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.
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.
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
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.
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
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