An Accelerated Accuracy-enhanced Randomized Singular Value Decomposition for Factorizing Matrices with Low-rank Structure
Joseph Roger Arhin,
Francis Sam,
Kenneth Coker,
Toufic Seini
Issue:
Volume 9, Issue 4, August 2020
Pages:
64-73
Received:
24 June 2020
Accepted:
13 July 2020
Published:
28 July 2020
Abstract: Big data has in recent years gained ground in many scientific and engineering problems. It seems to some extent prohibitive for traditional matrix decomposition methods (i.e. QR, SVD, EVD, etc.) to handle such large-scale problems involving data matrix. Many researchers have developed several algorithms to decompose such big data matrices. An accuracy-enhanced randomized singular value decomposition method (referred to as AE-RSVDM) with orthonormalization recently becomes the state-of-the-art to factorize large data matrices with satisfactory speed and accuracy. In our paper, low-rank matrix approximations based on randomization are studied, with emphasis on accelerating the computational efficiency on large data matrices. By this, we accelerate the AE-RSVDM with modified normalized power iteration to result in an accelerated version. The accelerated version is grounded on a two-stage scheme. The first stage seeks to find the range of a sketch matrix which involves a Gaussian random matrix. A low-dimensional space is then created from the high-dimensional data matrix via power iteration. Numerical experiments on matrices of different sizes demonstrate that our accelerated variant achieves speedups while attaining the same reconstruction error as the AE-RSVDM with orthonormalization. And with data from Google art project, we have made known the computational speed-up of the accelerated variant over the AE-RSVDM algorithm for decomposing large data matrices with low-rank form.
Abstract: Big data has in recent years gained ground in many scientific and engineering problems. It seems to some extent prohibitive for traditional matrix decomposition methods (i.e. QR, SVD, EVD, etc.) to handle such large-scale problems involving data matrix. Many researchers have developed several algorithms to decompose such big data matrices. An accur...
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Common Fixed Point for Two Multivalued Mappings on Proximinal Sets in Regular Modular Space
Nashat Faried,
Hany Abd-El Ghaffar,
Salwa Hamdy
Issue:
Volume 9, Issue 4, August 2020
Pages:
74-83
Received:
9 July 2020
Accepted:
3 August 2020
Published:
10 August 2020
Abstract: In 2010, Chistyakov, V. V., defined the theory of modular metric spaces. After that, in 2011, Mongkolkeha, C. et. al. studied and proved the new existence theorems of a fixed point for contraction mapping in modular metric spaces for one single-valued map in modular metric space. Also, in 2012, Chaipunya, P. introduced some fixed point theorems for multivalued mapping under the setting of contraction type in modular metric space. In 2014, Abdou, A. A. N., Khamsi, M. A. studied the existence of fixed points for contractive-type multivalued maps in the setting of modular metric space. In 2016, Dilip Jain et. al. presented a multivalued F-contraction and F-contraction of Hardy-Rogers-type in the case of modular metric space with specific assumptions. In this work, we extended these results into the case of a pair of multivalued mappings on proximinal sets in a regular modular metric space. This was done by introducing the notions of best approximation, proximinal set in modular metric space, conjoint F-proximinal contraction, and conjoint F-proximinal contraction of Hardy-Rogers-type for two multivalued mappings. Furthermore, we give an example showing the conditions of the theory which found a common fixed point of a pair of multivalued mappings on proximinal sets in a regular modular metric space. Also, the applications of the obtained results can be used in multiple fields of science like Electrorheological fluids and FORTRAN computer programming as shown in this communication.
Abstract: In 2010, Chistyakov, V. V., defined the theory of modular metric spaces. After that, in 2011, Mongkolkeha, C. et. al. studied and proved the new existence theorems of a fixed point for contraction mapping in modular metric spaces for one single-valued map in modular metric space. Also, in 2012, Chaipunya, P. introduced some fixed point theorems for...
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