Abstract
We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron, and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection. These problems at the interface of statistics and optimization are here examined from the perspective of convex algebraic geometry.
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References
Agler, J., Helton, J. W., McCullough, S., Rodman, S. (1988). Positive semidefinite matrices with a given sparsity pattern. Linear Algebra and its Applications, 107, 101–149.
Anderson, T. W. (1970). Estimation of covariance matrices which are linear combinations or whose inverses are linear combinations of given matrices. In R. C. Bose, I. M. Chakravati, P. C. Mahalanobis, C. R. Rao, K. J. C. Smith (Eds.), Essays in probability and statistics (pp. 1–24). Chapel Hill: University of North Carolina Press.
Barrett, W., Johnson, C., Tarazaga, P. (1993). The real positive definite completion problem for a simple cycle. Linear Algebra and its Applications, 192, 3–31.
Barrett, W., Johnson, C., Loewy, R. (1996). The real positive definite completion problem: Cycle completability. Memoirs of the American Mathematical Society, 584, 69.
Brown, L. D. (1986). Fundamentals of statistical exponential families. IMS monographs (Vol. IX). Hayward: Institute of Mathematical Statistics.
Brylawski, T. (1971). A combinatorial model for series-parallel networks. Transactions of the American Mathematical Society, 154, 1–22.
Buhl, S. L. (1993). On the existence of maximum likelihood estimators for graphical Gaussian models. Scandinavian Journal of Statistics, 20, 263–270.
Chardin, M., Eisenbud, D., Ulrich, B. (2009). Hilbert series of residual intersections. Manuscript.
Drton, M., Sturmfels, B., Sullivant, S. (2009). Lectures on algebraic statistics. Oberwolfach seminars (Vol. 40). Basel: Birkhäuser.
Eriksson, N., Fienberg, S., Rinaldo, A., Sullivant, S. (2006). Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models. Journal of Symbolic Computation, 41, 222–233.
Garcia, L., Stillman, M., Sturmfels, B. (2005). Algebraic geometry of Bayesian networks. Journal of Symbolic Computation 39, 331–355.
Grant, M., Boyd, S. CVX, a Matlab software for disciplined convex programming. Available at http://stanford.edu/boyd/cvx.
Grayson, D. R., Stillman, M. E. Macaulay2, a software system for research in algebraic geometry. Available athttp://www.math.uiuc.edu/Macaulay2/.
Greuel, G.-M., Pfister, G., Schönemann, H. Singular, a computer algebra system for polynomial computations. Available at http://www.singular.uni-kl.de .
Grone, R., Johnson, C., Sá, E., Wolkowicz, H. (1984). Positive definite completions of partial Hermitian matrices. Linear Algebra and its Applications, 58, 109–124.
Herzog, J., Vasconcelos, W., Villareal, R. (1985). Ideals with sliding depth. Nagoya Mathematical Journal, 99, 159–172.
Højsgaard, S., Lauritzen, S. (2008). Graphical Gaussian models with edge and vertex symmetries. Journal of the Royal Statistical Society, 70, 1005–1027.
Holzer, S., Labs, O. surfex 0.90. Available at www.surfex.AlgebraicSurface.net.
Jensen, S. T. (1988). Covariance hypothesis which are linear in both the covariance and the inverse covariance. Annals of Statistics, 16, 302–322.
Kotzev, B. (1991). Determinantal ideals of linear type of a generic symmetric matrix. Journal of Algebra, 139, 484–504.
Laurent, M. (2001a). On the sparsity order of a graph and its deficiency in chordality. Combinatorica, 21, 543–570.
Laurent, M. (2001b). Matrix completion problems. In The encyclopedia of optimization (pp. 221–229). Dordrecht: Kluwer.
Mardia, K. V., Kent, J. T., Bibby, J. M. (1979). Multivariate analysis. London: Academic Press.
Miller, E., Sturmfels, B. (2004). Combinatorial commutative algebra. Graduate texts in mathematics. New York: Springer.
Nie, J., Ranestad, K., Sturmfels, B. (2010) The algebraic degree of semidefinite programming. Mathematical Programming, 122, 379–405.
Pachter, L., Sturmfels, B. (2005). Algebraic statistics for computational biology. Cambridge: Cambridge University Press.
Proudfoot, N., Speyer, D. (2005). A broken circuit ring. Beiträge zur Algebra und Geometrie, 47, 161–166.
Stückrad, J. (1992). On quasi-complete intersections. Archiv der Mathematik 58, 529–538.
Sullivant, S. (2008). Algebraic geometry of Gaussian Bayesian networks. Advances in Applied Mathematics, 40, 482–513.
Sullivant, S., Talaska, K., Draisma, J. (2008). Trek separation for Gaussian graphical models. Annals of Statistics, to appear. arXiv:0812.1938.
Terao, H. (2002). Algebras generated by reciprocals of linear forms. Journal of Algebra, 250, 549–558.
Vandenberghe, L., Boyd, S., Wu, S.-P. (1996). Determinant maximization with linear matrix inequality constraints. SIAM Journal on Matrix Analysis and Applications, 19, 499–533.
White, N. (Ed.). (1992). Matroid applications. Cambridge: Cambridge University Press.
Zaslavsky, T. (1975). Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes. Memoirs of the American Mathematical Society, 1(1) 154 (vii+102 pp).
Ziegler, G. (1995). Lectures on polytopes. Graduate texts in mathematics (Vol. 152). New York: Springer.
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B. Sturmfels is supported in part by NSF grants DMS-0456960 and DMS-0757236. C. Uhler is supported by an International Fulbright Science and Technology Fellowship.
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Sturmfels, B., Uhler, C. Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry. Ann Inst Stat Math 62, 603–638 (2010). https://doi.org/10.1007/s10463-010-0295-4
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DOI: https://doi.org/10.1007/s10463-010-0295-4