ObjFun_FG_S_PCA_L2: Objective Function of Feature Grouping and Sparse PCA with L2...

In ipapercodes/FGSPCA: Feature Grouping and Sparse Principal Component Analysis (FGSPCA)

Description

Objective Function of Feature Grouping and Sparse PCA with L2 loss.

Usage

ObjFun_FG_S_PCA_L2(
  x,
  A,
  B,
  tau_S,
  lambda1,
  lambda2 = 0.1,
  lambda3 = 0.1,
  nnConstraint = FALSE,
  sparseTruncated = TRUE
)

Arguments

x	the data matrix X_{n\times p}, where n is the number of observations, p is the number of features.
A	matrix A in the FGSPCA problem
B	matrix B in the FGSPCA problem
tau_S	τ, a global τ, which is assigned to τ_1=τ_2=τ .
lambda1	λ_1, the tuning parameter corresponding to p_1(\cdot)
lambda2	λ_2, the tuning parameter corresponding to p_2(\cdot)
lambda3	λ_3, the tuning parameter corresponding to p_3(\cdot)
nnConstraint	Boolean, indicating the non-negative constraint is true or false, default FALSE
sparseTruncated	Boolean, indicating whether to use the truncated L1 penalty or not for sparsity, default TRUE

Details

The L2 loss is defined as follows

(X - X B A^T)^2 .

The penalty function with three parts is defined as follows,

Ψ(β) = λ_{1}p_1(β) + λ_{2}p_2(β) + λ_{3}p_3(β)

where

p_1(β) = ∑_{j=1}^p \min\{\frac{|β_j|}{τ_1}, 1\} ,

p_2(β) = ∑_{j < j', (j, j') \in E} \min\{\frac{|β_j - β_{j'}|}{τ_2}, 1\},

p_3(β) = ∑_{j=1}^p (\min\{β_j, 0\})^2 .

With the setting of lambda2=0, nnConstraint=FALSE, sparseTruncated=FALSE, it corresponds to the same objective function of sparse PCA; \

ObjFun_FG_S_PCA_L2(x, A, B, tau_S, lambda1, lambda2=0, lambda3=0.1, nnConstraint=FALSE, sparseTruncated=FALSE

With setting of lambda2=0, nnConstraint=FALSE, sparseTruncated=TRUE, corresponds to Objective function of truncated sparse PCA. \

ObjFun_FG_S_PCA_L2(x, A, B, tau_S, lambda1, lambda2=0, lambda3=0.1, nnConstraint=FALSE, sparseTruncated=TRUE

By tuning the different setting of lambda2, nnConstraint, sparseTruncated, we get different combination of models.

ipapercodes/FGSPCA documentation built on Dec. 20, 2021, 7:58 p.m.