On the characterization of totally nonpositive matrices

The final publication is available at Springer via http://dx.doi.org/10.1007/s40324-016-0073-1[EN] A nonpositive real matrix $A= (a_{ij})_{1 \leq i, j \leq n}$ is said to be totally nonpositive (negative) if all its minors are nonpositive (negative) and it is abbreviated as t.n.p. (t.n.). In this work a bidiagonal factorization of a nonsingular t.n.p. matrix $A$ is computed and it is stored in an matrix represented by $\mathcal{BD}_{(t.n.p.)}(A)$ when $a_{11}< 0$ (or $\mathcal{BD}_{(zero)}(A)$ when $a_{11}= 0$). As a converse result, an efficient algorithm to know if an matrix $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(zero)}(A)$) is the bidiagonal factorization of a t.n.p. matrix with $a_{11}<0$ ($a_{11}= 0$) is given. Similar results are obtained for t.n. matrices using the matrix $\mathcal{BD}_{(t.n.)}(A)$, and these characterizations are extended to rectangular t.n.p. (t.n.) matrices. Finally, the bidiagonal factorization of the inverse of a nonsingular t.n.p. (t.n.) matrix $A$ is directly obtained from $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(t.n.)}(A)$).This research was supported by the Spanish DGI grant MTM2013-43678-P and by the Chilean program FONDECYT 1100029Cantó Colomina, R.; Pelaez, MJ.; Urbano Salvador, AM. (2016). On the characterization of totally nonpositive matrices. SeMA Journal. 73(4):347-368. doi:10.1007/s40324-016-0073-1S347368734Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)Alonso, P., Peña, J.M., Serrano, M.L.: Almost strictly totally negative matrices: an algorithmic characterization. J. Comput. Appl. Math. 275, 238–246 (2015)Bapat, R.B., Raghavan, T.E.S.: Nonnegative Matrices and Applications. Cambridge University Press, New York (1997)Cantó, R., Koev, P., Ricarte, B., Urbano, A.M.: $$LDU$$ L D U -factorization of nonsingular totally nonpositive matrices. SIAM J. Matrix Anal. Appl. 30(2), 777–782 (2008)Cantó, R., Ricarte, B., Urbano, A.M.: Full rank factorization in echelon form of totally nonpositive (negative) rectangular matrices. Linear Algebra Appl. 431, 2213–2227 (2009)Cantó, R., Ricarte, B., Urbano, A.M.: Characterizations of rectangular totally and strictly totally positive matrices. Linear Algebra Appl. 432, 2623–2633 (2010)Cantó, R., Ricarte, B., Urbano, A.M.: Quasi- $$LDU$$ L D U factorization of nonsingular totally nonpositive matrices. Linear Algebra Appl. 439, 836–851 (2013)Cantó, R., Ricarte, B., Urbano, A.M.: Full rank factorization in quasi- $$LDU$$ L D U form of totally nonpositive rectangular matrices. Linear Algebra Appl. 440, 61–82 (2014)Fallat, S.M., Van Den Driessche, P.: On matrices with all minors negative. Electron. J. Linear Algebra 7, 92–99 (2000)Fallat, S.M.: Bidiagonal factorizations of totally nonnegative matrices. Am. Math. Mon. 108(8), 697–712 (2001)Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton University Press, New Jersey (2011)Gasca, M., Micchelli, C.A.: Total positivity and applications. Math. Appl. 359, Kluwer Academic Publishers, Dordrecht (1996)Gasca, M., Peña, J.M.: Total positivity, $$QR$$ Q R factorization and Neville elimination. SIAM J. Matrix Anal. Appl. 4, 1132–1140 (1993)Gasca, M., Peña, J.M.: A test for strict sign-regularity. Linear Algebra Appl. 197(198), 133–142 (1994)Gasca, M., Peña, J.M.: A matricial description of Neville elimination with applications to total positivity. Linear Algebra Appl. 202, 33–53 (1994)Gassó, M., Torregrosa, J.R.: A totally positive factorization of rectangular matrices by the Neville elimination. SIAM J. Matrix Anal. Appl. 25, 86–994 (2004)Huang, R., Chu, D.: Total nonpositivity of nonsingular matrices. Linear Algebra Appl. 432, 2931–2941 (2010)Huang, R., Chu, D.: Relative perturbation analysis for eigenvalues and singular values of totally nonpositive matrices. SIAM J. Matrix Anal. Appl. 36(2), 476–495 (2015)Karlin, S.: Total Nonpositivity. Stanford University Press, Stanford (1968)Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27(1), 1–23 (2005)Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29(3), 731–751 (2007)Parthasarathy, T.: $$N$$ N -matrices. Linear Algebra Appl. 139, 89–102 (1990)Peña, J.M.: Test for recognition of total positivity. SeMA J. 62(1), 61–73 (2013)Pinkus, A.: Totally Positive Matrices. Cambridge Tracts in Mathematics, vol. 181. Cambridge University Press (2009)Saigal, R.: On the class of complementary cones and Lemke’s algorithm. SIAM J. Appl. Math. 23, 46–60 (1972

Advances in Nonnegative Matrix Decomposition with Application to Cluster Analysis

Nonnegative Matrix Factorization (NMF) has found a wide variety of applications in machine learning and data mining. NMF seeks to approximate a nonnegative data matrix by a product of several low-rank factorizing matrices, some of which are constrained to be nonnegative. Such additive nature often results in parts-based representation of the data, which is a desired property especially for cluster analysis.  This thesis presents advances in NMF with application in cluster analysis. It reviews a class of higher-order NMF methods called Quadratic Nonnegative Matrix Factorization (QNMF). QNMF differs from most existing NMF methods in that some of its factorizing matrices occur twice in the approximation. The thesis also reviews a structural matrix decomposition method based on Data-Cluster-Data (DCD) random walk. DCD goes beyond matrix factorization and has a solid probabilistic interpretation by forming the approximation with cluster assigning probabilities only. Besides, the Kullback-Leibler divergence adopted by DCD is advantageous in handling sparse similarities for cluster analysis.  Multiplicative update algorithms have been commonly used for optimizing NMF objectives, since they naturally maintain the nonnegativity constraint of the factorizing matrix and require no user-specified parameters. In this work, an adaptive multiplicative update algorithm is proposed to increase the convergence speed of QNMF objectives.  Initialization conditions play a key role in cluster analysis. In this thesis, a comprehensive initialization strategy is proposed to improve the clustering performance by combining a set of base clustering methods. The proposed method can better accommodate clustering methods that need a careful initialization such as the DCD.  The proposed methods have been tested on various real-world datasets, such as text documents, face images, protein, etc. In particular, the proposed approach has been applied to the cluster analysis of emotional data

On the Geometric Interpretation of the Nonnegative Rank

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square

Using Underapproximations for Sparse Nonnegative Matrix Factorization

Nonnegative Matrix Factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g., text mining, image processing, microarray data analysis, collaborative filtering, etc. We introduce a novel approach to solve NMF problems, based on the use of an underapproximation technique, and show its effectiveness to obtain sparse solutions. This approach, based on Lagrangian relaxation, allows the resolution of NMF problems in a recursive fashion. We also prove that the underapproximation problem is NP-hard for any fixed factorization rank, using a reduction of the maximum edge biclique problem in bipartite graphs. We test two variants of our underapproximation approach on several standard image datasets and show that they provide sparse part-based representations with low reconstruction error. Our results are comparable and sometimes superior to those obtained by two standard Sparse Nonnegative Matrix Factorization techniques.Comment: Version 2 removed the section about convex reformulations, which was not central to the development of our main results; added material to the introduction; added a review of previous related work (section 2.3); completely rewritten the last part (section 4) to provide extensive numerical results supporting our claims. Accepted in J. of Pattern Recognitio