Complexity of Linear Circuits and Geometry
- 243 Downloads
We use algebraic geometry to study matrix rigidity and, more generally, the complexity of computing a matrix–vector product, continuing a study initiated in Kumar et al. (2009), Landsberg et al. (preprint). In particular, we (1) exhibit many non-obvious equations testing for (border) rigidity, (2) compute degrees of varieties associated with rigidity, (3) describe algebraic varieties associated with families of matrices that are expected to have super-linear rigidity, and (4) prove results about the ideals and degrees of cones that are of interest in their own right.
KeywordsMatrix rigidity Discrete Fourier transform Vandermonde matrix Cauchy matrix
Mathematics Subject Classification68Q17 15B05 65T50
We thank the anonymous referees for very careful reading and numerous useful suggestions.
- 1.Paolo Aluffi, Degrees of projections of rank loci, preprint arXiv:1408.1702.
- 2.E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985.Google Scholar
- 4.Ronald de Wolf, Lower bounds on matrix rigidity via a quantum argument, Automata, languages and programming. Part I, Lecture Notes in Comput. Sci., vol. 4051, Springer, Berlin, 2006, pp. 62–71.Google Scholar
- 7.William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998.Google Scholar
- 8.William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991, A first course, Readings in Mathematics.Google Scholar
- 9.Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995, A first course, Corrected reprint of the 1992 original.Google Scholar
- 11.Abhinav Kumar, Satyanarayana V. Lokam, Vijay M. Patankar, and Jayalal Sarma M. N., Using elimination theory to construct rigid matrices, Foundations of software technology and theoretical computer science—FSTTCS 2009, LIPIcs. Leibniz Int. Proc. Inform., vol. 4, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2009, pp. 299–310.Google Scholar
- 12.J. M. Landsberg, Tensors: geometry and applications, Graduate Studies in Mathematics, vol. 128, American Mathematical Society, Providence, RI, 2012.Google Scholar
- 13.J. M. Landsberg, J. Taylor, and N. K. Vishnoi, The geometry of matrix rigidity (preprint). https://smartech.gatech.edu/handle/1853/6514.
- 14.F. Thomson Leighton, Introduction to parallel algorithms and architectures, Morgan Kaufmann, San Mateo, CA, 1992, Arrays, trees, hypercubes.Google Scholar
- 17.Satyanarayana V. Lokam, Quadratic lower bounds on matrix rigidity, Theory and applications of models of computation, Lecture Notes in Comput. Sci., vol. 3959, Springer, Berlin, 2006, pp. 295–307.Google Scholar
- 18.Satyanarayana V. Lokam, Complexity lower bounds using linear algebra, Found. Trends Theor. Comput. Sci. 4 (2008), no. 1-2, front matter, 1–155 (2009).Google Scholar
- 20.David Mumford, Algebraic geometry. I, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Complex projective varieties, Reprint of the 1976 edition.Google Scholar
- 21.David Mumford, The red book of varieties and schemes, expanded ed., Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1999, Includes the Michigan lectures (1974) on curves and their Jacobians, With contributions by Enrico Arbarello.Google Scholar
- 25.Richard P. Stanley, Enumerative combinatorics. Volume 1, second ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012.Google Scholar
- 27.Leslie G. Valiant, Graph-theoretic arguments in low-level complexity, Mathematical foundations of computer science (Proc. Sixth Sympos., Tatranská Lomnica, 1977), Springer, Berlin, 1977, pp. 162–176. Lecture Notes in Comput. Sci., Vol. 53.Google Scholar