symconivol: symconivol

In damelunx/symconivol: A package for curvature measures of symmetric cones

Description

The symconivol package provides functions for analyzing intrinsic volumes and curvature measures of symmetric cones, as well as the Gaussian orthogonal ensembles conditioned on the index function.

Details

symconivol provides functions for analyzing intrinsic volumes and curvature measures of symmetric cones (positive semidefinite real/complex/quaternion matrices). These quantities can be estimated through the eigenvalue distribution of the Gaussian ensembles conditioned on the index, that is, the number of positive eigenvalues. The package provides functions for sampling from these conditioned eigenvalue distributions via Stan, and for reconstructing the curvature measures via MOSEK (second-order program). The package also provides several convenient functions for studiying these quantities, as well as a table of the algebraic degree of semidefinite programming. See the accompanying vignette for more information on how to use these functions.

Functions

• SDP_rnk_pred: produces the (estimated) probability vector for the rank of the solution of a random semidefinite program

• curv_meas_exact: gives the exact curvature measures for n=1,2,3

• pat_bnd: provides the Pataki inequalities for given beta and n

• leigh: produces a table and lookup functions for Leigh's curve (see vignette for definition)

• rate: produces a table and lookup function for the large deviation rate function of the index (see accompanying vignette for definition)

• mu: returns a pre-computed table and lookup functions for the estimated limit curve for dimension normalized curvature measures (see accompanying vignette for definition; we use C=0.2).

• constr_eigval: generates inputs for Stan (model string or external file) for sampling from the Gaussian orthogonal/unitary/symplectic ensemble conditioned on the index, the number of positive eigenvalues

• constr_eigval_to_bcbsq: converts a sample of eigenvalues produced by constr_eigval to a sample of the corresponding bivariate chi-bar-squared distribution

• prepare_em_cm: evaluates the sample data of the bivariate chi-bar-squared data (find the corresponding chi-squared density values)

• estim_em_cm: produces EM-type iterates to estimate the (normalized) curvature measures from a sample of the bivariate chi-bar-squared distribution

• alg_deg: looks up the algebraic degree of semidefinite programming from a table

Data

• ind_prob: A list of sample counts of a Bernoulli variable with (unnormalized) success and failure probabilities given by Prob{ind=r} and Prob{ind=r+1}.

• phi_ind: A list of reconstructed values of index constrained curvature measures; the constraints being of the form r<=ind(x)<=r+s.

• mu_data: A table of function values of the estimated limit curve of dimension normalized curvature measures mu.

• alg_deg_data: A list of the values of the algebraic degree of semidefinite programming delta(m,n,r) for n=2,3,...,14. The values are given as strings to avoid rounding errors.

See also

manual

damelunx.github.io/symconivol

sources

github.com/damelunx/symconivol

vignette

Studying curvature measures of symmetric cones: introduces curvature measures of symmetric cones, their relation to the Gaussian orthogonal/unitary/symplectic ensemble conditioned on the index function, explains the algorithms involved for estimating the curvature measures, gives some background and estimates involving limiting distributions and the algebraic degree of semidefinite programming

vignette

Studying curvature measures of symmetric cones - Technical details: a technical note to accompany the other vignette to give the commands for producing some figures in the main vignette, which are not computed on the fly

damelunx/symconivol documentation built on May 17, 2019, 7:01 p.m.