doc: use p instead of d for dimension

Author: Carol-seven

R/gen_prec_sbm.R

@@ -115,15 +115,15 @@
 #' The weighted adjacency matrix is symmetrized and used as the precision matrix
 #' \eqn{\Omega_0}. Since arbitrary block-structured weights may not be positive
 #' definite, a diagonal adjustment is applied to control the eigenvalue spectrum.
-#' Specifically, let \eqn{\lambda_{\max}} and \eqn{\lambda_{\min}} denote
+#' Specifically, let \eqn{\lambda_\max} and \eqn{\lambda_\min} denote
 #' the largest and smallest eigenvalues of a matrix. A non-negative numeric
 #' value \eqn{\tau} is added to the diagonal so that
 #' \deqn{
 #' \left\{
 #' \begin{array}{l}
-#' \dfrac{\lambda_{\max}(\Omega_0 + \tau I)}{\lambda_{\min}(\Omega_0 + \tau I)}
+#' \dfrac{\lambda_\max(\Omega_0 + \tau I)}{\lambda_\min(\Omega_0 + \tau I)}
 #' \leq \texttt{cond.target} \\[1em]
-#' \lambda_{\min}(\Omega_0 + \tau I) > 0 \\[.5em]
+#' \lambda_\min(\Omega_0 + \tau I) > 0 \\[.5em]
 #' \tau \geq 0
 #' \end{array}
 #' \right.

R/grasps.R

@@ -76,21 +76,21 @@
 #' \code{lambda} values to generate when \code{lambda = NULL}.
 #'
 #' @param lambda.min.ratio A numeric value > 0 (default = 0.01) specifying
-#' the fraction of the maximum \code{lambda} value \eqn{\lambda_{max}} to
-#' generate the minimum \code{lambda} \eqn{\lambda_{min}}.
+#' the fraction of the maximum \code{lambda} value \eqn{\lambda_\max} to
+#' generate the minimum \code{lambda} \eqn{\lambda_\min}.
 #' If \code{lambda = NULL}, a \code{lambda} grid of length \code{nlambda} is
-#' automatically generated on a log scale, ranging from \eqn{\lambda_{max}}
-#' down to \eqn{\lambda_{min}}.
+#' automatically generated on a log scale, ranging from \eqn{\lambda_\max}
+#' down to \eqn{\lambda_\min}.
 #'
 #' @param growiter.lambda An integer (default = 30) specifying the maximum
 #' number of exponential growth steps during the initial search for an
-#' admissible upper bound \eqn{\lambda_{\max}}.
+#' admissible upper bound \eqn{\lambda_\max}.
 #'
 #' @param tol.lambda A numeric value > 0 (default = 1e-03) specifying
 #' the relative tolerance for the bisection stopping rule on the interval width.
 #'
 #' @param maxiter.lambda An integer (default = 50) specifying the maximum number
-#' of bisection iterations in the line search for  \eqn{\lambda_{\max}}.
+#' of bisection iterations in the line search for  \eqn{\lambda_\max}.
 #'
 #' @param rho A numeric value > 0 (default = 2) specifying the ADMM
 #' augmented-Lagrangian penalty parameter (often called the ADMM step size).
@@ -150,7 +150,7 @@
 #' \item{iterations}{The number of ADMM iterations.}
 #' \item{lambda.grid}{The actual lambda grid used in the program.}
 #' \item{alpha.grid}{The actual alpha grid used in the program.}
-#' \item{lambda.safe}{The bisection-refined upper bound \eqn{\lambda_{\max}},
+#' \item{lambda.safe}{The bisection-refined upper bound \eqn{\lambda_\max},
 #' corresponding to \code{alpha.grid}, when \code{lambda = NULL}.}
 #' \item{loss}{The optimal k-fold loss when \code{crit = "CV"}.}
 #' \item{CV.loss}{Matrix of CV losses, with rows for parameter combinations and

R/line_search_lambda_max.R

@@ -3,7 +3,7 @@
 #' @description
 #' Given a list of off-diagonal blocks \code{blockList} from the sample
 #' covariance (or a related matrix), this routine finds a \strong{sharp}
-#' \eqn{\lambda_{\max}} by bisection so that the KKT condition
+#' \eqn{\lambda_\max} by bisection so that the KKT condition
 #' \deqn{\Vert\text{soft}(B, \lambda\alpha)\Vert_F \leq \lambda(1-\alpha)}
 #' holds for every block \eqn{B} in \code{blockList}, where
 #' \eqn{\text{soft}(\cdot, t)} denotes element-wise soft-thresholding with
@@ -13,15 +13,15 @@
 #' used in the group KKT check.
 #'
 #' @param lambda.safe A numeric value representing a safe (but possibly loose)
-#' upper bound for \eqn{\lambda_{\max}}. If the value is negative, the search
+#' upper bound for \eqn{\lambda_\max}. If the value is negative, the search
 #' starts from 1.
 #'
 #' @param alpha A numeric value in (0,1) specifying the mixture weight between
 #' the individual (L1) and group (L2) penalties in sparse-group formulations.
 #'
 #' @param growiter An integer specifying the maximum number of exponential
 #' growth steps during the initial search for an admissible upper bound
-#' \eqn{\lambda_{\max}}.
+#' \eqn{\lambda_\max}.
 #'
 #' @param tol A numeric value > 0 specifying the relative tolerance for
 #' the bisection stopping rule on the interval width.
@@ -30,7 +30,7 @@
 #' iterations.
 #'
 #' @return
-#' A numeric value giving the bisection-refined upper bound \eqn{\lambda_{\max}}.
+#' A numeric value giving the bisection-refined upper bound \eqn{\lambda_\max}.
 #'
 #' @details
 #' The procedure first grows an upper bound starting from \code{lambda.safe}

man/gen_prec_sbm.Rd

@@ -29,9 +29,9 @@ parameter $\lambda$ to balance sparsity (number of edges) and model fit
 ## Background: Negative Log-Likelihood
 
 
-In a Gaussian graphical model (GGM), the data matrix $X_{n \times d}$
+In a Gaussian graphical model (GGM), the data matrix $X_{n \times p}$
 consists of $n$ independent and identically distributed observations
-$X_1, \dots, X_n$ drawn from $N_d(\mu,\Sigma)$.
+$X_1, \dots, X_n$ drawn from $N_p(\mu,\Sigma)$.
 Let $\Omega = \Sigma^{-1}$ denote the precision matrix, and define the empirical
 covariance matrix as
 $S = n^{-1} \sum_{i=1}^n (X_i-\bar{X})(X_i-\bar{X})^\top$.
@@ -72,7 +72,7 @@ $$
 $$
 \hat{\Omega}_{\mathrm{EBIC}} = {\arg\min}_{\Omega} \left\{
 2\,\mathrm{nll}(\Omega) + \log(n)\,\lvert E(\Omega) \rvert +
-4\,\xi\,\log(d)\,\lvert E(\Omega) \rvert \right\},
+4\,\xi\,\log(p)\,\lvert E(\Omega) \rvert \right\},
 $$
 
 
@@ -85,7 +85,7 @@ the classic BIC.
 
 $$
 \hat{\Omega}_{\mathrm{HBIC}} = {\arg\min}_{\Omega} \left\{
-2\,\mathrm{nll}(\Omega) + \log[\log(n)]\,\log(d)\,\lvert E(\Omega) \rvert \right\}.
+2\,\mathrm{nll}(\Omega) + \log[\log(n)]\,\log(p)\,\lvert E(\Omega) \rvert \right\}.
 $$
 
 

man/grasps.Rd

@@ -24,16 +24,16 @@ Consider the following setting:
 
 
 - **Gaussian graphical model (GGM) assumption:** \
-  The data $X_{n \times d}$ consists of independent and identically distributed
-  samples $X_1, \dots, X_n \sim N_d(\mu,\Sigma)$.
+  The data $X_{p \times p}$ consists of independent and identically distributed
+  samples $X_1, \dots, X_n \sim N_p(\mu,\Sigma)$.
 
 
 - **Disjoint group structure:** \
-  The $d$ variables can be partitioned into disjoint groups.
+  The $p$ variables can be partitioned into disjoint groups.
 
 
 - **Goal:** \
-  Estimate the precision matrix $\Omega = \Sigma^{-1} = (\omega_{ij})_{d \times d}$.
+  Estimate the precision matrix $\Omega = \Sigma^{-1} = (\omega_{ij})_{p \times p}$.
 
 
 ## Sparse-Group Estimator
@@ -42,15 +42,16 @@ Consider the following setting:
 \begin{gather}
 \hat{\Omega}(\lambda,\alpha,\gamma)
 = {\arg\min}_{\Omega \succ 0} \left\{ -\log\det(\Omega) + \text{tr}(S\Omega)
-+ \lambda P_{\alpha,\gamma}(\Omega) \right\},
++ \mathcal{P}_{\lambda,\alpha,\gamma}(\Omega) \right\},
 \\[10pt]
-P_{\alpha,\gamma}(\Omega)
-= \alpha P^\text{idv}_\gamma(\Omega) + (1-\alpha) P^\text{grp}_\gamma(\Omega),
+\mathcal{P}_{\lambda,\alpha,\gamma}(\Omega)
+= \alpha \mathcal{P}^\text{idv}_{\lambda,\gamma}(\Omega) + (1-\alpha) \mathcal{P}^\text{grp}_{\lambda,\gamma}(\Omega),
 \\[10pt]
-P^\text{idv}_\gamma(\Omega) = \sum_{i,j} p_\gamma(\vert\omega_{ij}\vert),
+\mathcal{P}^\text{idv}_{\lambda,\gamma}(\Omega)
+= \sum_{i,j} P_{\lambda,\gamma}(\lvert\omega_{ij}\rvert),
 \\[5pt]
-P^\text{grp}_\gamma(\Omega)
-= \sum_{g,g^\prime} p_\gamma(\Vert\Omega_{gg^\prime}\Vert_F).
+\mathcal{P}^\text{grp}_{\lambda,\gamma}(\Omega)
+= \sum_{g,g^\prime} P_{\lambda,\gamma}(\lVert\Omega_{gg^\prime}\rVert_F).
 \end{gather}
 
 
@@ -60,6 +61,7 @@ where:
 - $S = n^{-1} \sum_{i=1}^n (X_i-\bar{X})(X_i-\bar{X})^\top$ is the empirical
   covariance matrix.
 
+
 - $\lambda \geq 0$ is the global regularization parameter controlling overall
   shrinkage.
 
@@ -68,41 +70,40 @@ where:
   element-wise and block-wise penalties.
 
 
-- $\gamma$ is the additional parameter controlling the curvature and effective
-  degree of nonconvexity of the penalty.
+- $\gamma$ is the additional parameter for non-convex penalties, controlling
+  the degree of nonconvexity (or concavity) of the penalty function.
 
 
-- $P_{\alpha,\gamma}(\Omega)$ is a generic bi-level penalty template that can
-  incorporate convex or non-convex regularizers while preserving the intrinsic
-  group structure among variables.
+- $\mathcal{P}_{\lambda,\alpha,\gamma}(\Omega)$ is a generic bi-level penalty
+  template that combines element-wise and block-wise regularization, allowing
+  convex or non-convex regularizers while preserving the intrinsic group
+  structure among variables.
 
 
-- $P^\text{idv}_\gamma(\Omega)$ is the element-wise individual penalty component.
+- $\mathcal{P}^\text{idv}_{\lambda,\gamma}(\Omega)$ is the element-wise
+  individual penalty component.
 
 
-- $P^\text{grp}_\gamma(\Omega)$ is the block-wise group penalty component.
+- $\mathcal{P}^\text{grp}_{\lambda,\gamma}(\Omega)$ is the block-wise
+  group penalty component.
 
 
-- $p_\gamma(\cdot)$ is a penalty kernel parameterized by $\gamma$.
+- $P_{\lambda,\gamma}(\cdot)$ is the penalty function.
 
 
 - $\Omega_{gg^\prime}$ is the submatrix of $\Omega$ with the rows from group $g$
   and columns from group $g^\prime$.
 
 
-- The Frobenius norm $\Vert\Omega\Vert_F$ is defined as
-  $\Vert\Omega\Vert_F = (\sum_{i,j} \vert\omega_{ij}\vert^2)^{1/2} = [\text{tr}(\Omega^\top\Omega)]^{1/2}$.
+- The Frobenius norm $\lVert\Omega\rVert_F$ is defined as
+  $\lVert\Omega\rVert_F = (\sum_{i,j} \lvert\omega_{ij}\rvert^2)^{1/2} = [\text{tr}(\Omega^\top\Omega)]^{1/2}$.
 
 
 <div class="note">
 **Note**:
 
-+ The regularization parameter $\lambda$ acts as the scale factor for the entire
-  penalty term $\lambda P_{\alpha,\gamma}(\Omega)$.
-
-
-+ The penalty kernel $p_\gamma(\cdot)$ is the shape function that governs
-  the fundamental characteristics of the regularization. </div>
++ The parameter $\gamma$ is only relevant for non-convex penalties. The Lasso
+  penalty can be viewed as a special case in which $\gamma$ is not required. </div>
 
 
 ## Penalties
@@ -112,14 +113,14 @@ where:
 [@tibshirani1996regression; @friedman2008sparse]
 
 
-$$\lambda p(\omega_{ij}) = \lambda\vert\omega_{ij}\vert.$$
+$$P_\lambda(\omega_{ij}) = \lambda\vert\omega_{ij}\vert.$$
 
 
 2. Adaptive lasso [@zou2006adaptive; @fan2009network]
 
 
 $$
-\lambda p_\gamma(\omega_{ij}) = \lambda\frac{\vert\omega_{ij}\vert}{v_{ij}},
+P_{\lambda,\gamma}(\omega_{ij}) = \lambda\frac{\vert\omega_{ij}\vert}{v_{ij}},
 $$
 where $V = (v_{ij})_{d \times d} = (\vert\tilde{\omega}_{ij}\vert^\gamma)_{d \times d}$
 is a matrix of adaptive weights, and $\tilde{\omega}_{ij}$ is the initial estimate
@@ -130,7 +131,7 @@ obtained using `penalty = "lasso"`.
 
 
 $$
-\lambda p_\gamma(\omega_{ij})
+P_{\lambda,\gamma}(\omega_{ij})
 = \lambda(\gamma+\frac{2}{\pi})
 \arctan\left(\frac{\vert\omega_{ij}\vert}{\gamma}\right),
 \quad \gamma > 0.
@@ -141,7 +142,7 @@ $$
 
 
 $$
-\lambda p_\gamma(\omega_{ij})
+P_{\lambda,\gamma}(\omega_{ij})
 = \lambda\left[1-\exp\left(-\frac{\vert\omega_{ij}\vert}{\gamma}\right)\right],
 \quad \gamma > 0.
 $$
@@ -151,7 +152,7 @@ $$
 
 
 $$
-\lambda p_\gamma(\omega_{ij}) = \lambda\vert\omega_{ij}\vert^\gamma,
+P_{\lambda,\gamma}(\omega_{ij}) = \lambda\vert\omega_{ij}\vert^\gamma,
 \quad 0 < \gamma < 1.
 $$
 
@@ -160,7 +161,7 @@ $$
 
 
 $$
-\lambda p_\gamma(\omega_{ij})
+P_{\lambda,\gamma}(\omega_{ij})
 = \lambda\log\left(1+\frac{\vert\omega_{ij}\vert}{\gamma}\right),
 \quad \gamma > 0.
 $$
@@ -170,7 +171,7 @@ $$
 
 
 $$
-\lambda p_\gamma(\omega_{ij})
+P_{\lambda,\gamma}(\omega_{ij})
 = \begin{cases}
 \lambda\vert\omega_{ij}\vert - \dfrac{\omega_{ij}^2}{2\gamma},
 & \text{if } \vert\omega_{ij}\vert \leq \gamma\lambda, \\
@@ -185,7 +186,7 @@ $$
 
 
 $$
-\lambda p_\gamma(\omega_{ij})
+P_{\lambda,\gamma}(\omega_{ij})
 = \begin{cases}
 \lambda\vert\omega_{ij}\vert
 & \text{if } \vert\omega_{ij}\vert \leq \lambda, \\
@@ -202,12 +203,8 @@ $$
 **Note**:
 
 + For Lasso, which is convex, the additional parameter $\gamma$ is not required,
-  and the penalty kernel $p_\gamma(\cdot)$ simplifies to $p(\cdot)$.
-
-
-+ For MCP and SCAD, $\lambda$ plays a dual role: it is the global regularization
-  parameter, but it is also implicitly contained within the kernel
-  $p_\gamma(\cdot)$. </div>
+  and the penalty function $P_{\lambda,\gamma}(\cdot)$ simplifies to
+  $P_\lambda(\cdot)$. </div>
 
 
 ## Illustrative Visualization