Sparse Recovery Using Conjugate Gradient and Orthogonal Triangular Decomposition

In this article, we propose the matching pursuit algorithm of combinatorial optimization based CGLS and LSQR. We use non-negative matrix factorization for measuring discrepancy of solution sequence between CGLS and LSQR, and represent combinatorial optimization based CGLS and LSQ to choose optimal solution sequences. The experiments indicate our method is extended to the case where target signal has been corrupted by noise, it demonstrate perfectly recovery ability of signal with noise.

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Combinatorial Optimization Based on Conjugate Gradient and Orthogonal Triangular Decomposition for Sparse Recovery

Journal of Information and Computational Science ◽

10.12733/jics20103643 ◽

2014 ◽

Vol 11 (4) ◽

pp. 1343-1356

Author(s):

Hailing Xiong

Keyword(s):

Combinatorial Optimization ◽

Conjugate Gradient ◽

Sparse Recovery ◽

Triangular Decomposition

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PROBABILISTIC NON-NEGATIVE MATRIX FACTORIZATION: THEORY AND APPLICATION TO MICROARRAY DATA ANALYSIS

Journal of Bioinformatics and Computational Biology ◽

10.1142/s0219720014500012 ◽

2014 ◽

Vol 12 (01) ◽

pp. 1450001 ◽

Cited By ~ 12

Author(s):

BELHASSEN BAYAR ◽

NIDHAL BOUAYNAYA ◽

ROMAN SHTERENBERG

Keyword(s):

Physical Interpretation ◽

Matrix Factorization ◽

Dna Microarrays ◽

Optimal Solution ◽

Microarray Data Analysis ◽

Factorization Problem ◽

The Stability ◽

Update Rules ◽

Negative Factors ◽

Non Negative Matrix Factorization

Non-negative matrix factorization (NMF) has proven to be a useful decomposition technique for multivariate data, where the non-negativity constraint is necessary to have a meaningful physical interpretation. NMF reduces the dimensionality of non-negative data by decomposing it into two smaller non-negative factors with physical interpretation for class discovery. The NMF algorithm, however, assumes a deterministic framework. In particular, the effect of the data noise on the stability of the factorization and the convergence of the algorithm are unknown. Collected data, on the other hand, is stochastic in nature due to measurement noise and sometimes inherent variability in the physical process. This paper presents new theoretical and applied developments to the problem of non-negative matrix factorization (NMF). First, we generalize the deterministic NMF algorithm to include a general class of update rules that converges towards an optimal non-negative factorization. Second, we extend the NMF framework to the probabilistic case (PNMF). We show that the Maximum a posteriori (MAP) estimate of the non-negative factors is the solution to a weighted regularized non-negative matrix factorization problem. We subsequently derive update rules that converge towards an optimal solution. Third, we apply the PNMF to cluster and classify DNA microarrays data. The proposed PNMF is shown to outperform the deterministic NMF and the sparse NMF algorithms in clustering stability and classification accuracy.

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