Nonnegative Matrix and Tensor Factorization [Lecture Notes]

@article{Cichocki2008NonnegativeMA,
  title={Nonnegative Matrix and Tensor Factorization [Lecture Notes]},
  author={Andrzej Cichocki and Rafał Zdunek and Shun‐ichi Amari},
  journal={IEEE Signal Processing Magazine},
  year={2008},
  volume={25},
  pages={142-145}
}
In these lecture notes, the authors have outlined several approaches to solve a NMF/NTF problem. The following main conclusions can be drawn: 1) Multiplicative algorithms are not necessary the best approaches for NMF, especially if data representations are not very redundant or sparse. 2) Much better performance can be achieved using the FP-ALS (especially for large-scale problems), IPC, and QN methods. 3) To achieve high performance it is quite important to use the multilayer structure with… 
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References

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Projected Gradient Methods for Nonnegative Matrix Factorization
TLDR
This letter proposes two projected gradient methods for nonnegative matrix factorization, both of which exhibit strong optimization properties and discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach.
Novel Multi-layer Non-negative Tensor Factorization with Sparsity Constraints
In this paper we present a new method of 3D non-negative tensor factorization (NTF) that is robust in the presence of noise and has many potential applications, including multi-way blind source
Generalized Nonnegative Matrix Approximations with Bregman Divergences
TLDR
This paper makes algorithmic progress by modeling and solving (using multiplicative updates) new generalized NNMA problems that minimize Bregman divergences between the input matrix and its low-rank approximation.
Hierarchical ALS Algorithms for Nonnegative Matrix and 3D Tensor Factorization
TLDR
This paper proposes to use local cost functions whose simultaneous or sequential (one by one) minimization leads to a very simple ALS algorithm which works under some sparsity constraints both for an under-determined and overdetermined model.
Nonnegative matrix factorization with constrained second-order optimization
Algorithms and applications for approximate nonnegative matrix factorization
Extended SMART Algorithms for Non-negative Matrix Factorization
TLDR
A family of new extended SMART (Simultaneous Multiplicative Algebraic Reconstruction Technique) algorithms for Non-negative Matrix Factorization (NMF) are derived by improved efficiency and convergence rate and can be applied for various distributions of data and additive noise.
Learning the parts of objects by non-negative matrix factorization
TLDR
An algorithm for non-negative matrix factorization is demonstrated that is able to learn parts of faces and semantic features of text and is in contrast to other methods that learn holistic, not parts-based, representations.
Nonnegative matrix factorization for rapid recovery of constituent spectra in magnetic resonance chemical shift imaging of the brain
TLDR
An algorithm for blindly recovering constituent source spectra from magnetic resonance (MR) chemical shift imaging (CSI) of the human brain is presented, showing that it can be used to recover tissue-specific spectra given a processing hierarchy that proceeds coarse-to-fine.
Metagenes and molecular pattern discovery using matrix factorization
TLDR
Nonnegative matrix factorization is described, an algorithm based on decomposition by parts that can reduce the dimension of expression data from thousands of genes to a handful of metagenes, and found less sensitive to a priori selection of genes or initial conditions and able to detect alternative or context-dependent patterns of gene expression in complex biological systems.
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