scholarly journals Alternate Low-Rank Matrix Approximation in Latent Semantic Analysis

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Fahrettin Horasan ◽  
Hasan Erbay ◽  
Fatih Varçın ◽  
Emre Deniz
Keyword(s):  
Latent Semantic Analysis ◽  
Semantic Analysis ◽  
Main Tool ◽  
Good Alternative ◽  
Irrelevant Information ◽  
Low Rank ◽  
Computational Time ◽  
Low Rank Approximation ◽  
Document Collection ◽  
Rank Approximation

The latent semantic analysis (LSA) is a mathematical/statistical way of discovering hidden concepts between terms and documents or within a document collection (i.e., a large corpus of text). Each document of the corpus and terms are expressed as a vector with elements corresponding to these concepts to form a term-document matrix. Then, the LSA uses a low-rank approximation to the term-document matrix in order to remove irrelevant information, to extract more important relations, and to reduce the computational time. The irrelevant information is called as “noise” and does not have a noteworthy effect on the meaning of the document collection. This is an essential step in the LSA. The singular value decomposition (SVD) has been the main tool obtaining the low-rank approximation in the LSA. Since the document collection is dynamic (i.e., the term-document matrix is subject to repeated updates), we need to renew the approximation. This can be done via recomputing the SVD or updating the SVD. However, the computational time of recomputing or updating the SVD of the term-document matrix is very high when adding new terms and/or documents to preexisting document collection. Therefore, this issue opened the door of using other matrix decompositions for the LSA as ULV- and URV-based decompositions. This study shows that the truncated ULV decomposition (TULVD) is a good alternative to the SVD in the LSA modeling.

Improved MR image denoising via low- rank approximation and Laplacian-of-Gaussian edge detector

2020 ◽  
Vol 14 (12) ◽  
pp. 2791-2798
Author(s):  
Xiaoqun Qiu ◽  
Zhen Chen ◽  
Saifullah Adnan ◽  
Hongwei He
Keyword(s):  
Image Denoising ◽  
Low Rank ◽  
Edge Detector ◽  
Low Rank Approximation ◽  
Mr Image ◽  
Rank Approximation

scholarly journals Fast Joint Multiband Reconstruction From Wideband Images Based on Low-Rank Approximation

2020 ◽  
Vol 6 ◽  
pp. 922-933
Author(s):  
M. Amine Hadj-Youcef ◽  
Francois Orieux ◽  
Alain Abergel ◽  
Aurelia Fraysse
Keyword(s):  
Low Rank ◽  
Low Rank Approximation ◽  
Rank Approximation

scholarly journals Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product

2021 ◽  
Vol 11 (10) ◽  
pp. 4582
Author(s):  
Kensuke Tanioka ◽  
Satoru Hiwa
Keyword(s):  
Dimensional Reduction ◽  
Low Rank ◽  
Inner Product ◽  
Low Rank Approximation ◽  
Correlation Matrices ◽  
Rank Matrix ◽  
Clustering Approach ◽  
Rank Approximation ◽  
The Difference ◽  
Low Rank Matrix

In the domain of functional magnetic resonance imaging (fMRI) data analysis, given two correlation matrices between regions of interest (ROIs) for the same subject, it is important to reveal relatively large differences to ensure accurate interpretation. However, clustering results based only on differences tend to be unsatisfactory and interpreting the features tends to be difficult because the differences likely suffer from noise. Therefore, to overcome these problems, we propose a new approach for dimensional reduction clustering. Methods: Our proposed dimensional reduction clustering approach consists of low-rank approximation and a clustering algorithm. The low-rank matrix, which reflects the difference, is estimated from the inner product of the difference matrix, not only from the difference. In addition, the low-rank matrix is calculated based on the majorize–minimization (MM) algorithm such that the difference is bounded within the range −1 to 1. For the clustering process, ordinal k-means is applied to the estimated low-rank matrix, which emphasizes the clustering structure. Results: Numerical simulations show that, compared with other approaches that are based only on differences, the proposed method provides superior performance in recovering the true clustering structure. Moreover, as demonstrated through a real-data example of brain activity measured via fMRI during the performance of a working memory task, the proposed method can visually provide interpretable community structures consisting of well-known brain functional networks, which can be associated with the human working memory system. Conclusions: The proposed dimensional reduction clustering approach is a very useful tool for revealing and interpreting the differences between correlation matrices, even when the true differences tend to be relatively small.

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.

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