FRANAGROUPLASSO Group LASSO regression in the TF-domain
Usage: tc = franagrouplasso(F,f,lambda)
tc = franagrouplasso(F,f,lambda,C,tol,maxit)
[tc,relres,iter,frec] = franagrouplasso(...)
Input parameters:
F : Frame definition
f : Input signal
lambda : Regularisation parameter, controls sparsity of the solution
C : Step size of the algorithm.
tol : Reative error tolerance.
maxit : Maximum number of iterations.
Output parameters:
tc : Thresholded coefficients
relres : Vector of residuals.
iter : Number of iterations done.
frec : Reconstructed signal
FRANAGROUPLASSO(F,f,lambda) solves the group LASSO regression problem
in the time-frequency domain: minimize a functional of the synthesis
coefficients defined as the sum of half the l^2 norm of the
approximation error and the mixed l^1 / l^2 norm of the coefficient
sequence, with a penalization coefficient lambda.
The matrix of time-frequency coefficients is labelled in terms of groups
and members. By default, the obtained expansion is sparse in terms of
groups, no sparsity being imposed to the members of a given group. This
is achieved by a regularization term composed of l^2 norm within a
group, and l^1 norm with respect to groups. See the help on
GROUPTHRESH for more information.
*Note* the involved frame F must support regular time-frequency
layout of coefficients.
[tc,relres,iter] = FRANAGROUPLASSO(...) returns the residuals relres in
a vector and the number of iteration steps done, maxit.
[tc,relres,iter,frec] = FRANAGROUPLASSO(...) returns the reconstructed
signal from the coefficients, frec. Note that this requires additional
computations.
The function takes the following optional parameters at the end of
the line of input arguments:
'freq' Group in frequency (search for tonal components). This is the
default.
'time' Group in time (search for transient components).
'C',cval Landweber iteration parameter: must be larger than
square of upper frame bound. Default value is the upper
frame bound.
'maxit',maxit
Stopping criterion: maximal number of iterations.
Default value is 100.
'tol',tol Stopping criterion: minimum relative difference between
norms in two consecutive iterations. Default value is
1e-2.
'print' Display the progress.
'quiet' Don't print anything, this is the default.
'printstep',p
If 'print' is specified, then print every p'th
iteration. Default value is 10;
In addition to these parameters, this function accepts all flags from
the GROUPTHRESH and THRESH functions. This makes it possible to
switch the grouping mechanism or inner thresholding type.
The parameters C, maxit and tol may also be specified on the
command line in that order: FRANAGROUPLASSO(F,x,lambda,C,tol,maxit).
The solution is obtained via an iterative procedure, called Landweber
iteration, involving iterative group thresholdings.
The relationship between the output coefficients is given by :
frec = frsyn(F,tc);
References:
M. Kowalski. Sparse regression using mixed norms. Appl. Comput. Harmon.
Anal., 27(3):303--324, 2009.
M. Kowalski and B. Torresani. Sparsity and persistence: mixed norms
provide simple signal models with dependent coefficients. Signal, Image
and Video Processing, 3(3):251--264, 2009.
Url: http://ltfat.github.io/doc/frames/franagrouplasso.html
See also: franalasso, framebounds.
Package: ltfat