In many contexts one encounters Hermitian operators @math on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such @math is positive semidefinite? Here we approach this problem by deriving asymptotically optimal bounds to the distance to the positive semidefinite cone in Schatten @math -norm for all integer @math , assuming that we know the moments @math up to a certain order @math . We then provide three methods to compute these bounds and relaxations thereof: the sos polynomial method (a semidefinite program), the Handelman method (a linear program relaxation), and the Chebyshev method (a relaxation not involving any optimization). We investigate the analytical and numerical performance of these methods and present a number of example computations, partly motivated by applications to tensor networks and to the theory of free spectrahedra.
There is only one work that we are aware of. Lasserre @cite_18 has investigated the smallest interval @math on which the support of a measure on @math can be shown to be contained, given only its first @math moments. One can think of this as providing a solution to the problems discussed above in the case @math . Lasserre found that the lower bound @math and upper bound @math are the optimal solutions of two simple semidefinite programs involving the moments.
@cite_18
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Given all moments of the marginals of a measure on Rn, one provides (a) explicit bounds on its support and (b), a numerical scheme to compute the smallest box that contains the support. It consists of solving a hierarchy of generalized eigenvalue problems associated with some Hankel matrices.