Recently projectedgradient (PG) approaches have found many applications in solving the minimization problems underlying nonnegative matrix factorization (NMF). NMF is a linear representation of data that could lead t...
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(纸本)9781424459490
Recently projectedgradient (PG) approaches have found many applications in solving the minimization problems underlying nonnegative matrix factorization (NMF). NMF is a linear representation of data that could lead to sparse result of natural images. To improve the parts-based representation of data some sparseness constraints have been proposed. In this paper the efficiency and execution time of five different PG algorithms and the basic multiplicative algorithm for NMF are compared. The factorization is done for an existing and proposed sparse NMF and the results are compared for all these PG methods. To compare the algorithms the resulted factorizations are used for a hand-written digit classifier
We consider nonmonotone projectedgradient methods based on non-Euclidean distances, which play the role of barrier for a given constraint set. Our basic scheme uses the resulting projection-like maps that produces in...
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We consider nonmonotone projectedgradient methods based on non-Euclidean distances, which play the role of barrier for a given constraint set. Our basic scheme uses the resulting projection-like maps that produces interior trajectories, and combines it with the recent nonmonotone line search technique originally proposed for unconstrained problems by Zhang and Hager. The combination of these two ideas leads to produce a nonmonotone scheme for constrained nonconvex problems, which is proven to converge to a stationary point. Some variants of this algorithm that incorporate spectral steplength are also studied and compared with classical nonmonotone schemes based on the usual Euclidean projection. To validate our approach, we report on numerical results solving bound constrained problems from the CUTEr library collection.
The most popular algorithms for Nonnegative Matrix Factorization (NMF) belong to a class of multiplicative Lee-Seung algorithms which have usually relative low complexity but are characterized by slow-convergence and ...
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The most popular algorithms for Nonnegative Matrix Factorization (NMF) belong to a class of multiplicative Lee-Seung algorithms which have usually relative low complexity but are characterized by slow-convergence and the risk of getting stuck to in local minima. In this paper, we present and compare the performance of additive algorithms based on three different variations of a projectedgradient approach. Additionally, we discuss a novel multilayer approach to NMF algorithms combined with multistart initializations procedure, which in general, considerably improves the performance of all the NMF algorithms. We demonstrate that this approach (the multilayer system with projected gradient algorithms) can usually give much better performance than standard multiplicative algorithms, especially, if data are ill-conditioned, badly-scaled, and/or a number of observations is only slightly greater than a number of nonnegative hidden components. Our new implementations of NMF are demonstrated with the simulations performed for Blind Source Separation (BSS) data.
This paper is devoted to the eigenvalue complementarity problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayl...
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This paper is devoted to the eigenvalue complementarity problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex (Queiroz et al., Math. Comput. 73, 1849-1863, 2004). We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigenvalue by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projectedgradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP.
We consider a dynamical system approach to solve finite-dimensional smooth optimization problems with a compact and connected feasible set. In fact, by the well-known technique of equalizing inequality constraints usi...
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We consider a dynamical system approach to solve finite-dimensional smooth optimization problems with a compact and connected feasible set. In fact, by the well-known technique of equalizing inequality constraints using quadratic slack variables, we transform a general optimization problem into an associated problem without inequality constraints in a higher-dimensional space. We compute the projectedgradient for the latter problem and consider its projection on the feasible set in the original, lower-dimensional space. In this way, we obtain an ordinary differential equation in the original variables, which is specially adapted to treat inequality constraints (for the idea, see Jongen and Stein, Frontiers in Global Optimization, pp. 223-236, Kluwer Academic, Dordrecht, 2003). The article shows that the derived ordinary differential equation possesses the basic properties which make it appropriate to solve the underlying optimization problem: the longtime behavior of its trajectories becomes stationary, all singularities are critical points, and the stable singularities are exactly the local minima. Finally, we sketch two numerical methods based on our approach.
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