325 research outputs found
Descent methods for Nonnegative Matrix Factorization
In this paper, we present several descent methods that can be applied to
nonnegative matrix factorization and we analyze a recently developped fast
block coordinate method called Rank-one Residue Iteration (RRI). We also give a
comparison of these different methods and show that the new block coordinate
method has better properties in terms of approximation error and complexity. By
interpreting this method as a rank-one approximation of the residue matrix, we
prove that it \emph{converges} and also extend it to the nonnegative tensor
factorization and introduce some variants of the method by imposing some
additional controllable constraints such as: sparsity, discreteness and
smoothness.Comment: 47 pages. New convergence proof using damped version of RRI. To
appear in Numerical Linear Algebra in Signals, Systems and Control. Accepted.
Illustrating Matlab code is included in the source bundl
A multilevel approach for nonnegative matrix factorization
Nonnegative Matrix Factorization (NMF) is the problem of approximating a nonnegative matrix with the product of two low-rank nonnegative matrices and has been shown to be particularly useful in many applications, e.g., in text mining, image processing, computational biology, etc. In this paper, we explain how algorithms for NMF can be embedded into the framework of multi- level methods in order to accelerate their convergence. This technique can be applied in situations where data admit a good approximate representation in a lower dimensional space through linear transformations preserving nonnegativity. A simple multilevel strategy is described and is experi- mentally shown to speed up significantly three popular NMF algorithms (alternating nonnegative least squares, multiplicative updates and hierarchical alternating least squares) on several standard image datasets.nonnegative matrix factorization, algorithms, multigrid and multilevel methods, image processing
Bi-Objective Nonnegative Matrix Factorization: Linear Versus Kernel-Based Models
Nonnegative matrix factorization (NMF) is a powerful class of feature
extraction techniques that has been successfully applied in many fields, namely
in signal and image processing. Current NMF techniques have been limited to a
single-objective problem in either its linear or nonlinear kernel-based
formulation. In this paper, we propose to revisit the NMF as a multi-objective
problem, in particular a bi-objective one, where the objective functions
defined in both input and feature spaces are taken into account. By taking the
advantage of the sum-weighted method from the literature of multi-objective
optimization, the proposed bi-objective NMF determines a set of nondominated,
Pareto optimal, solutions instead of a single optimal decomposition. Moreover,
the corresponding Pareto front is studied and approximated. Experimental
results on unmixing real hyperspectral images confirm the efficiency of the
proposed bi-objective NMF compared with the state-of-the-art methods
Clustering and Latent Semantic Indexing Aspects of the Nonnegative Matrix Factorization
This paper provides a theoretical support for clustering aspect of the
nonnegative matrix factorization (NMF). By utilizing the Karush-Kuhn-Tucker
optimality conditions, we show that NMF objective is equivalent to graph
clustering objective, so clustering aspect of the NMF has a solid
justification. Different from previous approaches which usually discard the
nonnegativity constraints, our approach guarantees the stationary point being
used in deriving the equivalence is located on the feasible region in the
nonnegative orthant. Additionally, since clustering capability of a matrix
decomposition technique can sometimes imply its latent semantic indexing (LSI)
aspect, we will also evaluate LSI aspect of the NMF by showing its capability
in solving the synonymy and polysemy problems in synthetic datasets. And more
extensive evaluation will be conducted by comparing LSI performances of the NMF
and the singular value decomposition (SVD), the standard LSI method, using some
standard datasets.Comment: 28 pages, 5 figure
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