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A Scalable CUR Matrix Decomposition Algorithm: Lower Time Complexity and Tighter Bound [article]

Shusen Wang, Zhihua Zhang, Jian Li
2012 arXiv   pre-print
The proposed algorithm has the advantages over the existing relative-error CUR algorithms that it possesses tighter theoretical bound and lower time complexity, and that it can avoid maintaining the whole  ...  The CUR matrix decomposition is an important extension of Nyström approximation to a general matrix. It approximates any data matrix in terms of a small number of its columns and rows.  ...  The time cost of the fast CUR algorithm is the sum of Stage 1, Stage 2, and the Moore-  ... 
arXiv:1210.1461v1 fatcat:xhuvuacmczeithgapiqpxgpac4

Improving CUR Matrix Decomposition and the Nyström Approximation via Adaptive Sampling [article]

Shusen Wang, Zhihua Zhang
2013 arXiv   pre-print
The proposed CUR and Nyström algorithms also have low time complexity and can avoid maintaining the whole data matrix in RAM.  ...  In this paper we establish a more general error bound for the adaptive column/row sampling algorithm, based on which we propose more accurate CUR and Nyström algorithms with expected relative-error bounds  ...  Acknowledgments This work has been supported in part by the Natural Science Foundations of China (No. 61070239) and the Scholarship Award for Excellent Doctoral Student granted by Chinese Ministry of Education  ... 
arXiv:1303.4207v7 fatcat:7bsqa2je7fedfedyeg5ogkvpsy

Mathematical and Algorithmic Analysis of Network and Biological Data [article]

Charalampos E. Tsourakakis
2014 arXiv   pre-print
This dissertation contributes to mathematical and algorithmic problems that arise in the analysis of network and biological data.  ...  Upon running a standard probe selection algorithm based on Singular Value Decomposition (SVD), we obtain a 295×1000 matrix.  ...  demonstrated an exact algorithm with time complexity Θ(s 4 ) and a (1+ ) approximate method with complexity O(s + 1/ 6 ).Below, we examine the present definition and show that Programs 11.1 and 11.2 have  ... 
arXiv:1407.0375v1 fatcat:6s2qka5fazbl7bzxz4k3u75hzm

Quantum Algorithms for Unsupervised Machine Learning and Neural Networks [article]

Jonas Landman
2021 arXiv   pre-print
For this, we introduce an algorithm to perform a quantum convolution product on images, as well as a new way to perform a fast tomography on quantum states.  ...  solve tasks such as matrix product or distance estimation.  ...  Remerciements This Ph.D. was a long and wonderfully rewarding experience, half of which happened during the Covid-19 global pandemic.  ... 
arXiv:2111.03598v1 fatcat:k7b5oct53zcrdewj4xybvem3je

A Distance-preserving Matrix Sketch [article]

Leland Wilkinson, Hengrui Luo
2021 arXiv   pre-print
To ameliorate this bias and to make visualizations of very large datasets feasible, we introduce two new algorithms that respectively select a subset of rows and columns of a rectangular matrix.  ...  We compare our matrix sketch to more traditional alternatives on a variety of artificial and real datasets.  ...  The CUR decomposition is similar to the Singular Value Decomposition (SVD) (Stewart and Stewart, 1998; Drineas et al., 2008) : X = CUR (3) For X np the CUR components are dimensioned as C nk , U km and  ... 
arXiv:2009.03979v3 fatcat:jheh4moigjdpxgzccfghtz6r3q

Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning

Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, Chunhao Wang
2022 Journal of the ACM  
Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén, Su, Low, and Wiebe [STOC'19], we develop classical algorithms for SVT that run in  ...  in time independent of their dimension.  ...  Given SQ() and SQ( † ), we give an algorithm to output a CUR decomposition approximating (SV) (). .  ... 
doi:10.1145/3549524 fatcat:6dqprm7apbbttjehaiu64q3clu

Approximate Kernel Selection via Matrix Approximation

Lizhong Ding, Shizhong Liao, Yong Liu, Li Liu, Fan Zhu, Yazhou Yao, Ling Shao, Xin Gao
2020 IEEE Transactions on Neural Networks and Learning Systems  
to conduct kernel selection in a linear or quasi-linear complexity.  ...  We introduce two selection criteria based on error estimation and prove the approximate consistency of the multilevel circulant matrix (MCM) approximation and Nyström approximation under these criteria  ...  [38] , [39] , sparse greedy approximations [40] , matrix least squares approximation [41] and CUR matrix decomposition [42] , [43] .  ... 
doi:10.1109/tnnls.2019.2958922 pmid:31945003 fatcat:fvmkbxqnj5hvln3zmjc2uj74tu

Optimal Randomized Approximations for Matrix based Renyi's Entropy [article]

Yuxin Dong and Tieliang Gong and Shujian Yu and Chen Li
2022 arXiv   pre-print
a O(n^3) time complexity that is prohibitive for large-scale applications.  ...  Specifically, we develop random approximations for integer order α cases and polynomial series approximations (Taylor and Chebyshev) for non-integer α cases, leading to a O(n^2sm) overall time complexity  ...  ) time complexity with traditional eigenvalue algorithms including eigenvalue decomposition, singular value decomposition, CUR decomposition and QR factorization [14] , [15] , greatly hampering its  ... 
arXiv:2205.07426v1 fatcat:5werz5taj5d6ffig6ttcof27jy

Lecture Notes on Randomized Linear Algebra [article]

Michael W. Mahoney
2016 arXiv   pre-print
These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.  ...  Running time. Let's say a few words about the running time of these (1 + ǫ) relative-error CX and CUR matrix decompositions.  ...  CX and CUR Decompositions of General Matrices Next, we will use the generalized ℓ 2 regression result to get very fine relative-error bounds on CX and CUR decompositions.  ... 
arXiv:1608.04481v1 fatcat:qoatb2jq3rd4tk5mo5jokydmju

A continuous analogue of the tensor-train decomposition [article]

Alex A. Gorodetsky, Sertac Karaman, Youssef M. Marzouk
2018 arXiv   pre-print
To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks  ...  Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR  ...  Acknowledgments This work was supported by the National Science Foundation through grant IIS-1452019, and by the US Department of Energy, Office of Advanced Scientific Computing Research under award number  ... 
arXiv:1510.09088v3 fatcat:lwtp43p4nzf5dpmsngddfdzw5y

Low Rank Approximation of a Sparse Matrix Based on LU Factorization with Column and Row Tournament Pivoting

Laura Grigori, Sebastien Cayrols, James W. Demmel
2018 SIAM Journal on Scientific Computing  
In this paper we present an algorithm for computing a low rank approximation of a sparse matrix based on a truncated LU factorization with column and row permutations.  ...  We also compare the computational complexity of our algorithm with randomized algorithms and show that for sparse matrices and high enough but still modest accuracies, our approach is faster.  ...  The authors thank the anonymous reviewers and the editor for their very useful comments that helped us improve the paper.  ... 
doi:10.1137/16m1074527 fatcat:jljawclhh5bw7ghxtvkg5ostje

State of the Art in Ray Tracing Animated Scenes

Ingo Wald, William R. Mark, Johannes Günther, Solomon Boulos, Thiago Ize, Warren Hunt, Steven G. Parker, Peter Shirley
2009 Computer graphics forum (Print)  
With faster hardware and algorithmic improvements this has recently changed, and real-time ray tracing is finally within reach.  ...  This bottleneck has received much recent attention by researchers that has resulted in a multitude of different algorithms, data structures, and strategies for handling animated scenes.  ...  Acknowledgments We are grateful to a large number of people that have provided feedback and/or insight into their respective papers and systems.  ... 
doi:10.1111/j.1467-8659.2008.01313.x fatcat:ezggrka36feb5bamnbljpvxzle

Decomposition and reformulation of integer linear programming problems

Fabio Furini
2011 4OR  
A feasible solution to Model (4.24)-(4.28) is then obtained using set S ′ and a Branch-and-Bound algorithm; the value of the resulting solution is a valid lower bound to the AGVDP.  ...  used to generate a valid lower bound to the AGVDP.  ... 
doi:10.1007/s10288-011-0178-4 fatcat:cp6c7sxqg5gudftye4ek5bedo4

Compressing Graphs and Indexes with Recursive Graph Bisection

Laxman Dhulipala, Igor Kabiljo, Brian Karrer, Giuseppe Ottaviano, Sergey Pupyrev, Alon Shalita
2016 Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining - KDD '16  
We design and implement a novel theoretically sound reordering algorithm that is based on recursive graph bisection.  ...  Our experiments show a significant improvement of the compression rate of graph and indexes over existing heuristics.  ...  , and Sebastiano Vigna for helping with the WebGraph library.  ... 
doi:10.1145/2939672.2939862 dblp:conf/kdd/DhulipalaKKOPS16 fatcat:ibwib444xfcxha3uccpdx3mnqy

Recursive Sampling for the Nyström Method [article]

Cameron Musco, Christopher Musco
2017 arXiv   pre-print
Empirically we show that it finds more accurate, lower rank kernel approximations in less time than popular techniques such as uniformly sampled Nyström approximation and the random Fourier features method  ...  The algorithm projects the kernel onto a set of s landmark points sampled by their *ridge leverage scores*, requiring just O(ns) kernel evaluations and O(ns^2) additional runtime.  ...  We would also like to thank Michael Cohen for pointing out (and fixing) an error in our original manuscript and generally for his close collaboration in our work on leverage score sampling algorithms.  ... 
arXiv:1605.07583v5 fatcat:tvkl2txnr5c5zbnincix3qbrxy
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