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Multilinear Algebra for Distributed Storage [article]

Iwan Duursma and Xiao Li and Hsin-Po Wang
2020 arXiv   pre-print
We establish a multilinear algebra foundation to assemble $(n, k, d, \alpha, \beta, M)$-ERRCs for all meaningful $(n, k, d)$ tuples.  ...  For a file of total size $M$, each node stores $\alpha$ symbols, any $k$ nodes recover the file, and any $d$ nodes repair any other node via sending out $\beta$ symbols.  ...  However, finite fields-especially those with characteristic 2-are assumed for applications (distributed storage as in the title).  ... 
arXiv:2006.08911v1 fatcat:43o2m6lv3ffbnkbzljqgjc7cui

Parallel Minimum Spanning Forest Computation using Sparse Matrix Kernels [article]

Tim Baer, Raghavendra Kanakagiri, Edgar Solomonik
2021 arXiv   pre-print
We implement this MSF algorithm with Cyclops, a distributed-memory library for generalized sparse tensor algebra.  ...  Formulations of graph algorithms using sparse linear algebra have yielded highly scalable distributed algorithms for problems such as connectivity and shortest path computation.  ...  We mainly rely on the data distribution and algebraic primitives supported by CTF. We optimize vector distribution/transpose operations, and use them to implement the multilinear kernel.  ... 
arXiv:2110.04865v2 fatcat:3shkieljwfdmjcfwnxucfs2dqa

Limits and Applications of Group Algebras for Parameterized Problems [chapter]

Ioannis Koutis, Ryan Williams
2009 Lecture Notes in Computer Science  
The algebraic framework introduced in [Koutis, Proc. of the 35 th ICALP 2008] reduces several combinatorial problems in parameterized complexity to the problem of detecting multilinear degree-k monomials  ...  algebra.  ...  We use the group algebra R[Z k 2 ]. Proof. Applying the algorithm of Lemma 4.1 to the circuits for P and Q, we can assume that P, Q contain only multilinear terms.  ... 
doi:10.1007/978-3-642-02927-1_54 fatcat:v2yxdzzilrautoovjprqdivd2e

LIMITS and Applications of Group Algebras for Parameterized Problems

Ioannis Koutis, Ryan Williams
2016 ACM Transactions on Algorithms  
The algebraic framework introduced in [Koutis, Proc. of the 35 th ICALP 2008] reduces several combinatorial problems in parameterized complexity to the problem of detecting multilinear degree-k monomials  ...  algebra.  ...  We use the group algebra R[Z k 2 ]. Proof. Applying the algorithm of Lemma 4.1 to the circuits for P and Q, we can assume that P, Q contain only multilinear terms.  ... 
doi:10.1145/2885499 fatcat:jo5zfmif7badrkvtu5kb75qe5a

Secure Remote Data Processing in Cloud Computing

Mohd Rizuan Baharon, Qi Shi, David Llewellyn-Jones, Madjid Merabti
2013 Journal of clean energy technologies  
The scheme should support the n-multilinear map so as to allow for arbitrary multiplications on encrypted data.  ...  Cloud computing provides solutions as a service to meet customers' needs such as massive data storage and a lot of computing resources to process customer data efficiently.  ...  Primitive encryption technique like AES is ideal to be used for storage purposes as this encryption technique provides a strong protection to the data in the cloud storage [3] .  ... 
doi:10.7763/ijcte.2013.v5.823 fatcat:zz27daf4u5bihgresvkme7crla

Finite element assembly [chapter]

Anders Logg, Kent-Andre Mardal, Garth N. Wells
2012 Lecture Notes in Computational Science and Engineering  
In this chapter, we describe the standard algorithm for computing the discrete operator (tensor) A defined in Chapter [kirby-5]. This algorithm is known as finite element assembly.  ...  The finite element method may be viewed as a method for forming a discrete linear system AU = b or nonlinear system b(U) = 0 corresponding to the discretization of the variational form of a differential  ...  T b ι b T (i) + = b K i end for end for Parallel Assembly The assembly algorithms remain unchanged in a distributed parallel environment if the linear algebra backend supports distributed matrices and  ... 
doi:10.1007/978-3-642-23099-8_6 fatcat:udnf53zhwvepxezumu2elb4wj4

Access control mechanisms for outsourced data in public cloud using polynomial interpolation and multilinear map

B.R. Purushothama, B.B. Amberker
2017 International Journal of Cloud Computing  
The security of multilinear map based schemes relies on the multilinear Diffie-Hellman assumption. The key generated for data encryption is regulated by the access authorisations holding on the data.  ...  Outsourcing the user's sensitive data on to honest-but-curious cloud storage servers pose several data security and access control challenges.  ...  We define a multilinear map generator. We cannot treat these groups as abstract algebraic objects as we study the computational problems on the groups G 1 and G 2 .  ... 
doi:10.1504/ijcc.2017.083901 fatcat:tfjppuk4rraazfw7jemllk3s2a

Joint tensor compression for coupled canonical polyadic decompositions

Jeremy Emile Cohen, Rodrigo Cabral Farias, Pierre Comon
2016 2016 24th European Signal Processing Conference (EUSIPCO)  
In particular, for tensors, increasing dimensions of all modes may result in an enormous increase in computation time and data storage requirements.  ...  To handle dimensionality reduction without destroying the multilinear structure of the data, standard tools of matrix algebra can be used [1] , [8] as explained in section II.  ... 
doi:10.1109/eusipco.2016.7760656 dblp:conf/eusipco/CohenFC16 fatcat:fthxz6sz6nekhhiyu5gdhhvxwq

Tensor Numerical Methods: Actual Theory and Recent Applications

Ivan Gavrilyuk, Boris N. Khoromskij
2018 Computational Methods in Applied Mathematics  
At present, both tensor numerical methods and multilinear algebra of big data continue to expand actively to further theoretical and applied research topics.  ...  We appreciate the authors of all articles published in this special issue for the excellent contributions as well as the reviewers for their work on refereeing the manuscripts.  ...  Acknowledgment: We would like to thank Professor Carsten Carstensen, the editor-in-chief of the Journal of Computational Methods in Applied Mathematics, for his kind support of this special issue and for  ... 
doi:10.1515/cmam-2018-0014 fatcat:jtjt6j77kfd4fk4lem3zvgfhpq

A survey of multilinear subspace learning for tensor data

Haiping Lu, Konstantinos N. Plataniotis, Anastasios N. Venetsanopoulos
2011 Pattern Recognition  
This paper surveys the field of multilinear subspace learning (MSL) for dimensionality reduction of multidimensional data directly from their tensorial representations.  ...  It discusses the central issues of MSL, including establishing the foundations of the field via multilinear projections, formulating a unifying MSL framework for systematic treatment of the problem, examining  ...  Acknowledgment The authors would like to thank the anonymous reviewers for their insightful comments, which have helped to improve the quality of this paper.  ... 
doi:10.1016/j.patcog.2011.01.004 fatcat:6puqzxrohfawraeiyid6633wdm

Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis

Andrzej Cichocki, Danilo Mandic, Lieven De Lathauwer, Guoxu Zhou, Qibin Zhao, Cesar Caiafa, HUY ANH PHAN
2015 IEEE Signal Processing Magazine  
We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated  ...  Benefiting fromthe power ofmultilinear algebra as theirmathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that  ...  His research interests include multilinear algebra, tensor computation, blind source separation, and brain-computer interfaces.  ... 
doi:10.1109/msp.2013.2297439 fatcat:gcyacttbvbhtjcjfr4hdfzdciy

Tensor-train approximation of the chemical master equation and its application for parameter inference [article]

Ion Gabriel Ion, Christian Wildner, Dimitrios Loukrezis, Heinz Koeppl, Herbert De Gersem
2021 arXiv   pre-print
Since all linear algebra operations are performed in the tensor-train format, a significant reduction of the computational time is observed as well.  ...  In this work, we perform Bayesian inference tasks for the chemical master equation in the tensor-train format.  ...  It should be noted that the TT-ranks grow after a multilinear algebraic operation is performed in the TT-format.  ... 
arXiv:2106.15188v1 fatcat:srucb4onbncwlk6y372xpaf6eu

Protecting Big Data Privacy Using Randomized Tensor Network Decomposition and Dispersed Tensor Computation [article]

Jenn-Bing Ong, Wee-Keong Ng, Ivan Tjuawinata, Chao Li, Jielin Yang, Sai None Myne, Huaxiong Wang, Kwok-Yan Lam, C.-C. Jay Kuo
2021 arXiv   pre-print
for compressed and distributed / dispersed computation.  ...  Experiments show that the proposed randomization techniques are helpful for big data anonymization and efficient for big data storage and computation.  ...  , multilinear operations to reduce the tensor-rank complexity for storage and computational efficiency.  ... 
arXiv:2101.04194v1 fatcat:pay7hgk24jddnmvxrexefhd7di

Multilinear Motion Synthesis with Level-of-Detail Controls

Tomohiko Mukai, Shigeru Kuriyama
2007 15th Pacific Conference on Computer Graphics and Applications (PG'07)  
This paper proposes a LOD control method of motion synthesis with a multilinear model.  ...  Our method provides a practical tool for creating an interactive animation of many characters while ensuring accurate and flexible controls at a modest level of computational cost.  ...  Acknowledgements We would like to thank anonymous reviewers for helpfull comments.  ... 
doi:10.1109/pg.2007.36 dblp:conf/pg/MukaiK07 fatcat:p7ntlwruvnavplyni4oekloqaq

QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations

Venera Khoromskaia, Boris Khoromskij, Reinhold Schneider
2011 Computational Methods in Applied Mathematics  
The numerical examples for some volumetric organic molecules confirm that the QTT ranks of these potentials are nearly independent of the one-dimension grid size n.  ...  We present numerical illustrations of the QTT approximation to the Hartree and exchange integral operators of some moderate size molecules for the grid-size in the range n 2 14 .  ...  Hence, the complexity of the 3D grid-based tensor operations for evaluation of the Coulomb and exchange matrices, J(C) and K(C), becomes almost independent of n, being regulated only by the tensor ranks  ... 
doi:10.2478/cmam-2011-0018 fatcat:hc6iy7knobhhvapz4nrhrrdqkm
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