Random Determinants, Mixed Volumes of Ellipsoids, and Zeros of Gaussian Random Fields
Random Determinants, Mixed Volumes of Ellipsoids, and Zeros of Gaussian Random Fields
复制标题
随机行列式、椭球混合体积和高斯随机场的零点
DOI：
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发表时间：
2014
影响因子：
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通讯作者：
Z. Kabluchko
中科院分区：
文献类型：
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作者：
D. Zaporozhets;Z. Kabluchko
Consider a d × d matrix M whose rows are independent, centered, nondegenerate Gaussian vectors ξ1,…,ξd with covariance matrices Σ1,…,Σd. Denote by εi the dispersion ellipsoid of documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$ {xi_{mathrm{i}}}:{varepsilon_i}=left{ {mathbf{x}in {{mathbb{R}}^d}:{{mathbf{x}}^{ op }}sum
olimits_i^{-1 } {mathbf{x}leq 1} }
ight} $end{document}. We show that documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathbb{E}left| {det M}
ight|=frac{d! }{{{{{left( {2pi }
ight)}}^{{{d left/ {2}
ight.}}}}}}{V_d}left( {{varepsilon_1}ldots {varepsilon_d}}
ight), $$end{document} where Vd (·,…,·) denotes the mixed volume. We also generalize this result to the case of rectangular matrices. As a direct corollary, we get an analytic expression for the mixed volume of d arbitrary ellipsoids in ℝd. As another application, we consider a smooth, centered, nondegenerate Gaussian random field X = (X1,…,Xk)⊤ : ℝd → ℝk. Using the Kac-Rice formula, we obtain a geometric interpretation of the intensity of zeros of X in terms of the mixed volume of dispersion ellipsoids of the gradients of documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$ {{{{X_i}}} left/ {{sqrt{{mathbf{Var}{X_i}}}}}
ight.} $end{document}. This relates zero sets of equations to mixed volumes in a way which is reminiscent of the well-known Bernstein theorem about the number of solutions of a typical system of algebraic equations.
Consider a d × d matrix M whose rows are independent, centered, nondegenerate Gaussian vectors ξ1,…,ξd with covariance matrices Σ1,…,Σd. Denote by εi the dispersion ellipsoid of documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$ {xi_{mathrm{i}}}:{varepsilon_i}=left{ {mathbf{x}in {{mathbb{R}}^d}:{{mathbf{x}}^{ op }}sum
olimits_i^{-1 } {mathbf{x}leq 1} }
ight} $end{document}. We show that documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathbb{E}left| {det M}
ight|=frac{d! }{{{{{left( {2pi }
ight)}}^{{{d left/ {2}
ight.}}}}}}{V_d}left( {{varepsilon_1}ldots {varepsilon_d}}
ight), $$end{document} where Vd (·,…,·) denotes the mixed volume. We also generalize this result to the case of rectangular matrices. As a direct corollary, we get an analytic expression for the mixed volume of d arbitrary ellipsoids in ℝd. As another application, we consider a smooth, centered, nondegenerate Gaussian random field X = (X1,…,Xk)⊤ : ℝd → ℝk. Using the Kac-Rice formula, we obtain a geometric interpretation of the intensity of zeros of X in terms of the mixed volume of dispersion ellipsoids of the gradients of documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$ {{{{X_i}}} left/ {{sqrt{{mathbf{Var}{X_i}}}}}
ight.} $end{document}. This relates zero sets of equations to mixed volumes in a way which is reminiscent of the well-known Bernstein theorem about the number of solutions of a typical system of algebraic equations.