# Nonnegative matrix factorization : complexity, algorithms and applications

@inproceedings{Gillis2011NonnegativeMF, title={Nonnegative matrix factorization : complexity, algorithms and applications}, author={Nicolas Gillis}, year={2011} }

Linear dimensionality reduction techniques such as principal component analysis are powerful tools for the analysis of high-dimensional data. In this thesis, we explore a closely related problem, namely nonnegative matrix factorization (NMF), a low-rank matrix approximation problem with nonnegativity constraints. More precisely, we seek to approximate a given nonnegative matrix with the product of two low-rank nonnegative matrices. These nonnegative factors can be interpreted in the same way as…

## Figures and Tables from this paper

figure 1.2 figure 3.8 figure 3.9 table 4.1 figure 4.12 figure 4.13 figure 4.14 figure 4.15 figure 4.16 figure 4.17 figure 4.18 figure 4.19 figure 4.2 table 4.2 figure 4.3 table 4.4 figure 4.5 table 4.5 figure 4.6 table 4.6 figure 4.7 table 4.7 figure 4.8 table 5.2 figure 5.3 figure 5.4 table 6.1 table 6.2 figure 6.3 table 6.3 table 6.4 figure 6.5 figure 6.6 figure 7.11 figure 7.12 figure 7.14 figure 7.16 figure 7.17 figure 7.19 figure 7.2 figure 7.20 figure 7.22 figure 7.6 figure 7.8 figure 7.9

## 58 Citations

Nonnegative matrix and tensor factorizations, least squares problems, and applications

- Computer Science
- 2011

An accelerated block principal pivoting method is proposed to solve the NLS problems, thereby significantly speeding up the NMF and NTF computation and proposing mixed-norm regularization to promote group-level sparsity.

A multilevel approach for nonnegative matrix factorization

- Computer ScienceJ. Comput. Appl. Math.
- 2012

Nonnegative Matrix Factorization Using Nonnegative Polynomial Approximations

- Computer ScienceIEEE Signal Processing Letters
- 2017

A novel algorithm for nonnegative matrix factorization, in which the factors are modeled by nonnegative polynomials, to obtain an optimization problem without external nonnegativity constraints, which can be solved using conventional quasi-Newton or nonlinear least-squares methods.

The Why and How of Nonnegative Matrix Factorization

- Computer ScienceArXiv
- 2014

A recent subclass of NMF problems is presented, referred to as near-separable NMF, that can be solved efficiently (that is, in polynomial time), even in the presence of noise.

Sparse and unique nonnegative matrix factorization through data preprocessing

- Computer ScienceJ. Mach. Learn. Res.
- 2012

A completely new way to obtaining more well-posed NMF problems whose solutions are sparser is introduced, based on the preprocessing of the nonnegative input data matrix, and relies on the theory of M-matrices and the geometric interpretation of NMF.

Efficient and Non-Convex Coordinate Descent for Symmetric Nonnegative Matrix Factorization

- Computer ScienceIEEE Transactions on Signal Processing
- 2016

This paper proposes simple and very efficient coordinate descent schemes, which solve a series of fourth-order univariate subproblems exactly and derive convergence guarantees for these schemes, and shows that they perform favorably compared to recent state-of-the-art methods on synthetic and real-world datasets.

Nonnegative Matrix Factorization: Algorithms, Complexity and Applications

- Computer ScienceISSAC
- 2015

Recent progress on the question of how quickly can the authors compute the nonnegative rank (r) of an m x n matrix is surveyed.

Two algorithms for orthogonal nonnegative matrix factorization with application to clustering

- Computer ScienceNeurocomputing
- 2014

Generalized Low Rank Models

- Computer ScienceFound. Trends Mach. Learn.
- 2016

This work extends the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types, and proposes several parallel algorithms for fitting generalized low rank models.

## References

SHOWING 1-10 OF 139 REFERENCES

Using underapproximations for sparse nonnegative matrix factorization

- Computer SciencePattern Recognit.
- 2010

A multilevel approach for nonnegative matrix factorization

- Computer ScienceJ. Comput. Appl. Math.
- 2012

Nonnegative Factorization and The Maximum Edge Biclique Problem

- Computer Science
- 2008

This paper shows that when the matrix to be factored is not required to be nonnegative, the corresponding problem (R1NF) becomes NP-hard and designs a new type of biclique finding algorithm based on the application of a block-coordinate descent scheme to R1NF.

Nonnegative Matrix Approximation: Algorithms and Applications

- Computer Science
- 2006

Generic methods for minimizingeralized divergences between the input and its low rank approximant and interesting extensions such as the use of penalty function, non-linear relationships via “link” functions, weighted errors, and multi-factor approximations are considered.

NONNEGATIVE RANK FACTORIZATION VIA RANK REDUCTION

- Computer Science, Mathematics
- 2008

The proposed algorithm is to recurrently extract a rank-one nonnegative portion from the previous matrix, starting with A, while keeping the residual nonnegative and lowering its rank by one, and can equally be applied to another important class of completely positive matrices.

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.

Nonnegative matrix factorization via rank-one downdate

- Computer ScienceICML '08
- 2008

An algorithm called rank-one downdate (R1D) is proposed for computing an NMF that is partly motivated by the singular value decomposition, and establishes a theoretical result that maximizing this objective function corresponds to correctly classifying articles in a nearly separable corpus.

Fast Local Algorithms for Large Scale Nonnegative Matrix and Tensor Factorizations

- Computer ScienceIEICE Trans. Fundam. Electron. Commun. Comput. Sci.
- 2009

A class of optimized local algorithms which are referred to as Hierarchical Alternating Least Squares (HALS) algorithms, which work well for NMF-based blind source separation (BSS) not only for the over-determined case but also for an under-d determined (over-complete) case if data are sufficiently sparse.

On the Equivalence of Nonnegative Matrix Factorization and K-means - Spectral Clustering

- Computer Science
- 2005

A systematic analysis of nonnegative matrix factorization (NMF) relating to data cluster- ing and the importance of orthogonality in NMF and soft clustering nature of NMF is emphasized.