Low-rank approximation of large-scale matrices via randomized methods

2017 ◽  
Vol 74 (2) ◽  
pp. 830-844 ◽  
Author(s):  
Sarvenaz Hatamirad ◽  
Mir Mohsen Pedram
Keyword(s):  
Large Scale ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Rank Approximation ◽  
Randomized Methods

hIPPYlib

2021 ◽  
Vol 47 (2) ◽  
pp. 1-34
Author(s):  
Umberto Villa ◽  
Noemi Petra ◽  
Omar Ghattas
Keyword(s):  
Inverse Problems ◽  
Large Scale ◽  
Low Rank ◽  
Scalable Algorithms ◽  
Low Rank Approximation ◽  
Dimensional Parameter ◽  
Infinite Dimensional ◽  
Bayesian Inverse Problems ◽  
Rank Approximation ◽  
Extensible Software

We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with (possibly) infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitively expensive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYlib is that the solution of the inverse problem is computed at a cost, measured in linearized forward PDE solves, that is independent of the parameter dimension. The mean of the posterior is approximated by the MAP point, which is found by minimizing the negative log-posterior with an inexact matrix-free Newton-CG method. The posterior covariance is approximated by the inverse of the Hessian of the negative log posterior evaluated at the MAP point. The construction of the posterior covariance is made tractable by invoking a low-rank approximation of the Hessian of the log-likelihood. Scalable tools for sample generation are also discussed. hIPPYlib makes all of these advanced algorithms easily accessible to domain scientists and provides an environment that expedites the development of new algorithms.

A low-rank approximation for large-scale 3D controlled-source electromagnetic Gauss-Newton inversion

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. E211-E225 ◽  
Author(s):  
Manuel Amaya ◽  
Jan Petter Morten ◽  
Linus Boman
Keyword(s):  
Large Scale ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Controlled Source ◽  
Newton Equation ◽  
Rank Approximation ◽  
Memory Complexity ◽  
Simultaneous Source ◽  
The Impact ◽  
Matrix Free

We have developed an approximation to the Hessian for inversion of 3D controlled source electromagnetic data. Our approach can considerably reduce the numerical complexity in terms of the number of forward solutions as well as the size and complexity of the calculations required to compute the update direction from the Gauss-Newton equation. The approach makes use of “supershots,” in which several source positions are combined for simultaneous-source simulations. The resulting Hessian can be described as a low-rank approximation to the Gauss-Newton Hessian. The structure of the approximate Hessian facilitates a matrix-free direct solver for the Gauss-Newton equation, and the reduced memory complexity allows the use of a large number of unknowns. We studied the crosstalk introduced in the approximation, and we determined how the dissipative nature of marine electromagnetic field propagation reduces the impact of this noise. Inversion results from recent field data demonstrated the numerical and practical feasibility of the approach.

scholarly journals LOW-RANK APPROXIMATION BASED NON-NEGATIVE MULTI-WAY ARRAY DECOMPOSITION ON EVENT-RELATED POTENTIALS

2014 ◽  
Vol 24 (08) ◽  
pp. 1440005 ◽  
Author(s):  
FENGYU CONG ◽  
GUOXU ZHOU ◽  
PIIA ASTIKAINEN ◽  
QIBIN ZHAO ◽  
QIANG WU ◽  
...  
Keyword(s):  
Large Scale ◽  
Event Related Potentials ◽  
Least Square ◽  
Low Rank ◽  
Tensor Factorization ◽  
Low Rank Approximation ◽  
Significant Group ◽  
Time Frequency ◽  
Related Potentials ◽  
Rank Approximation

Non-negative tensor factorization (NTF) has been successfully applied to analyze event-related potentials (ERPs), and shown superiority in terms of capturing multi-domain features. However, the time-frequency representation of ERPs by higher-order tensors are usually large-scale, which prevents the popularity of most tensor factorization algorithms. To overcome this issue, we introduce a non-negative canonical polyadic decomposition (NCPD) based on low-rank approximation (LRA) and hierarchical alternating least square (HALS) techniques. We applied NCPD (LRAHALS and benchmark HALS) and CPD to extract multi-domain features of a visual ERP. The features and components extracted by LRAHALS NCPD and HALS NCPD were very similar, but LRAHALS NCPD was 70 times faster than HALS NCPD. Moreover, the desired multi-domain feature of the ERP by NCPD showed a significant group difference (control versus depressed participants) and a difference in emotion processing (fearful versus happy faces). This was more satisfactory than that by CPD, which revealed only a group difference.

scholarly journals Image Inpainting Algorithm Based on Low-Rank Approximation and Texture Direction

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Jinjiang Li ◽  
Mengjun Li ◽  
Hui Fan
Keyword(s):  
Level Set ◽  
Large Scale ◽  
Image Inpainting ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Rank Matrix ◽  
Rank Approximation ◽  
Rank Decomposition ◽  
The Impact ◽  
Low Rank Decomposition

Existing image inpainting algorithm based on low-rank matrix approximation cannot be suitable for complex, large-scale, damaged texture image. An inpainting algorithm based on low-rank approximation and texture direction is proposed in the paper. At first, we decompose the image using low-rank approximation method. Then the area to be repaired is interpolated by level set algorithm, and we can reconstruct a new image by the boundary values of level set. In order to obtain a better restoration effect, we make iteration for low-rank decomposition and level set interpolation. Taking into account the impact of texture direction, we segment the texture and make low-rank decomposition at texture direction. Experimental results show that the new algorithm is suitable for texture recovery and maintaining the overall consistency of the structure, which can be used to repair large-scale damaged image.

scholarly journals Low rank approximation methods for MR fingerprinting with large scale dictionaries

2017 ◽  
Vol 79 (4) ◽  
pp. 2392-2400 ◽  
Author(s):  
Mingrui Yang ◽  
Dan Ma ◽  
Yun Jiang ◽  
Jesse Hamilton ◽  
Nicole Seiberlich ◽  
...  
Keyword(s):  
Large Scale ◽  
Low Rank ◽  
Approximation Methods ◽  
Low Rank Approximation ◽  
Rank Approximation

scholarly journals HPOAnnotator: improving large-scale prediction of HPO annotations by low-rank approximation with HPO semantic similarities and multiple PPI networks

2019 ◽  
Vol 12 (S10) ◽  
Author(s):  
Junning Gao ◽  
Lizhi Liu ◽  
Shuwei Yao ◽  
Xiaodi Huang ◽  
Hiroshi Mamitsuka ◽  
...  
Keyword(s):  
Hierarchical Structure ◽  
Matrix Factorization ◽  
Large Scale ◽  
Low Rank ◽  
Superior Performance ◽  
Low Rank Approximation ◽  
Human Phenotype ◽  
Ppi Networks ◽  
The Hierarchical Structure ◽  
Rank Approximation

Abstract Background As a standardized vocabulary of phenotypic abnormalities associated with human diseases, the Human Phenotype Ontology (HPO) has been widely used by researchers to annotate phenotypes of genes/proteins. For saving the cost and time spent on experiments, many computational approaches have been proposed. They are able to alleviate the problem to some extent, but their performances are still far from satisfactory. Method For inferring large-scale protein-phenotype associations, we propose HPOAnnotator that incorporates multiple Protein-Protein Interaction (PPI) information and the hierarchical structure of HPO. Specifically, we use a dual graph to regularize Non-negative Matrix Factorization (NMF) in a way that the information from different sources can be seamlessly integrated. In essence, HPOAnnotator solves the sparsity problem of a protein-phenotype association matrix by using a low-rank approximation. Results By combining the hierarchical structure of HPO and co-annotations of proteins, our model can well capture the HPO semantic similarities. Moreover, graph Laplacian regularizations are imposed in the latent space so as to utilize multiple PPI networks. The performance of HPOAnnotator has been validated under cross-validation and independent test. Experimental results have shown that HPOAnnotator outperforms the competing methods significantly. Conclusions Through extensive comparisons with the state-of-the-art methods, we conclude that the proposed HPOAnnotator is able to achieve the superior performance as a result of using a low-rank approximation with a graph regularization. It is promising in that our approach can be considered as a starting point to study more efficient matrix factorization-based algorithms.

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