Low-Rank Approximation Task
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A Low-Rank Approximation Task is an approximation task in which the loss function measures the fit between a given matrix dataset and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced matrix rank.
- Context:
- It can be solved by a Low-Rank Approximation System (that implements a low-rank approximation algorithm).
- See: Weighted Matrix Factorization, Orthogonal Regression, Data Compression, Nonnegative Matrix, Hankel Matrix, Principal Component Analysis, Factor Analysis, Total Least Squares, Latent Semantic Analysis.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/low-rank_approximation Retrieved:2015-2-15.
- In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure.
Low-rank approximation is closely related to:
- In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure.
2004
- (Ye, 2004) ⇒ Jieping Ye. (2004). “Generalized Low Rank Approximations of Matrices.” In: Proceedings of the twenty-first International Conference on Machine learning. ISBN:1-58113-838-5 doi:10.1145/1015330.1015347
- QUOTE: We consider the problem of computing low rank approximations of matrices. The novelty of our approach is that the low rank approximations are on a sequence of matrices. Unlike the problem of low rank approximations of a single matrix, which was well studied in the past, the proposed algorithm in this paper does not admit a closed form solution in general.