Essential Matrix Decomposition without Dependence of Singular Value Decomposition

2015 ◽  
Vol 12 (8) ◽  
pp. 3299-3309
Author(s):  
Chunfu Wu
Keyword(s):  
Singular Value Decomposition ◽  
Matrix Decomposition ◽  
Singular Value ◽  
Essential Matrix ◽  
Value Decomposition

scholarly journals Split-and-Combine Singular Value Decomposition for Large-Scale Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jengnan Tzeng
Keyword(s):  
Singular Value Decomposition ◽  
Large Scale ◽  
Semantic Analysis ◽  
Computational Cost ◽  
Matrix Decomposition ◽  
Singular Value ◽  
Fast Method ◽  
The Matrix ◽  
Scale Matrix ◽  
Value Decomposition

The singular value decomposition (SVD) is a fundamental matrix decomposition in linear algebra. It is widely applied in many modern techniques, for example, high- dimensional data visualization, dimension reduction, data mining, latent semantic analysis, and so forth. Although the SVD plays an essential role in these fields, its apparent weakness is the order three computational cost. This order three computational cost makes many modern applications infeasible, especially when the scale of the data is huge and growing. Therefore, it is imperative to develop a fast SVD method in modern era. If the rank of matrix is much smaller than the matrix size, there are already some fast SVD approaches. In this paper, we focus on this case but with the additional condition that the data is considerably huge to be stored as a matrix form. We will demonstrate that this fast SVD result is sufficiently accurate, and most importantly it can be derived immediately. Using this fast method, many infeasible modern techniques based on the SVD will become viable.

Rec-CFSVD++: Implementing Recommendation System Using Collaborative Filtering and Singular Value Decomposition (SVD)++

2021 ◽  
pp. 1-19
Author(s):  
Taushif Anwar ◽  
V. Uma ◽  
Gautam Srivastava
Keyword(s):  
Singular Value Decomposition ◽  
Collaborative Filtering ◽  
Error Rate ◽  
Recommendation System ◽  
Recommendation Systems ◽  
Matrix Decomposition ◽  
Cold Start ◽  
Singular Value ◽  
Decomposition Approach ◽  
Value Decomposition

In recommender systems, Collaborative Filtering (CF) plays an essential role in promoting recommendation services. The conventional CF approach has limitations, namely data sparsity and cold-start. The matrix decomposition approach is demonstrated to be one of the effective approaches used in developing recommendation systems. This paper presents a new approach that uses CF and Singular Value Decomposition (SVD)[Formula: see text] for implementing a recommendation system. Therefore, this work is an attempt to extend the existing recommendation systems by (i) finding similarity between user and item from rating matrices using cosine similarity; (ii) predicting missing ratings using a matrix decomposition approach, and (iii) recommending top-N user-preferred items. The recommender system’s performance is evaluated considering Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). Performance evaluation is accomplished by comparing the systems developed using CF in combination with six different algorithms, namely SVD, SVD[Formula: see text], Co-Clustering, KNNBasic, KNNBaseline, and KNNWithMeans. We have experimented using MovieLens 100[Formula: see text]K, MovieLens 1[Formula: see text]M, and BookCrossing datasets. The results prove that the proposed approach gives a lesser error rate when cross-validation ([Formula: see text]) is performed. The experimental results show that the lowest error rate is achieved with MovieLens 100[Formula: see text]K dataset ([Formula: see text], [Formula: see text]). The proposed approach also alleviates the sparsity and cold-start problems and recommends the relevant items.

DESIGN OF 2-D DIGITAL FILTERS BASED ON THE OPTIMAL DECOMPOSITION OF MAGNITUDE SPECIFICATIONS

1993 ◽  
Vol 03 (03) ◽  
pp. 733-756 ◽  
Author(s):  
T.-B. DENG ◽  
M. KAWAMATA ◽  
T. HIGUCHI
Keyword(s):  
Singular Value Decomposition ◽  
Digital Filters ◽  
Matrix Decomposition ◽  
Decomposition Methods ◽  
Singular Value ◽  
The Other ◽  
Design Technique ◽  
One Dimensional ◽  
Rms Error ◽  
Value Decomposition

The optimal decomposition (OD) technique for decomposing 2-D magnitude specifications into 1-D ones is proposed. Differing from the conventional matrix decomposition methods such as the singular value decomposition (SVD), the OD of a 2-D magnitude specification matrix results in 1-D magnitude specifications which are always non-negative while the root mean-squared (rms) error in decomposition is minimum. Therefore, the OD is more suitable for designing 2-D digital filters (2DDFs) through designing one-dimensional digital filters (1DDFs) than the conventional matrix decomposition methods. Based on the OD, the problem of designing a recursive 2DDF can be reduced to one of designing a pair of 1DDFs, one 1-input/multi-output, and the other multi-input/ 1-output. Consequently, 2DDFs can easily be obtained by designing 1DDFs using the well-established 1-D design techniques. Three design examples are presented to illustrate that this OD-based 2DDF design technique is extremely efficient.

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