lambda_max: Analytic solution for the minimum lambda_K that gives the...

In genscore: Generalized Score Matching Estimators

Analytic solution for the minimum \lambda_{\mathbf{K}} that gives the empty graph.

Description

Analytic solution for the minimum \lambda_{\mathbf{K}} that gives the empty graph. In the non-centered setting the bound is not tight, as it is such that both \mathbf{K} and \boldsymbol{\eta} are empty. The bound is also not tight if symmetric == "and".

Usage

lambda_max(elts, symmetric, lambda_ratio = Inf)

Arguments

elts	A list, elements necessary for calculations returned by get_elts().
symmetric	A string. If equals "symmetric", estimates the minimizer \mathbf{K} over all symmetric matrices; if "and" or "or", use the "and"/"or" rule to get the support.
lambda_ratio	A positive number (or Inf), the fixed ratio \lambda_{\mathbf{K}} and \lambda_{\boldsymbol{\eta}}, if \lambda_{\boldsymbol{\eta}}\neq 0 (non-profiled) in the non-centered setting.

Value

A number, the smallest lambda that produces the empty graph in the centered case, or that gives zero solutions for \mathbf{K} and \boldsymbol{\eta} in the non-centered case. If symmetric == "and", it is not a tight bound for the empty graph.

Examples

# Examples are shown for Gaussian truncated to R+^p only. For other distributions
#   on other types of domains, please refer to \code{gen()} or \code{get_elts()},
#   as the way to call this function (\code{lambda_max()}) is exactly the same in those cases.
n <- 50
p <- 30
domain <- make_domain("R+", p=p)
mu <- rep(0, p)
K <- diag(p)
x <- tmvtnorm::rtmvnorm(n, mean = mu, sigma = solve(K),
       lower = rep(0, p), upper = rep(Inf, p), algorithm = "gibbs",
       burn.in.samples = 100, thinning = 10)

dm <- 1 + (1-1/(1+4*exp(1)*max(6*log(p)/n, sqrt(6*log(p)/n))))
h_hp <- get_h_hp("min_pow", 1, 3)
elts_gauss_np <- get_elts(h_hp, x, setting="gaussian", domain=domain,
                centered=FALSE, profiled=FALSE, diag=dm)

# Exact analytic solution for the smallest lambda such that K and eta are both zero,
#  but not a tight bound for K ONLY
lambda_max(elts_gauss_np, "symmetric", 2)
# Use the upper bound as a starting point for numerical search
test_lambda_bounds2(elts_gauss_np, "symmetric", lambda_ratio=2, lower = FALSE,
     lambda_start = lambda_max(elts_gauss_np, "symmetric", 2))

# Exact analytic solution for the smallest lambda such that K and eta are both zero,
#  but not a tight bound for K ONLY
lambda_max(elts_gauss_np, "or", 2)
# Use the upper bound as a starting point for numerical search
test_lambda_bounds2(elts_gauss_np, "or", lambda_ratio=2, lower = FALSE,
     lambda_start = lambda_max(elts_gauss_np, "or", 2))

# An upper bound, not tight.
lambda_max(elts_gauss_np, "and", 2)
# Use the upper bound as a starting point for numerical search
test_lambda_bounds2(elts_gauss_np, "and", lambda_ratio=2, lower = FALSE,
     lambda_start = lambda_max(elts_gauss_np, "and", 2))


elts_gauss_p <- get_elts(h_hp, x, setting="gaussian", domain=domain,
              centered=FALSE, profiled=TRUE, diag=dm)
# Exact analytic solution
lambda_max(elts_gauss_p, "symmetric")
# Numerical solution, should be close to the analytic solution
test_lambda_bounds2(elts_gauss_p, "symmetric", lambda_ratio=Inf, lower = FALSE,
     lambda_start = NULL)

# Exact analytic solution
lambda_max(elts_gauss_p, "or")
# Numerical solution, should be close to the analytic solution
test_lambda_bounds2(elts_gauss_p, "or", lambda_ratio=Inf, lower = FALSE,
     lambda_start = NULL)

# An upper bound, not tight
lambda_max(elts_gauss_p, "and")
# Use the upper bound as a starting point for numerical search
test_lambda_bounds2(elts_gauss_p, "and", lambda_ratio=Inf, lower = FALSE,
     lambda_start = lambda_max(elts_gauss_p, "and"))

genscore documentation built on May 29, 2024, 9 a.m.