tc = franagrouplasso(F,f,lambda) tc = franagrouplasso(F,f,lambda,C,tol,maxit) [tc,relres,iter,frec] = franagrouplasso(...)
| 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. |
| 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);
M. Kowalski. Sparse regression using mixed norms. Appl. Comput. Harmon. Anal., 27(3):303--324, 2009. [ http ]
M. Kowalski and B. Torrésani. Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients. Signal, Image and Video Processing, 3(3):251--264, 2009. [ http ]