Citation Context

...error and the number of endmembers to be extracted), only applies to a restricted noise model, and cannot deal with repeated columns of W in the data set (i.e., repeated endmembers). • Bittorf et al. =-=[6]-=- proposed a method based on the resolution of a single convex optimization problem in n2 variables (cf. Section 5.2). In order to deal with large-scale problems (m ∼ 106, n ∼ 105), a fast incremental ...

Citation Context

... another heuristic approach without any theoretical justification. • Our formulation uses a data-driven modeling approach by regressing each data point against all others. As noted by Bittorf et. al. =-=[14]-=- this type of approach appears in a number of other 25 factorization problems. In particular, [7] and recent variations [7,14] use a convex formulation very similar to SSC for the purpose of non-negat...

Citation Context

...procedures that lack optimality guarantees beyond convergence to a stationary point of the objective function for approximate NMF. Very recently, in a series of elegant papers (Arora et al., 2012a;b; =-=Bittorf et al., 2012-=-; Gillis & Vavasis, 2012; Esser et al., 2012; Elhamifar et al., 2012), promising alternative approaches have been developed based on certain separability assumption on the data which enables the NMF p...

Citation Context

...]Π where W and H ′ are nonnegative matrices, Π is a permutation matrix and N is the noise, find a set K of r indices such that M̃(:,K) ≈W . Several algorithms have been proposed to solve this problem =-=[2, 3, 5, 9, 10, 13, 15]-=-. In this paper, our focus is on the linear programming (LP) model proposed by Bittorf, Recht, Ré and Tropp [5] and referred to as Hottopixx. It is described in the next section. Remark 1 (Nonnegativi...

Citation Context

...negative matrix factorization (NMF) problem. To address the scenario where only the realization X is known and not A, several papers (Lee & Seung, 1999; Donoho & Stodden, 2004; Cichocki et al., 2009; =-=Recht et al., 2012-=-) attempt to minimize a regularized cost function. Nevertheless, this joint optimization is non-convex and sub-optimal strategies have been used. Unfortunately, when N ≪ W which is often the case, man...

Citation Context

... as the pure-pixel assumption; see [7] and the references therein. In practice, the input separable matrixM is perturbed with some noise and it is therefore desirable to design robust algorithms; see =-=[1, 2, 3, 4, 5, 7, 8]-=-. In fact, in the noiseless case, the problem is rather easy and reduces to identifying the vertices of the convex hull of a set of points. In this paper, we will focus on the algorithm of Bittorf, Re...

Citation Context

...ily focuses on mixed signals drawn from the superposition model (1), recent extensions to nonlinear mixing models arise in blind deconvolution, source separation, and nonnegative matrix factorization =-=[10, 11, 12]-=-. We will see that the same techniques that let us demix superimposed signals reappear in nonlinear demixing problems. The role of convexity Convex optimization provides a unifying theme for all of th...

Citation Context

...f noise (only asymptotic results were proved, that is, when the noise level goes to zero and when no duplicates are present [40]). A rather different approach to enforce row sparsity was suggested in =-=[9]-=-, and later improved in [54]. Row sparsity of Y is enforced by (i) minimizing a weighted sum of the diagonal entries of Y hence enforcing diag(Y ) to be sparse (in fact, this is nothing but a weighted...

Citation Context

...ynomial-time in general, and most algorithms for NMF (e.g., [34]) perform a form a block coordinate descent with no guarantees (see hardness results and particular situations of actual solvability in =-=[3, 46]-=-). In this paper, we follow a convex relaxation approach. 1hal-00861118, version 1 - 12 Sep 2013 We impose some structure on each column of U and V and consider a general convex framework which amoun...

Citation Context

...arch or sampling methods [12, 18, 19], which may suffer from local optima. Much recent progress has been made on developing spectral decomposition [1, 2, 3] and nonnegative matrix factorization (NMF) =-=[4, 5, 6, 7]-=- methods to infer latent topic-word distributions. Instead of finding MLE estimates, which is a known NP-hard problem [6], these methods assume that the documents are i.i.d. sampled from a topic model...