On the Complexity of Nonnegative Matrix Factorization

  title={On the Complexity of Nonnegative Matrix Factorization},
  author={Stephen A. Vavasis},
  journal={SIAM J. Optim.},
  • S. Vavasis
  • Published 30 August 2007
  • Computer Science
  • SIAM J. Optim.
Nonnegative matrix factorization (NMF) has become a prominent technique for the analysis of image databases, text databases, and other information retrieval and clustering applications. The problem is most naturally posed as continuous optimization. In this report, we define an exact version of NMF. Then we establish several results about exact NMF: (i) that it is equivalent to a problem in polyhedral combinatorics; (ii) that it is NP-hard; and (iii) that a polynomial-time local search… 

Figures from this paper

Nonnegative Matrix Factorization: Algorithms, Complexity and Applications
Recent progress on the question of how quickly can the authors compute the nonnegative rank (r) of an m x n matrix is surveyed.
The Why and How of Nonnegative Matrix Factorization
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.
A multilevel approach for nonnegative matrix factorization
Sparse and unique nonnegative matrix factorization through data preprocessing
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.
Convex analysis of Nonnegative Matrix Factorization
Different ways in which NMF can be relaxed to a convex program are summarized, and how matrix under-approximation constraints can be used to give sparser factorizations than what the general NMF delivers which is practically useful in many cases.
Nonnegative Matrix Factorization
SVD is a classical method for matrix factorization, which gives the optimal low-rank approximation to a real-valued matrix in terms of the squared error.
On Rationality of Nonnegative Matrix Factorization
It is shown that state minimization of labeled Markov chains can require the introduction of irrational transition probabilities, and complements this result with an NP-complete version of NMF for which rational numbers suffice.
Nonnegative Factorization and The Maximum Edge Biclique Problem
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.
Using underapproximations for sparse nonnegative matrix factorization
Computing a nonnegative matrix factorization -- provably
This work gives an algorithm that runs in time polynomial in n, m and r under the separablity condition identified by Donoho and Stodden in 2003, and is the firstPolynomial-time algorithm that provably works under a non-trivial condition on the input matrix.


Nonnegative matrix factorization via rank-one downdate
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.
Nonnegative ranks, decompositions, and factorizations of nonnegative matrices
SVD based initialization: A head start for nonnegative matrix factorization
Sparse non-negative matrix factorizations via alternating non-negativity-constrained least squares for microarray data analysis
The experimental results illustrate that the proposed sparse NMF algorithm often achieves better clustering performance with shorter computing time compared to other existing NMF algorithms.
Learning the parts of objects by non-negative matrix factorization
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.
Probabilistic Latent Semantic Analysis
This work proposes a widely applicable generalization of maximum likelihood model fitting by tempered EM, based on a mixture decomposition derived from a latent class model which results in a more principled approach which has a solid foundation in statistics.
Complexity and Real Computation
  • L. Blum
  • Mathematics, Computer Science
    Springer New York
  • 1998
This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.
Using Rank-One Biclusters to Classify Microarray Data
This paper proposes a novel algorithm for learning a microarray classier by reducing the dimensionality of the data matrix using biclusters, where each bicluster is a subset of genes andA subset of samples whose expression values have similar patterns.
Systems of distinct representatives and linear algebra
So me purposes of thi s paper are: (1) To take se riously the term , " term rank. " (2) To ma ke an issue of not " rea rra nging rows a nd colu mns" by not "a rranging" the m in the firs t place. (3)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge