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Bounded Matrix Low Rank Approximation

Ramakrishnan Kannan, Mariya Ishteva, Haesun Park
2012 2012 IEEE 12th International Conference on Data Mining  
In this paper, we propose a new matrix lower rank approximation called Bounded Matrix Low Rank Approximation (BMA) which imposes a lower and an upper bound on every element of a lower rank matrix that  ...  BMA is different from NMF as it imposes bounds on the approximation itself rather than on each of the low rank factors.  ...  Matrix Low Rank Approximation (BMA) Figure 1 : 1 Block Bounded Matrix Low Rank Approximation the ground truth.  ... 
doi:10.1109/icdm.2012.131 dblp:conf/icdm/KannanIP12 fatcat:bt2i6dfhwjcnvioc6fmgo4k4xu

Bounded Matrix Low Rank Approximation [chapter]

Ramakrishnan Kannan, Mariya Ishteva, Barry Drake, Haesun Park
2015 Signals and Communication Technology  
In this paper, we propose a new matrix lower rank approximation called Bounded Matrix Low Rank Approximation (BMA) which imposes a lower and an upper bound on every element of a lower rank matrix that  ...  BMA is different from NMF as it imposes bounds on the approximation itself rather than on each of the low rank factors.  ...  Matrix Low Rank Approximation (BMA) Figure 1 : 1 Block Bounded Matrix Low Rank Approximation the ground truth.  ... 
doi:10.1007/978-3-662-48331-2_4 fatcat:wxtdcaqrqza6fkvut3ynmcbtu4

Low Rank Matrix-Valued Chernoff Bounds and Approximate Matrix Multiplication [article]

Avner Magen, Anastasios Zouzias
2010 arXiv   pre-print
To handle similar bounds for row sampling, we develop a novel matrix-valued Chernoff bound inequality which we call low rank matrix-valued Chernoff bound.  ...  In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A∈^n× m and B∈^n × p be two matrices and >0.  ...  , ℓ 2 -regression, and low rank matrix approximation.  ... 
arXiv:1005.2724v3 fatcat:wk2qxpcpsjgd3p7wv6cm45wawe

Lower bounds for the low-rank matrix approximation

Jicheng Li, Zisheng Liu, Guo Li
2017 Journal of Inequalities and Applications  
Low-rank matrix recovery is an active topic drawing the attention of many researchers. It addresses the problem of approximating the observed data matrix by an unknown low-rank matrix.  ...  Suppose that A is a low-rank matrix approximation of D, where D and A are m × n matrices.  ...  There are some reasons for the study of lower bound of a low-rank matrix approximation problem.  ... 
doi:10.1186/s13660-017-1564-z pmid:29200797 pmcid:PMC5696467 fatcat:gfkv7rhfjrb7pjgc6xsc3svpbu

Low Rank Matrix-valued Chernoff Bounds and Approximate Matrix Multiplication [chapter]

Avner Magen, Anastasios Zouzias
2011 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms  
Second we give improved approximation algorithms for the low rank matrix approximation problem with respect to the spectral norm.  ...  To handle similar bounds for row sampling, we develop a novel matrix-valued Chernoff bound inequality which we call low rank matrixvalued Chernoff bound.  ...  , 2 -regression, and low rank matrix approximation.  ... 
doi:10.1137/1.9781611973082.109 dblp:conf/soda/MagenZ11 fatcat:nkiq6xfprvc7vg77ihrz7bndbq

Least upper bound of truncation error of low-rank matrix approximation algorithm using QR decomposition with pivoting

Haruka Kawamura, Reiji Suda
2021 Japan journal of industrial and applied mathematics  
AbstractLow-rank approximation by QR decomposition with pivoting (pivoted QR) is known to be less accurate than singular value decomposition (SVD); however, the calculation amount is smaller than that  ...  The least upper bound of the ratio of the truncation error, defined by $$\Vert A-BC\Vert _2$$ ‖ A - B C ‖ 2 , using pivoted QR to that using SVD is proved to be $$\sqrt{\frac{4^k-1}{3}(n-k)+1}$$ 4 k -  ...  Introduction Low-rank approximation Low-rank matrix approximation involves approximating a matrix by a matrix whose rank is less than that of the original matrix.  ... 
doi:10.1007/s13160-021-00459-x fatcat:jhqosv7c5bf6lp4i4fagxuhoka

Bilateral Random Projections [article]

Tianyi Zhou, Dacheng Tao
2011 arXiv   pre-print
In this paper, we show a dense matrix X's low-rank approximation can be rapidly built from its left and right random projections Y_1=XA_1 and Y_2=X^TA_2, or bilateral random projection (BRP).  ...  The effectiveness and the efficiency of BRP based low-rank approximation is empirically verified on both artificial and real datasets.  ...  Approximation error bounds We analyze the error bounds of the BRP based low-rank approximation (1) and its power scheme modification (5) .  ... 
arXiv:1112.5215v1 fatcat:aqnc33djofelrkylotjaehq7bq

Bilateral random projections

Tianyi Zhou, Dacheng Tao
2012 2012 IEEE International Symposium on Information Theory Proceedings  
In this paper, we show a dense matrix X's low-rank approximation can be rapidly built from its left and right random projections Y 1 = XA 1 and Y 2 = X T A 2 , or bilateral random projection (BRP).  ...  The effectiveness and the efficiency of BRP based low-rank approximation is empirically verified on both artificial and real datasets.  ...  APPROXIMATION ERROR BOUNDS We analyze the error bounds of the BRP based low-rank approximation (1) and its power scheme modification (5) .  ... 
doi:10.1109/isit.2012.6283064 dblp:conf/isit/ZhouT12 fatcat:j2dm3ucd3bglfivskt3ze2cacy

Why are Big Data Matrices Approximately Low Rank? [article]

Madeleine Udell, Alex Townsend
2018 arXiv   pre-print
Hence any sufficiently large matrix from such a latent variable model can be approximated, up to a small entrywise error, by a low rank matrix.  ...  However, we show that we can approximate every entry of an m × n matrix drawn from this model to within a fixed absolute error by a low rank matrix whose rank grows as O((m + n)).  ...  It is useful to know when a dataset can be approximated by a low rank matrix. A low rank approximation can be used to make filtering and statistics either computationally feasible or more efficient.  ... 
arXiv:1705.07474v2 fatcat:hqwyluzpf5hd3faxmfuq6lmjzi

Matrix Coherence and the Nystrom Method [article]

Ameet Talwalkar, Afshin Rostamizadeh
2010 arXiv   pre-print
The Nystrom method is an efficient technique to speed up large-scale learning applications by generating low-rank approximations.  ...  Making use of related work in the compressed sensing and the matrix completion literature, we derive novel coherence-based bounds for the Nystrom method in the low-rank setting.  ...  a low-rank approximation.  ... 
arXiv:1004.2008v1 fatcat:bojflagkffaijf2mdjcra4txse

Matrix Coherence and the Nystrom Method [article]

Ameet Talwalkar, Afshin Rostamizadeh
2014 arXiv   pre-print
Making use of related work in the compressed sensing and the matrix completion literature, we derive novel coherence-based bounds for the Nystrom method in the low-rank setting.  ...  The Nystrom method is an efficient technique used to speed up large-scale learning applications by generating low-rank approximations.  ...  This is done by generating low-rank approximations of G using a subset of the columns of the matrix.  ... 
arXiv:1408.2044v1 fatcat:qnhwnoqcsreghbrp2hvzuvky2y

Practical Sketching Algorithms for Low-Rank Matrix Approximation

Joel A. Tropp, Alp Yurtsever, Madeleine Udell, Volkan Cevher
2017 SIAM Journal on Matrix Analysis and Applications  
This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch.  ...  These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank.  ...  Proof of Theorem 4.2: Spectral Error Bound. In this section, we establish a spectral-norm error bound for the low-rank approximation (4.3).  ... 
doi:10.1137/17m1111590 fatcat:nnntlyw3nray5c3cha5ay452gi

Clustered low rank approximation of graphs in information science applications [chapter]

Berkant Savas, Inderjit S. Dhillon
2011 Proceedings of the 2011 SIAM International Conference on Data Mining  
In this paper we present a fast and accurate procedure called clustered low rank matrix approximation for massive graphs.  ...  Further, we generalize stochastic algorithms to the clustered low rank approximation framework and present theoretical bounds for the approximation error.  ...  Contributions Clustered low rank matrix approximation. We are interested in a low rank approximation of a given matrix A arising from a graph.  ... 
doi:10.1137/1.9781611972818.15 dblp:conf/sdm/SavasD11 fatcat:stqgahy4zrdv7i3agofx6al4bi

Robust and Sample Optimal Algorithms for PSD Low-Rank Approximation [article]

Ainesh Bakshi, Nadiia Chepurko, David P. Woodruff
2021 arXiv   pre-print
Bakshi and Woodruff (NeurIPS, 2018) showed a bi-criteria, relative-error low-rank approximation which queries O(nk/ϵ^2.5) entries and outputs a rank-(k+4) matrix.  ...  Recently, Musco and Woodruff (FOCS, 2017) showed that given an n × n positive semidefinite (PSD) matrix A, it is possible to compute a (1+ϵ)-approximate relative-error low-rank approximation to A by querying  ...  Lower Bound for Robust PSD Low-Rank Approximation In this subsection, we show a query lower bound of Ω( 2 2 / 2 ) = Ω( 2 max / ) for any algorithm that outputs a low rank approximation up to additive-error  ... 
arXiv:1912.04177v5 fatcat:746322w7njalrlc2a2p2huhub4

Computing Structured Singular Values for Sturm-Liouville Problems

2019 International Journal of Analysis and Applications  
The low rank ODE's based technique is used for the approximation of the bounds of SSV. The lower bounds of SSV discuss the instability analysis of linear system in system theory.  ...  The numerical experimentation show the comparison of bounds of SSV computed by low rank ODE'S technique with the well-known MATLAB routine mussv available in MATLAB Control Toolbox.  ...  routine mussv is very fast compare to Low rank ODE's based technique. • The MATLAB routine mussv additionally approximate an upper bounds of SSV which is not possible while making use of Low rank ODE's  ... 
doi:10.28924/2291-8639-17-2019-879 fatcat:yvvzi3r6indoxa3cfiqsqtq7fu
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