# NeNMF: An Optimal Gradient Method for Nonnegative Matrix Factorization

@article{Guan2012NeNMFAO, title={NeNMF: An Optimal Gradient Method for Nonnegative Matrix Factorization}, author={Naiyang Guan and Dacheng Tao and Zhigang Luo and Bo Yuan}, journal={IEEE Transactions on Signal Processing}, year={2012}, volume={60}, pages={2882-2898} }

Nonnegative matrix factorization (NMF) is a powerful matrix decomposition technique that approximates a nonnegative matrix by the product of two low-rank nonnegative matrix factors. It has been widely applied to signal processing, computer vision, and data mining. Traditional NMF solvers include the multiplicative update rule (MUR), the projected gradient method (PG), the projected nonnegative least squares (PNLS), and the active set method (AS). However, they suffer from one or some of the…

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## References

SHOWING 1-10 OF 38 REFERENCES

Alternating projected Barzilai-Borwein methods for nonnegative matrix factorization.

- Computer Science
- 2009

Four algorithms for solving the nonsmooth nonnegative matrix factorization (nsNMF) problems are proposed and a numerical comparison between the APBB2 method and the Hierarchical Alternating Least Squares (HAL S)/Rank-one Residue Iteration (RRI) method is provided.

Non-negative Matrix Factorization with Quasi-Newton Optimization

- Computer ScienceICAISC
- 2006

This work derived a relatively simple second-order quasi-Newton method for NMF: so-called Amari alpha divergence, which has been extensively tested for blind source separation problems, both for signals and images.

Toward Faster Nonnegative Matrix Factorization: A New Algorithm and Comparisons

- Computer Science2008 Eighth IEEE International Conference on Data Mining
- 2008

This paper presents a novel algorithm for NMF based on the ANLS framework that builds upon the block principal pivoting method for the nonnegativity constrained least squares problem that overcomes some limitations of active set methods.

Fast Newton-type Methods for the Least Squares Nonnegative Matrix Approximation Problem

- Computer ScienceSDM
- 2007

New and improved algorithms for the least-squares NNMA problem are presented which are not only theoretically well-founded, but also overcome many of the deficiencies of other methods, and use non-diagonal gradient scaling to obtain rapid convergence.

Non-negative Matrix Factorization on Manifold

- Computer Science2008 Eighth IEEE International Conference on Data Mining
- 2008

This paper construct an affinity graph to encode the geometrical information and seek a matrix factorization which respects the graph structure and demonstrates the success of this novel algorithm by applying it on real world problems.

Manifold Regularized Discriminative Nonnegative Matrix Factorization With Fast Gradient Descent

- Computer ScienceIEEE Transactions on Image Processing
- 2011

The manifold regularization and the margin maximization to NMF are introduced and the manifold regularized discriminative NMF (MD-NMF) is obtained to overcome the aforementioned problems.

On the Convergence of Multiplicative Update Algorithms for Nonnegative Matrix Factorization

- Computer Science, MathematicsIEEE Transactions on Neural Networks
- 2007

This paper proposes slight modifications of existing updates and proves their convergence, and techniques invented in this paper may be applied to prove the convergence for other bound-constrained optimization problems.

Projected Gradient Methods for Nonnegative Matrix Factorization

- Computer ScienceNeural Computation
- 2007

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.

An accelerated gradient method for trace norm minimization

- Computer ScienceICML '09
- 2009

This paper exploits the special structure of the trace norm, based on which it is proposed an extended gradient algorithm that converges as O(1/k) and proposes an accelerated gradient algorithm, which achieves the optimal convergence rate of O( 1/k2) for smooth problems.

Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values†

- Mathematics
- 1994

A new variant ‘PMF’ of factor analysis is described. It is assumed that X is a matrix of observed data and σ is the known matrix of standard deviations of elements of X. Both X and σ are of…