Moments Tensors, Hilbert's Identity, and k-wise Uncorrelated Random Variables

In this paper, we introduce a notion to be called k-wise uncorrelated random variables, which is similar but not identical to the so-called k-wise independent random variables in the literature. We show how to construct k-wise uncorrelated random variables by a simple procedure. The constructed random variables can be applied, e.g., to express the quartic polynomial (xTQx)2, where Q is an n × n positive semidefinite matrix, by a sum of fourth powered polynomial terms, known as Hilbert's identity. By virtue of the proposed construction, the number of required terms is no more than 2n4 + n. This implies that it is possible to find a (2n4 + n)-point distribution whose fourth moments tensor is exactly the symmetrization of Q ⊗ Q. Moreover, we prove that the number of required fourth powered polynomial terms to express (xTQx)2 is at least n(n + 1)/2. The result is applied to prove that computing the matrix 2 ↦ 4 norm is NP-hard. Extensions of the results to complex random variables are discussed as well.

[1]  W. J. Ellison.  Waring's Problem , 1971 .

[2]  Yuan Zhou,et al.  Hypercontractivity, sum-of-squares proofs, and their applications , 2012, STOC '12.

[3]  A. Joffe.  On a Set of Almost Deterministic $k$-Independent Random Variables , 1974 .

[4]  Aditya Bhaskara,et al.  Approximating Matrixp-norms , 2011 .

[5]  Julien M. Hendrickx,et al.  Matrix p-Norms Are NP-Hard to Approximate If p!=q1, 2, INFINITY , 2010, SIAM J. Matrix Anal. Appl..

[6]  C. Carathéodory.  Über den variabilitätsbereich der fourier’schen konstanten von positiven harmonischen funktionen , 1911 .

[7]  Y. Nesterov.  Random walk in a simplex and quadratic optimization over convex polytopes , 2003 .

[8]  Aditya Bhaskara,et al.  Approximating matrix p-norms , 2010, SODA '11.

[9]  D. Hilbert,et al.  Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n ter Potenzen (Waringsches Problem) , 1932 .

[10]  Melvyn B. Nathanson,et al.  Additive Number Theory The Classical Bases , 1996 .

[11]  B. Reznick.  Sums of Even Powers of Real Linear Forms , 1992 .

[12]  D. Hilbert,et al.  Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem). Dem Andenken an Hermann Minkowski gewidmet. , 1909 .

[13]  Paul Pollack,et al.  Central European Journal of Mathematics On Hilbert ’ s solution of Waring ’ s problem Research Article , 2011 .

[14]  P. McMullen.  A COURSE IN CONVEXITY (Graduate Studies in Mathematics 54) By ALEXANDER BARVINOK: 366 pp., US$59.00, ISBN 0-8218-2968-8 (American Mathematical Society, Providence, RI, 2002) , 2003 .

[15]  Melvyn B. Nathanson,et al.  Additive Number Theory , 1996 .

[16]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[17]  Daureen Steinberg.  COMPUTATION OF MATRIX NORMS WITH APPLICATIONS TO ROBUST OPTIMIZATION , 2007 .

[18]  Alexander Olshevsky,et al.  Matrix P-norms are NP-hard to approximate if p ≠1,2,∞ , 2009 .

[19]  Noga Alon,et al.  Approximating the cut-norm via Grothendieck's inequality , 2004, STOC '04.

[20]  Alexander Barvinok,et al.  A course in convexity , 2002, Graduate studies in mathematics.

[21]  B. Reznick.  Uniform denominators in Hilbert's seventeenth problem , 1995 .

[22]  Y. Mansour,et al.  On construction of k-wise independent random variables , 1994, STOC '94.

[23]  N. Higham.  Estimating the matrixp-norm , 1992 .