A survey of some tensor analysis techniques for biological systems
Farzane Yahyanejad, Réka Albert, Bhaskar DasGupta
A survey of some tensor analysis techniques for biological systems
Background: Since biological systems are complex and often involve multiple types of genomic relationships, tensor analysis methods can be utilized to elucidate these hidden complex relationships. There is a pressing need for this, as the interpretation of the results of highthroughput experiments has advanced at a much slower pace than the accumulation of data.
Results: In this review we provide an overview of some tensor analysis methods for biological systems.
Conclusions: Tensors are natural and powerful generalizations of vectors and matrices to higher dimensions and play a fundamental role in physics, mathematics and many other areas. Tensor analysis methods can be used to provide the foundations of systematic approaches to distinguish significant higher order correlations among the elements of a complex systems via finding ensembles of a small number of reduced systems that provide a concise and representative summary of these correlations.
biological systems / tensor analysis / biological and statistical validation
[1] 
DasGupta, B. and Liang, J. (2016) Models and Algorithms for Biomolecules and Molecular Networks. John Wiley and Sons, Inc.

[2] 
Alter, O. and Golub, G. H. (2005) Reconstructing the pathways of a cellular system from genomescale signals by using matrix and tensor computations. Proc. Natl. Acad. Sci. USA, 102, 17559–17564
CrossRef
Pubmed
Google scholar

[3] 
Dyrby, M., Baunsgaard, D., Bro, R. and Engelsen, S. B. (2005) Multiway chemometric analysis of the metabolic response to toxins monitored by NMR. Chemom. Intell. Lab. Syst., 76, 79–89
CrossRef
Google scholar

[4] 
Hore, V., Viñuela, A., Buil, A., Knight, J., McCarthy, M. I., Small, K. and Marchini, J. (2016) Tensor decomposition for multipletissue gene expression experiments. Nat. Genet., 48, 1094–1100
CrossRef
Pubmed
Google scholar

[5] 
Omberg, L., Golub, G. H. and Alter, O. (2007) A tensor higherorder singular value decomposition for integrative analysis of DNA microarray data from different studies. Proc. Natl. Acad. Sci. USA, 104, 18371–18376
CrossRef
Pubmed
Google scholar

[6] 
Yener, B., Acar, E., Aguis, P., Bennett, K., Vandenberg, S. L. and Plopper, G. E. (2008) Multiway modeling and analysis in stem cell systems biology. BMC Syst. Biol., 2, 63
CrossRef
Pubmed
Google scholar

[7] 
Harshman, R. A. (1970) Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimodal factor analysis. UCLA Working Papers in Phonetics, 16, 1–84

[8] 
Miwakeichi, F., MartínezMontes, E., ValdésSosa, P. A., Nishiyama, N., Mizuhara, H. and Yamaguchi, Y. (2004) Decomposing EEG data into spacetimefrequency components using Parallel Factor Analysis. Neuroimage, 22, 1035–1045
CrossRef
Pubmed
Google scholar

[9] 
Mørup. M. (2011) Applications of tensor (multiway array) factorizations and decompositions in data mining. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 1, 24–40

[10] 
Mørup, M., Hansen, L. K. and Arnfred, S. M. (2007) ERPWAVELAB a toolbox for multichannel analysis of timefrequency transformed event related potentials. J. Neurosci. Methods, 161, 361–368
Pubmed

[11] 
Carroll, J. D. and Chang, J. J. (1970) Analysis of individual differences in multidimensional scaling via an Nway generalization of “EckartYoung” decomposition. Psychometrika, 35, 283–319
CrossRef
Google scholar

[12] 
Kroonenberg, P. M. (2008) Applied Multiway Data Analysis. John Wiley & Sons

[13] 
Murakami, T. and Kroonenberg, P. M. (2003) Threemode models and individual differences in semantic differential data. Multivariate Behav. Res., 38, 247–283
CrossRef
Google scholar

[14] 
Appellof, C. J. and Davidson, E. R. (1981) Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents. Anal. Chem., 53, 2053–2056
CrossRef
Google scholar

[15] 
Kolda, T. G. and Bader, B. W. (2009) Tensor decompositions and applications. SIAM Rev., 51, 455–500
CrossRef
Google scholar

[16] 
De Lathauwer, L., De Moor, B. and Vandewalle, J. (2000) A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl., 21, 1253–1278
CrossRef
Google scholar

[17] 
Miettinen, P. (2011) Boolean tensor factorizations. In: 11th IEEE International Conference on Data Mining, 447–456

[18] 
Kiers, H. A. L. (2000) Towards a standardized notation and terminology in multiway analysis. J. Chemometr., 14, 105–122
CrossRef
Google scholar

[19] 
Golub, G. H. and Van Loan, C. F. (1996) Matrix Computations, 3rd edition. The Johns Hopkins University Press

[20] 
Kruskal, J. B. (1977) Threeway arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra Appl., 18, 95–138
CrossRef
Google scholar

[21] 
Kruskal, J. B. (1989) Rank, decomposition, and uniqueness for 3way and Nway arrays. In: Multiway Data Analysis, Coppi, R. and Bolasco, S. (Eds.), pp. 7–18. NorthHolland, Amsterdam

[22] 
Levin, J. (1963) ThreeMode Factor Analysis. Dissertation for the Doctoral Degree, University of Illinois, Urbana

[23] 
Tucker, L. R. (1963) Implications of factor analysis of threeway matrices for measurement of change. In: Problems in Measuring Change, Harris, C. W. (ed.), University of Wisconsin Press, 122–137

[24] 
Tucker, L. R. (1964) The extension of factor analysis to threedimensional matrices. In: Contributions to Mathematical Psychology, Gulliksen, H. and Frederiksen, N. (eds.), Holt, Rinehardt & Winston, New York, 110–127

[25] 
Tucker, L. R. (1966) Some mathematical notes on threemode factor analysis. Psychometrika, 31, 279–311
CrossRef
Pubmed
Google scholar

[26] 
Kroonenberg, P. M. and De Leeuw, J. (1980) Principal component analysis of threemode data by means of alternating least squares algorithms. Psychometrika, 45, 69–97
CrossRef
Google scholar

[27] 
Håstad , H. (1990) Tensor rank is NPcomplete. J. Algorithms, 11, 644–654

[28] 
Hillar, C. J. and Lim, L.H. (2013) Most tensor problems are NPhard J. ACM, 60, 45:1–45:38

[29] 
Barak, B., Kelner, J. A. and Steurer, D. (2015) Dictionary learning and tensor decomposition via the sumofsquares method. In: Proceedings of the 47th ACM Symposium on Theory of Computing, pp. 143–151

[30] 
Barak, B. and Moitra, A. (2016) Noisy tensor completion via the sumofsquares hierarchy. In: Proceedings of Machine Learning Research, 49, 417–445

[31] 
Ge, R. and Ma, T. (2015) Decomposing overcomplete 3rd order tensors using sumofsquares algorithms. In: Proceedings of the 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp. 829–849

[32] 
Hopkins, S. B., Shi, J. and Steurer, D. (2015) Tensor principal component analysis via sumofsquare proofs. Proceedings of Machine Learning Research, 40, 956–1006

[33] 
Ma, T., Shi, J. and Steurer, D. (2016) Polynomialtime tensor decompositions with sumofsquares. In: Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science, pp. 438–446

[34] 
Potechin, A. and Steurer, D. (2017) Exact tensor completion with sumofsquares. In: Proceedings of Machine Learning Research, 65, 1–54

[35] 
Barak, B. and Steurer, D. (2014) Sumofsquares proofs and the quest toward optimal algorithms. In: Proceedings of International Congress of Mathematicians

[36] 
Laurent, M. (2009) Sums of Squares, Moment Matrices and Optimization Over Polynomials. In: Emerging Applications of Algebraic Geometry, The IMA Volumes in Mathematics and its Applications, Putinar M. and Sullivant S., (eds.), 149, 157–270

[37] 
Papadimitriou, C. H. and Steiglitz, K. (1988) Combinatorial Optimization: Algorithms and Complexity. Dover Publications

[38] 
Sun, J., Tao, D. and Faloutsos, C. (2006) Beyond Streams and Graphs: Dynamic Tensor Analysis. In: Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 374–383

[39] 
Chelliah, V., Juty, N., Ajmera, I., Ali, R., Dumousseau, M., Glont, M., Hucka, M., Jalowicki, G., Keating, S., KnightSchrijver, V.,
CrossRef
Pubmed
Google scholar

[40] 
Helikar, T., Kowal, B., McClenathan, S., Bruckner, M., Rowley, T., Madrahimov, A., Wicks, B., Shrestha, M., Limbu, K. and Rogers, J. A. (2012) The Cell Collective: toward an open and collaborative approach to systems biology. BMC Syst. Biol., 6, 96
CrossRef
Pubmed
Google scholar

[41] 
Chaouiya, C., Naldi, A. and Thieffry, D. (2012) Logical modelling of gene regulatory networks with GINsim. Methods Mol. Biol., 804, 463–479
CrossRef
Pubmed
Google scholar

[42] 
Kolesnikov, N., Hastings, E., Keays, M., Melnichuk, O., Tang, Y. A., Williams, E., Dylag, M., Kurbatova, N., Brandizi, M., Burdett, T.,
CrossRef
Pubmed
Google scholar

[43] 
Cho, R. J., Campbell, M. J., Winzeler, E. A., Steinmetz, L., Conway, A., Wodicka, L., Wolfsberg, T. G., Gabrielian, A. E., Landsman, D., Lockhart, D. J.,
CrossRef
Pubmed
Google scholar

[44] 
Li, S., Assmann, S. M. and Albert, R. (2006) Predicting essential components of signal transduction networks: a dynamic model of guard cell abscisic acid signaling. PLoS Biol., 4, e312
CrossRef
Pubmed
Google scholar

[45] 
Sun, Z., Jin, X., Albert, R. and Assmann, S. M. (2014) Multilevel modeling of lightinduced stomatal opening offers new insights into its regulation by drought. PLOS Comput. Biol., 10, e1003930
CrossRef
Pubmed
Google scholar

[46] 
Kannan, R., Tetali, P. and Vempala, S. (1999) Markovchain algorithms for generating bipartite graphs and tournaments. Random Struct. Algor., 14, 293–308
CrossRef
Google scholar

[47] 
Girvan, M. and Newman, M. E. J. (2002) Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA, 99, 7821–7826
CrossRef
Pubmed
Google scholar

[48] 
Leicht, E. A. and Newman, M. E. J. (2008) Community structure in directed networks. Phys. Rev. Lett., 100, 118703
CrossRef
Pubmed
Google scholar

[49] 
Newman, M. E. J. (2004) Fast algorithm for detecting community structure in networks. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 69, 066133
CrossRef
Pubmed
Google scholar

[50] 
Newman, M. E. J. (2003) The structure and function of complex networks. SIAM Rev., 45, 167–256
CrossRef
Google scholar

[51] 
Newman, M. E. J. and Girvan, M. (2004) Finding and evaluating community structure in networks. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 69, 026113
CrossRef
Pubmed
Google scholar

[52] 
Meijer, E. (2005) Matrix algebra for higher order moments. Linear Algebra Appl., 410, 112–134
CrossRef
Google scholar

[53] 
Mendes, P. (1997) Biochemistry by numbers: simulation of biochemical pathways with Gepasi 3. Trends Biochem. Sci., 22, 361–363
CrossRef
Pubmed
Google scholar

[54] 
Larsen, R. J. and Marx, M. L. (2000) An Introduction to Mathematical Statistics and Its Applications. Third Edition, Pearson Education

[55] 
Anandkumar, A., Ge, R., Hsu, D., Kakade, S. M. and Telgarsky, M. (2014) Tensor decompositions for learning latent variable models. J. Mach. Learn. Res., 15, 2773–2832

[56] 
Bhaskara, A., Charikar, M., Moitra, A. and Vijayaraghavan, A. (2014) Smoothed analysis of tensor decompositions. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, 594–603

[57] 
Goyal, N., Vempala, S. and Xiao, Y. (2014) Fourier PCA and robust tensor decomposition. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, 584–593

[58] 
Spielman, D. A. and Teng, S. H. (2004) Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. Assoc. Comput. Mach., 51, 385–463
CrossRef
Google scholar

[59] 
Spielman, D. A. and Teng, S. H. (2009) Smoothed Analysis: An Attempt to Explain the Behavior of Algorithms in Practice. Commun. ACM, 52, 76–84
CrossRef
Google scholar

[60] 
Albert, R., DasGupta, B., Hegde, R., Sivanathan, G. S., Gitter, A., G�rsoy, G., Paul, P. and Sontag, E. (2011) Computationally efficient measure of topological redundancy of biological and social networks. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 84, 036117
CrossRef
Pubmed
Google scholar

[61] 
Albert, R., DasGupta, B. and Mobasheri, N. (2014) Topological implications of negative curvature for biological and social networks. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 89, 032811
CrossRef
Pubmed
Google scholar

[62] 
Comon, P. (2004) Canonical Tensor Decompositions. Research report ISRN I3S/RR200417FR

[63] 
Comon, P. and Mourrain, B. (1996) Decomposition of quantics in sums of powers of linear forms. Signal Processing, 53, 93–107
CrossRef
Google scholar

[64] 
Anandkumar, A., Hsu, D. and Kakade, S. M. (2012) A method of moments for mixture models and hidden markov models. In: Proceedings of the Conference on Learning Theory, 33.1–33.34

[65] 
Qi, L. (2007) Eigenvalues and invariants of tensors. J. Math. Anal. Appl., 325, 1363–1377
CrossRef
Google scholar

[66] 
Reiss, D. J., Baliga, N. S. and Bonneau, R. (2006) Integrated biclustering of heterogeneous genomewide datasets for the inference of global regulatory networks. BMC Bioinformatics, 7, 280
CrossRef
Pubmed
Google scholar

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