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Tighter Low-rank Approximation via Sampling the Leveraged Element [article]

Srinadh Bhojanapalli, Prateek Jain, Sujay Sanghavi
2014 arXiv   pre-print
weighted alternating minimization over the factored form of the intended low-rank matrix, to minimize error only on these samples.  ...  Taking an approach different from existing literature, our method first involves a specific biased sampling, with an element being chosen based on the leverage scores of its row and column, and then involves  ...  Tighter Low-rank Approximation via Sampling the Leveraged Element  ... 
arXiv:1410.3886v1 fatcat:rfmvfs2benaj7cdtvzkty3qm5y

Tighter Low-rank Approximation via Sampling the Leveraged Element [chapter]

Srinadh Bhojanapalli, Prateek Jain, Sujay Sanghavi
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
In this work, we propose a new randomized algorithm for computing a low-rank approximation to a given matrix.  ...  Taking an approach different from existing literature, our method first involves a specific biased sampling, with an element being chosen based on the leverage scores of its row and column, and then involves  ...  As shown in Theorem 3.1, our method requires Algorithm 1 LELA: Leveraged Element Low-rank Approximation input matrix: M ∈ R n×d , rank: r, number of samples: m, number of iterations:T 1: Sample Ω ⊆ [n]  ... 
doi:10.1137/1.9781611973730.62 dblp:conf/soda/Bhojanapalli0S15 fatcat:uddz5bu33nbd5ju66vgazenas4

Recovering PCA from Hybrid-(ℓ_1,ℓ_2) Sparse Sampling of Data Elements [article]

Abhisek Kundu, Petros Drineas, Malik Magdon-Ismail
2015 arXiv   pre-print
We prove that the hybrid algorithm recovers a near-PCA reconstruction of the data from a sublinear sample-size: hybrid-(ℓ_1,ℓ_2) inherits the ℓ_2-ability to sample the important elements as well as the  ...  We present a randomized algorithm that element-wise sparsifies the data, retaining only a few its elements.  ...  Also, we use element-wise leverage scores (Chen et al. (2014) ) for sparsification of low-rank data.  ... 
arXiv:1503.00547v1 fatcat:dyzfvwlntjgwdg6ynczjwob43y

Fast and Accurate Randomized Algorithms for Low-rank Tensor Decompositions [article]

Linjian Ma, Edgar Solomonik
2021 arXiv   pre-print
Low-rank Tucker and CP tensor decompositions are powerful tools in data analytics.  ...  Theoretical sketch size upper bounds are provided to achieve O(ϵ) relative error for each subproblem with two sketching techniques, TensorSketch and leverage score sampling.  ...  Algorithm 5 RSVD-LRLS: Low-rank approximation of least squares solution via randomized SVD 1: Input: Left-hand-side matrix Z ∈ R m×r , right-hand-side matrix Y ∈ R m×s , rank R 2: Initialize S ∈ R s×O(  ... 
arXiv:2104.01101v2 fatcat:rfljacrakvcktbnt24ugs4246u

The Effect of Coherence on Sampling from Matrices with Orthonormal Columns, and Preconditioned Least Squares Problems [article]

Ilse C. F. Ipsen, Thomas Wentworth
2014 arXiv   pre-print
For uniform sampling with replacement we derive a potentially tighter condition number bound in terms of the leverage scores of Q.  ...  In particular, sampled matrices of full rank tend to have two-norm condition numbers of at most 10. We derive a bound on the condition number of the sampled matrices in terms of the coherence μ of Q.  ...  Uniform sampling without replacement can be implemented via random permutations 3 .  ... 
arXiv:1203.4809v3 fatcat:owcdcv574ffzzcr34enspytopq

Recovering PCA and Sparse PCA via Hybrid-(l1, l2) Sparse Sampling of Data Elements

Abhisek Kundu, Petros Drineas, Malik Magdon-Ismail
2017 Journal of machine learning research  
Hybrid-( 1 , 2 ) inherits the 2 -ability to sample the important elements, as well as the regularization properties of 1 sampling, and maintains strictly better quality than either 1 or 2 on their own.  ...  Our new algorithm independently samples the data using probabilities that depend on both squares ( 2 sampling) and absolute values ( 1 sampling) of the entries.  ...  or the U.S.  ... 
dblp:journals/jmlr/KunduDM17 fatcat:57mfcflfzbaszn7uuqpu6gk5k4

The Effect of Coherence on Sampling from Matrices with Orthonormal Columns, and Preconditioned Least Squares Problems

Ilse C. F. Ipsen, Thomas Wentworth
2014 SIAM Journal on Matrix Analysis and Applications  
For uniform sampling with replacement we derive a potentially tighter condition number bound in terms of the leverage scores of Q.  ...  If Q has low coherence then, in the context of sampling, all rows are equally important. Hence any sampled matrix SQ with sufficiently many rows is likely to have full rank.  ... 
doi:10.1137/120870748 fatcat:xeeeyomlzbap3mosrub5ua27cm

Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds [article]

John T. Holodnak, Ilse C. F. Ipsen
2014 arXiv   pre-print
The bounds depend on the stable rank or the rank of A, but not on the matrix dimensions.  ...  Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A.  ...  The matrices have the same dimension, and similar high ranks and low stable ranks, see Table 4 .1. Note that only for low stable ranks can Algorithm 3.1 achieve any accuracy.  ... 
arXiv:1310.1502v3 fatcat:2nddnilmc5eljct7doc57atdp4

Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds

John T. Holodnak, Ilse C. F. Ipsen
2015 SIAM Journal on Matrix Analysis and Applications  
The bounds depend on the stable rank or the rank of A, but not on the matrix dimensions.  ...  Given a real matrix A with n columns, the problem is to approximate the Gram product AA T by c n weighted outer products of columns of A.  ...  We thank Petros Drineas and Michael Mahoney for useful discussions, and the four anonymous reviewers whose suggestions helped us to improve the quality of the paper.  ... 
doi:10.1137/130940116 fatcat:ogck6fmhazgeloljn4rzzamp4a

Sketching Matrix Least Squares via Leverage Scores Estimates [article]

Brett W. Larsen, Tamara G. Kolda
2022 arXiv   pre-print
The subset of rows is determined via random sampling based on leverage score estimates for each row.  ...  We prove that the number of samples required for an ϵ-accurate solution is O(r/(βϵ)) where β∈ (0,1] is a measure of the quality of the leverage score estimates.  ...  The first is that it provides the foundation for leverage-based sampling for low-rank tensor decomposition as described in [5] .  ... 
arXiv:2201.10638v1 fatcat:gibyc6oxojd7nesrcemt2fjjt4

Near Optimal Linear Algebra in the Online and Sliding Window Models [article]

Vladimir Braverman, Petros Drineas, Cameron Musco, Christopher Musco, Jalaj Upadhyay, David P. Woodruff, Samson Zhou
2022 arXiv   pre-print
for low-rank approximation/projection-cost preservation.  ...  Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and  ...  We first prove that low-rank approximation is not smooth for any meaningful parameters (α, β) in Definition A.1, even when the best low-rank approximations are nonzero. Lemma A.3.  ... 
arXiv:1805.03765v5 fatcat:kxk2frvlirhczix666jlczd344

Toward Efficient and Accurate Covariance Matrix Estimation on Compressed Data

Xixian Chen, Michael R. Lyu, Irwin King
2017 International Conference on Machine Learning  
In contrast to previous data-oblivious compression schemes, we leverage a data-aware weighted sampling method to construct lowdimensional data for such estimation.  ...  an efficient and accurate covariance matrix estimation method via data compression.  ...  Acknowledgments We truly thank Akshay Krishnamurthy for the fruitful discussions and interpretations on (Azizyan et al., 2015) . We also thank Yuxin Su for the help on the experiments.  ... 
dblp:conf/icml/ChenLK17 fatcat:guzydfmrejdw5c6jo6w56cc6qu

Approximate Euclidean lengths and distances beyond Johnson-Lindenstrauss [article]

Aleksandros Sobczyk, Mathieu Luisier
2022 arXiv   pre-print
We finally show how these results can be extended to estimate the Euclidean distances between data points and to approximate the statistical leverage scores of a tall-and-skinny data matrix, which are  ...  We prove element-wise probabilistic bounds that are at least as good as standard JL approximations in the worst-case, but are asymptotically better for matrices with decaying spectrum.  ...  We can finally conclude that there exist corner cases where O(d) samples are needed to achieve -accuracy for element-wise Euclidean norm estimation based on low rank projections.  ... 
arXiv:2205.12307v1 fatcat:crewgnb4uvg2njeftk573xsvty

Effective Stiffness: Generalizing Effective Resistance Sampling to Finite Element Matrices [article]

Haim Avron, Sivan Toledo
2014 arXiv   pre-print
In particular, we show that sampling O(n n) elements according to probabilities derived from effective stiffnesses yields a high quality preconditioner that can be used to solve the linear system in a  ...  Solving finite elements problems is of considerably more interest than the solution of SDD linear systems, since the finite element method is frequently used to numerically solve PDEs arising in scientific  ...  Sivan Toledo was supported by grant 1045/09 from the Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities) and by grant 2010231 from the US-Israel Binational Science Foundation  ... 
arXiv:1110.4437v2 fatcat:nlirtmqdanfwrf7bodl4vpnstq

Sublinear Time Low-Rank Approximation of Positive Semidefinite Matrices [article]

Cameron Musco, David P. Woodruff
2019 arXiv   pre-print
We prove time lower bounds for low-rank approximation of PSD matrices, showing that our algorithm is close to optimal.  ...  Finally, we extend our techniques to give sublinear time algorithms for low-rank approximation of A in the (often stronger) spectral norm metric A-B_2^2 and for ridge regression on PSD matrices.  ...  Acknowledgements The authors thank IBM Almaden where part of this work was done. David Woodruff also thanks the Simons Institute program on Machine Learning and the XDATA program of DARPA for support.  ... 
arXiv:1704.03371v3 fatcat:6slwgwwyhbdkrpva6ero4637oe
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