calc_q_function function

Function to fit a solution with q active groups of an RKHS Group Lasso problem.

Fits a solution of the group lasso problem based on RKHS, with $q$ active groups in the obtained solution for the Gaussian regression model. It determines $\mu_{g}(q)$, for which the number of active groups in the solution of the RKHS group lasso problem is equal to $q$, and returns the RKHS meta model associated with $\mu_{g}(q)$.

grplasso_q(Y, Kv, q, rat, Num)

Arguments

• Y: Vector of response observations of size $n$.
• Kv: List of eigenvalues and eigenvectors of positive definite Gram matrices $K_v$ and their associated group names. It should have the same format as the output of the function calc_Kv (see details).
• q: Integer, the number of active groups in the obtained solution.
• rat: Positive scalar, used to restrict the minimum value of $\mu_g$, to be evaluted in the RKHS Group Lasso algorithm, $\mu_{min}=\mu_{max}/rat$. The value $\mu_{max}$ is calculated inside the program, see function mu_max.
• Num: Integer, used to restrict the number of different values of the penalty parameter $\mu_g$ to be evaluated in the RKHS Group Lasso algorithm, until it achieves $\mu_g(q)$: for Num $= 1$ the program is done for $3$ values of $\mu_g$, $\mu_{1}=(\mu_{min}+\mu_{max})/2$, $\mu_{2}=(\mu_{min}+\mu_{1})/2$ or $\mu_{2}=(\mu_{1}+\mu_{max})/2$ depending on the value of $q$ associated with $\mu_{1}$, $\mu_{3}=\mu_{min}$.

Details

Input Kv should contain the eigenvalues and eigenvectors of positive definite Gram matrices $K_v$. It is necessary to set input "correction" in the function calc_Kv equal to "TRUE".

Returns

List of $4$ components: "mus", "qs", "mu", "res": - mus: Vector, values of the evaluated penalty parameters $\mu_g$ in the RKHS group lasso algorithm until it achieves $\mu_{g}(q)$.

• qs: Vector, number of active groups associated with each value of $\mu_g$ in mus.

• mu: Scalar, value of $\mu_{g}(q)$.

• res: An RKHS Group Lasso object:

• intercept: Scalar, estimated value of intercept.

• teta: Matrix with vMax rows and $n$ columns. Each row of the matrix is the estimated vector $\theta_{v}$ for $v=1,...,$vMax.

• fit.v: Matrix with $n$ rows and vMax columns. Each row of the matrix is the estimated value of $f_{v}=K_{v}\theta_{v}$.

• fitted: Vector of size $n$, indicates the estimator of $m$.

• Norm.H: Vector of size vMax, estimated values of the penalty norm.

• supp: Vector of active groups.

• Nsupp: Vector of the names of the active groups.

• SCR: Scalar, equals to $\Vert Y-f_{0}-\sum_{v}K_{v}\theta_{v}\Vert ^{2}$.

• crit: Scalar, indicates the value of the penalized criteria.

• MaxIter: Integer, number of iterations until convergence is reached.

• convergence: TRUE or FALSE. Indicates whether the algorithm has converged or not.

• RelDiffCrit: Scalar, value of the first convergence criteria at the last iteration, $\frac{crit_{lastIter}-crit_{lastIter-1}}{crit_{lastIter-1}}$.

• RelDiffPar: Scalar, value of the second convergence criteria at the last iteration, $\Vert\frac{\theta_{lastIter}-\theta_{lastIter-1}}{\theta_{lastIter-1}}\Vert ^{2}$.

References

Kamari, H., Huet, S. and Taupin, M.-L. (2019) RKHSMetaMod : An R package to estimate the Hoeffding decomposition of an unknown function by solving RKHS Ridge Group Sparse optimization problem. arXiv:1905.13695

Author(s)

Halaleh Kamari

Note

Note.

See Also

calc_Kv, mu_max

Examples

d <- 3
n <- 50
library(lhs)
X <- maximinLHS(n, d)
c <- c(0.2,0.6,0.8)
F <- 1;for (a in 1:d) F <- F*(abs(4*X[,a]-2)+c[a])/(1+c[a])
epsilon <- rnorm(n,0,1);sigma <- 0.2 
Y <- F + sigma*epsilon
Dmax <- 3
kernel <- "matern"
Kv <- calc_Kv(X, kernel, Dmax, TRUE, TRUE)
result <- grplasso_q(Y,Kv,5,100 ,Num=10)
result$mu
result$res$Nsupp