Filters








39,181 Hits in 5.7 sec

Smoothed analysis of the condition number under low-rank perturbations [article]

Rikhav Shah, Sandeep Silwal
2021 arXiv   pre-print
Let M be an arbitrary n by n matrix of rank n-k. We study the condition number of M plus a low-rank perturbation UV^T where U, V are n by k random Gaussian matrices.  ...  Lastly, barriers in applying low-rank noise to other problems studied in the smoothed analysis framework are discussed.  ...  We also thank Piotr Indyk and Arsen Vasilyan for helpful feedback on a draft of the paper.  ... 
arXiv:2009.01986v2 fatcat:tan26tiqqvflvgd5xvrjtj3p5q

Efficient Tensor Decomposition [article]

Aravindan Vijayaraghavan
2020 arXiv   pre-print
This chapter studies the problem of decomposing a tensor into a sum of constituent rank one tensors.  ...  We will see how to design efficient algorithms with provable guarantees under mild assumptions, and using beyond worst-case frameworks like smoothed analysis.  ...  Acknowledgments I thank Aditya Bhaskara for many discussions related to the chapter, and Tim Roughgarden, Aidao Chen, Rong Ge and Paul Valiant for their comments on a preliminary draft of this chapter.  ... 
arXiv:2007.15589v1 fatcat:o3hyfndu7nfydk57uackk5e6yi

Sensitivity of low-rank matrix recovery [article]

Paul Breiding, Nick Vannieuwenhoven
2021 arXiv   pre-print
In addition, we study the condition number of the rank-r matrix approximation problem.  ...  A special case covered by our analysis is approximating an incomplete matrix by a low-rank matrix.  ...  The base-10 logarithm of the condition number of low-rank recovery at (A t , Y ) is visualized in Fig. 3 .  ... 
arXiv:2103.00531v2 fatcat:xsll5mlseje57gisv7j2lyc4yy

Smoothed analysis of the low-rank approach for smooth semidefinite programs [article]

Thomas Pumir, Samy Jelassi, Nicolas Boumal
2018 arXiv   pre-print
To this end, and under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square  ...  root of the number of constraints (up to log factors).  ...  Under these assumptions, we proved using smoothed analysis that, provided k =Ω( √ m) where m is the number of constraints, if the cost matrix is perturbed randomly, with high probability, approximate second-order  ... 
arXiv:1806.03763v2 fatcat:nkfxe4gfkbf67mn2xf22jqozum

On the Hardness and Smoothed Complexity of Quasi-Concave Minimization

Jonathan A. Kelner, Evdokia Nikolova
2007 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)  
From this, we obtain the first randomized fully polynomialtime approximation scheme for low-rank quasi-concave minimization under broad conditions.  ...  The analysis is based on a smoothed bound for the number of extreme points of the projection of the feasible polytope onto a k-dimensional subspace, where k is the rank (informally, the dimension of nonconvexity  ...  low-rank concave functions under certain conditions.  ... 
doi:10.1109/focs.2007.68 dblp:conf/focs/Tao07 fatcat:mufih5aftngfzdia2rme5svhfe

On the Hardness and Smoothed Complexity of Quasi-Concave Minimization

Jonathan A. Kelner, Evdokia Nikolova
2007 Foundations of Computer Science (FOCS), IEEE Annual Symposium on  
From this, we obtain the first randomized fully polynomialtime approximation scheme for low-rank quasi-concave minimization under broad conditions.  ...  The analysis is based on a smoothed bound for the number of extreme points of the projection of the feasible polytope onto a k-dimensional subspace, where k is the rank (informally, the dimension of nonconvexity  ...  low-rank concave functions under certain conditions.  ... 
doi:10.1109/focs.2007.4389517 fatcat:gdxlrq6fr5cxzh7zzjfg2bqcrq

Unsupervised feature selection under perturbations: meeting the challenges of biological data

Roy Varshavsky, Assaf Gottlieb, David Horn, Michal Linial
2007 Computer applications in the biosciences : CABIOS  
Method: Information loss is incorporated into a perturbation scheme, testing which features are stable under it. This method is applied to data analysis by unsupervised feature filtering (UFF).  ...  The latter has been shown to be a very successful method in analysis of gene-expression data. Results: We find that the UFF quality degrades smoothly with information loss.  ...  R.V. is awarded a fellowship by the SCCB, the Sudarsky Center for Computational Biology of the Hebrew University of Jerusalem.  ... 
doi:10.1093/bioinformatics/btm528 pmid:17989091 fatcat:cjv6cwd7g5hixcs5tzohbjzon4

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.  ...  A number λ belongs to epsilon-pseudo-spectrum of an operator A, denoted by Λ (A) and satisfies the following equivalent conditions (i) λ ∈ Λ(A + E) for some perturbation E having E ≤ ; (ii) ∃ u ∈ C n,n  ... 
doi:10.28924/2291-8639-17-2019-879 fatcat:yvvzi3r6indoxa3cfiqsqtq7fu

Smooth input preparation for quantum and quantum-inspired machine learning [article]

Zhikuan Zhao, Jack K. Fitzsimons, Patrick Rebentrost, Vedran Dunjko, Joseph F. Fitzsimons
2019 arXiv   pre-print
classical algorithms in the low-rank cases.  ...  Here we prove using smoothed analysis, that if the data-analysis algorithm is robust against small entry-wise input perturbation, state preparation can always be achieved with constant queries.  ...  analysis of data processing.  ... 
arXiv:1804.00281v2 fatcat:uwq5zubt5bbjfmjazpizn3er3q

Joint Estimation of Low-Rank Components and Connectivity Graph in High-Dimensional Graph Signals: Application to Brain Imaging [article]

Rui Liu, Hossein Nejati, Ngai-Man Cheung
2018 arXiv   pre-print
In our problem formulation, we assume that the perturbation on the low-rank components is sparse and the signal is smooth on the graph.  ...  Moreover, we perform a mathematical analysis to understand and quantify the impact of the inexact graph on the low-rank estimation, justifying our scheme with graph refinement as an integrated step in  ...  ACKNOWLEDGMENT The authors would like to thank Pavitra Krishnaswamy of Institute for Infocomm Research, A*STAR for helpful discussion.  ... 
arXiv:1801.02303v1 fatcat:quei4nliczdizl3rkfdpdmhg2i

Smoothed analysis

Daniel A. Spielman, Shang-Hua Teng
2009 Communications of the ACM  
Smoothed analysis [36] is a step towards a theory that explains the behavior of algorithms in practice.  ...  A concrete example of such a smoothed analysis is a proof that the simplex algorithm for linear programming usually runs in polynomial time, when its input is subject to modeling or measurement noise.  ...  See [9, 13] for smoothed analysis of the condition numbers of other problems.  ... 
doi:10.1145/1562764.1562785 fatcat:7ohrugpymnffdcybqntslgtkom

Smooth input preparation for quantum and quantum-inspired machine learning

Zhikuan Zhao, Jack K. Fitzsimons, Patrick Rebentrost, Vedran Dunjko, Joseph F. Fitzsimons
2021 Quantum Machine Intelligence  
classical algorithms in the low-rank cases.  ...  Here we prove using smoothed analysis that if the data analysis algorithm is robust against small entry-wise input perturbation, state preparation can always be achieved with constant queries.  ...  of efficient 2 -sampling leads to equally efficient classical algorithms in the low-rank cases.  ... 
doi:10.1007/s42484-021-00045-x fatcat:nfev36vcivalxnlwossfeuxkba

Graph Structure Learning for Robust Graph Neural Networks [article]

Wei Jin, Yao Ma, Xiaorui Liu, Xianfeng Tang, Suhang Wang, Jiliang Tang
2020 arXiv   pre-print
For example, many real-world graphs are low-rank and sparse, and the features of two adjacent nodes tend to be similar.  ...  Therefore, developing robust algorithms to defend adversarial attacks is of great significance. A natural idea to defend adversarial attacks is to clean the perturbed graph.  ...  ACKNOWLEDGEMENTS This research is supported by the National Science Foundation (NSF) under grant numbers IIS1907704, IIS1928278, IIS1714741, IIS1715940, IIS1845081, IIS1909702 and CNS1815636.  ... 
arXiv:2005.10203v3 fatcat:6imhudo6rvhtxiwfamxzgqpdni

Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form [article]

Srinadh Bhojanapalli, Nicolas Boumal, Prateek Jain, Praneeth Netrapalli
2018 arXiv   pre-print
Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices.  ...  In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs.  ...  Acknowledgment NB thanks Dustin Mixon for many interesting conversations on the applicability of smoothed analysis to low-rank SDPs. NB was supported in part by NSF award DMS-1719558.  ... 
arXiv:1803.00186v1 fatcat:apprxmvetfgnljpoo2wolhng6q

Typical properties of winners and losers in discrete optimization

Rene Beier, Berthold Vöcking
2004 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04  
Our analysis covers various probability distributions for the choice of the stochastic numbers and includes smoothed analysis with Gaussian and other kinds of perturbation models as a special case.  ...  In fact, we can exactly characterize the smoothed complexity of optimization problems in terms of their random worstcase complexity.  ...  Smoothed Analysis. The framework of smoothed analysis was introduced by Spielman and Teng in [24] .  ... 
doi:10.1145/1007352.1007409 dblp:conf/stoc/BeierV04 fatcat:2bwg7ny4yfcu3kzt3sne3p5f5u
« Previous Showing results 1 — 15 out of 39,181 results