A function for fitting generalized fused-Lasso problems

This function fits the standard version of the fused lasso. It can take a general matrix x and provides for possible weights on the lambda1 and lambda2 penalties.

Usage

fusedlasso(
  x,
  y,
  lambda1 = NULL,
  lambda2 = 1,
  pen_fused = c("L1", "L2", "Huber"),
  penscale = rep(1, ncol(x)),
  struct = NULL,
  intercept = TRUE,
  normalize = TRUE,
  nlambda1 = ifelse(is.null(lambda1), 50, length(lambda1)),
  minratio = 0.01,
  maxfeat = ifelse(lambda2 < 1, min(nrow(x), ncol(x)), min(2 * nrow(x), ncol(x))),
  beta0 = rep(0, ncol(x)),
  control = list()
)

Arguments

x

matrix of features, possibly sparsely encoded (experimental). Do NOT include intercept. When normalized os TRUE, coefficients will then be rescaled to the original scale.

y

response vector.

lambda1

sequence of decreasing \(\ell_1\)-penalty levels. If NULL (the default), a vector is generated with nlambda1 entries, starting from a guessed level lambda1.max where only the intercept is included, then shrunken to minratio*lambda1.max.

lambda2

real scalar; tunes the \(\ell_2\) penalty in the Elastic-net. Default is 0.01. Set to 0 to recover the Lasso.

pen_fused

penalty used for fusing the variables (either L1, L2 or Huber). Default is L1

penscale

vector with real positive values that weight the \(\ell_1\)-penalty of each feature. Default set all weights to 1.

struct

Description of the graph that corresponds to the lambda2 penalty structure. If NULL (the default) a chain graph is assumed, like in the standard fused-lasso. If a matrix is given, interpreted as a symmetric adjacency matrix

intercept

logical; indicates if an intercept should be included in the model. Default is TRUE.

normalize

logical; indicates if variables should be normalized to have unit L2 norm before fitting. Default is TRUE.

nlambda1

integer that indicates the number of values to put in the lambda1 vector. Ignored if lambda1 is provided.

minratio

minimal value of \(\ell_1\)-part of the penalty that will be tried, as a fraction of the maximal lambda1 value. A too small value might lead to unstability at the end of the solution path corresponding to small lambda1 combined with \(\lambda_2=0\). The default value tries to avoid this, adapting to the '\(n<p\)' context. Ignored if lambda1 is provided.

maxfeat

integer; limits the number of features ever to enter the model; i.e., non-zero coefficients for the Elastic-net: the algorithm stops if this number is exceeded and lambda1 is cut at the corresponding level. Default is min(nrow(x),ncol(x)) for small lambda2 (<0.01) and min(4*nrow(x),ncol(x)) otherwise. Use with care, as it considerably changes the computation time.

beta0

a starting point for the vector of parameter. By default, will initialized zero. May save time in some situation.

control

list of argument controlling low level options of the algorithm:

verbose: logical; ener verbose mode
timer: logical; use to record the timing of the algorithm. Default is FALSE.
maxiterin Maximum number of iterations in the inner loop to run.
maxiterout Maximum number of iterations in the outer loop to run.
maxactivation Maximum number of previously inactive variables to activate at the same time
accuracy Accuracy at which the algorithm will stop.
fusioncheck Should the fused sets be checked for breaking up?
verbose Should the function give some output what it is doing?

Value

an object with class FusedLassoFit, inheriting from QuadrupenFit.

Details

The optimized criterion is the following:

β^hat_λ₁,λ₂ = argmin_β 1/2 RSS(&beta) + λ₁ | D β |₁ + λ/2 ₂ β^T S β, where \(D\) is a diagonal matrix, whose diagonal terms are provided as a vector by the penscale argument. The \(\ell_1\) fusion penalty is structured by a possibily weighted graph \(G\) provided via the struct argument, as a symmetrix (undirected) adjacency matrix.

Author

Original code by Holger Hoefling, refactoring by Julien Chiquet

Examples

beta <- rep(c(0,1,0,-1,0), c(25,10,25,10,25))
cor <- 0.75
Soo <- toeplitz(cor^(0:(25-1))) ## Toeplitz correlation for irrelevant variables
Sww  <- matrix(cor,10,10) ## bloc correlation between active variables
Sigma <- Matrix::bdiag(Soo,Sww,Soo,Sww,Soo)
diag(Sigma) <- 1
n <- 50
x <- as.matrix(matrix(rnorm(95*n),n,95) %*% chol(Sigma))
y <- 10 + x %*% beta + rnorm(n,0,10)

res <- fusedlasso(x, y, lambda2=5)
G <- igraph::make_ring(ncol(x)) |> igraph::as_adjacency_matrix(sparse = FALSE)
resG <- fusedlasso(x, y, lambda2=5, struct = G)
plot(res)

plot(resG)