C. P. Musco, A. Sidford
2017 SIAM journal on computing (Print)  
While O( 1 2 n polylog(n)) space algorithms are known for computing cut sparsifiers in dynamic streams [AGM12b, GKP12] and spectral sparsifiers in insertion-only streams [KL11], prior to our work, the  ...  To achieve our result, we show that, using a coarse sparsifier of G and a linear sketch of G's incidence matrix, it is possible to sample edges by effective resistance, obtaining a spectral sparsifier  ...  2150701, AFOSR grants FA9550-13-1-0042 and FA9550-12-1-0411, MADALGO center, Simons Foundation, and the Defense Advanced Research Projects Agency (DARPA).  ... 
doi:10.1137/141002281 fatcat:nehqritumnhtnoh4edudgvzjaq

COPT: Coordinated Optimal Transport for Graph Sketching [article]

Yihe Dong, Will Sawin
2020 arXiv   pre-print
This gives an unsupervised way to learn general-purpose graph representation, applicable to both graph sketching and graph comparison.  ...  Empirically, COPT outperforms state of the art methods in graph classification on both synthetic and real datasets.  ...  Given a query graph vector v q , we take its predicted class to be the class of its nearest neighbor, where the distance is determined with l 1 distance for COPT sketches, and l 2 for spectral projections  ... 
arXiv:2003.03892v2 fatcat:ahhszoaqfrbaxbssxhuxid63n4

On Differentially Private Graph Sparsification and Applications

Raman Arora, Jalaj Upadhyay
2019 Neural Information Processing Systems  
We also give an efficient private algorithm to learn Laplacian eigenmap on a graph.  ...  This is exemplified with application of the proposed meta-algorithm to graph algorithms for privately answering cut-queries, as well as practical algorithms for computing MAX-CUT and SPARSEST-CUT with  ...  Authors would like to thank Adam Smith, Lorenzo Orecchia, Cynthia Steinhardt, and Sarvagya Upadhyay for insightful discussions during the early stages of the project.  ... 
dblp:conf/nips/AroraU19 fatcat:fxeqp2sfrrhexfrarn3m3dipwm

Spectral Sparsification via Bounded-Independence Sampling [article]

Dean Doron, Jack Murtagh, Salil Vadhan, David Zuckerman
2020 arXiv   pre-print
w_max and w_min are the maximum and minimum edge weights in G, and produces a weighted graph H with Õ(n^1+2/k/ϵ^2) edges that spectrally approximates G, in the sense of Spielmen and Teng [ST04], up to  ...  We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs.  ...  The first author would like to thank Tselil Schramm and Amnon Ta-Shma for interesting conversations.  ... 
arXiv:2002.11237v2 fatcat:4t3hu4424rerrgeajkrnkdhkgy

Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation

Francois Fouss, Alain Pirotte, Jean-michel Renders, Marco Saerens
2007 IEEE Transactions on Knowledge and Data Engineering  
It turns out that the square root of the average commute time is a Euclidean distance and that the pseudoinverse of the Laplacian matrix is a kernel (it contains inner-products closely related to commute  ...  More precisely, we compute quantities (the average commute time, the pseudoinverse of the Laplacian matrix of the graph, etc) that provide similarities between any pair of nodes, having the nice property  ...  It also derives a number of interesting properties of the Laplacian pseudoinverse.  ... 
doi:10.1109/tkde.2007.46 fatcat:eps2m5uqfvhgzheuegi3t3jk5u

Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations [article]

Vasseur Xavier
2016 Zenodo  
of the Habilitation manuscript: Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications  ...  Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions.  ...  Duff and Xavier Pinel for fruitful discussions and comments.  ... 
doi:10.5281/zenodo.5813538 fatcat:jhsbgwinxngcpe53ewid6ygo7i

Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectral [article]

Minh Tang and Joshua Cape and Carey E. Priebe
2017 arXiv   pre-print
We establish asymptotic normality results for estimation of the block probability matrix B in stochastic blockmodel graphs using spectral embedding when the average degrees grows at the rate of ω(√(n))  ...  in n, the number of vertices.  ...  both estimate rk(B) and incorporate it in the subsequent estimation of B are equally flexible and generally more efficient.  ... 
arXiv:1710.10936v1 fatcat:jq7rq5ghubhy5a4liqvno2gkwm

Optimal Lower Bounds for Sketching Graph Cuts [article]

Charles Carlson and Alexandra Kolla and Nikhil Srivastava and Luca Trevisan
2017 arXiv   pre-print
We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs.  ...  Ω(n/ϵ^2) bound of Andoni et al. and matching the best known upper bound achieved by spectral sparsifiers.  ...  We will use lg and ln to denote binary and natural logarithms. We prove our lower bound for spectral sketching in Section 2, and the lower bound for cut sketching in Section 3.  ... 
arXiv:1712.10261v1 fatcat:7rrwsbu4erbmfca5szmrmpj23a

Dynamic Streaming Spectral Sparsification in Nearly Linear Time and Space [article]

Michael Kapralov and Navid Nouri and Aaron Sidford and Jakab Tardos
2019 arXiv   pre-print
This work provides both the first efficient ℓ_2-sparse recovery algorithm for graphs and new primitives for manipulating the effective resistance embedding of a graph, both of which we hope have further  ...  Our algorithm first buckets vertices of the graph by performing ball-carving using (an approximation to) its effective resistance metric, and then recovers the high effective resistance edges from a sketched  ...  Note that L = B ⊤ W B = B T n W B n , is the Laplacian matrix of G. Let L + denote the Moore-Penrose pseudoinverse of L. Also, for a real valued variable s, we define s + := max{0, s}.  ... 
arXiv:1903.12150v1 fatcat:55r4ia2nuzdbvjiwhaudzz4xse
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