Update 11/06/17: FISTA with backtracking is tested with lasso, lasso_weighted, and Elastic net.

A simple implementation of FISTA

A MATLAB FISTA implementation based on the paper:

A. Beck and M. Teboulle, "A fast iterative shrinkage-thresholding algorithm for linear inverse problems", SIAM Journal on Imaging Sciences, vol. 2, no. 1, pp. 183–202, 2009. View the paper.

Tiep Vu, Penn State, Sep 2016

If you find any issue, please let me know via this. I would really appreciate. Thank you.

Note: Results in this repo are compared with those obtained by the SPAMS toolbox. You need to install spams and put the generated 'build' folder under the 'spams' folder of this repo.

Table of content

General Optimization problem
Algorithms
- If L(f) is easy to calculate,
- In case L(f) is hard to find,
Usage
- fista_general.m
- fista_backtracking
Examples

General Optimization problem

where:

g: R^n -> R: a continuous convex function which is possibly nonsmooth.

f: R^n -> R: a smooth convex function of the type C^{1, 1}, i.e., continuously differentiable with Lipschitz continuous gradient L(f): ||grad_f(x) - grad_f(y)|| <= L(f)||x - y|| for every x, y \in R^n

Note: this implementation also work on nonnegativity constrained problems.

Algorithms

If `L(f)` is easy to calculate,

We use the following algorithm: where pL(y) is a proximal function defined as:

For a new problem, our job is to implement two functions: grad_f(x) and pL(y) which are often simpler than the original optimization stated in (1).

In case `L(f)` is hard to find,

We can alternatively use the following algorithm:

where QL(x, y) is defined as:

Usage

`fista_general.m`

[X, iter, min_cost] = fista_general(grad, proj, Xinit, L, opts, calc_F)

See fista_general.m.

where:

    INPUT:
        grad   : a function calculating gradient of f(X) given X.
        proj   : a function calculating pL(x) -- projection
        Xinit  : a matrix -- initial guess.
        L      : a scalar the Lipschitz constant of the gradient of f(X).
        opts   : a struct
            opts.lambda  : a regularization parameter, can be either a scalar or
                            a weighted matrix.
            opts.max_iter: maximum iterations of the algorithm. 
                            Default 300.
            opts.tol     : a tolerance, the algorithm will stop if difference 
                            between two successive X is smaller than this value. 
                            Default 1e-8.
            opts.verbose : showing F(X) after each iteration or not. 
                            Default false. 
        calc_F: optional, a function calculating value of F at X 
                via feval(calc_F, X). 
   OUTPUT:
       X        : solution
       iter     : number of run iterations
       min_cost : the achieved cost

`fista_backtracking`

function X = fista_backtracking(calc_f, grad, Xinit, opts, calc_F)

See fista_backtracking.m where:

 INPUT:
      calc_f  : a function calculating f(x) in F(x) = f(x) + g(x) 
      grad   : a function calculating gradient of f(X) given X.
      Xinit  : a matrix -- initial guess.
      opts   : a struct
          opts.lambda  : a regularization parameter, can be either a scalar or
                          a weighted matrix.
          opts.max_iter: maximum iterations of the algorithm. 
                          Default 300.
          opts.tol     : a tolerance, the algorithm will stop if difference 
                          between two successive X is smaller than this value. 
                          Default 1e-8.
          opts.verbose : showing F(X) after each iteration or not. 
                          Default false. 
          opts.L0 : a positive scalar. 
          opts.eta: (must be > 1). eta in the algorithm (page 194)

      calc_F: optional, a function calculating value of F at X 
              via feval(calc_F, X). 
 OUTPUT:
     X        : solution

Examples

Lasso (and weighted) problems

Optimization problem: This function solves the l1 Lasso problem:

if lambda is a scalar, or :

if lambda is a matrix. In case lambda is a vector, it will be converted to a matrix with same columns and its # of columns = # of columns of X.

MATLAB function:

function X = lasso_fista(Y, D, Xinit, opts)
    opts = initOpts(opts);
    lambda = opts.lambda;
    if numel(Xinit) == 0
        Xinit = zeros(size(D,2), size(Y,2));
    end
    %% cost f
    function cost = calc_f(X)
        cost = 1/2 *normF2(Y - D*X);
    end 
    %% cost function 
    function cost = calc_F(X)
        if numel(lambda) == 1 % scalar 
            cost = calc_f(X) + lambda*norm1(X);
        elseif numel(lambda) == numel(X)
            cost = calc_f(X) + norm1(lambda.*X);
        end
    end 
    %% gradient
    DtD = D'*D;
    DtY = D'*Y;
    function res = grad(X) 
        res = DtD*X - DtY;
    end 
    %% Checking gradient 
    if nargin == 0 && opts.check_grad
        check_grad(@calc_f, @grad, Xinit);
    end 
    %% Lipschitz constant 
    L = max(eig(DtD));
    %% Use fista 
    [X, ~, ~] = fista_general(@grad, @proj_l1, Xinit, L, opts, @calc_F);
end

(See [])

Example:

1. L1 minimization (lambda is a scalar)

See demo_lasso.m

function test_lasso()
    clc
    d      = 300;   % data dimension
    N      = 70;    % number of samples 
    k      = 100;   % dictionary size 
    lambda = 0.01;
    Y      = normc(rand(d, N));
    D      = normc(rand(d, k));
    %% cost function 
    function c = calc_F(X)
        c = 0.5*normF2(Y - D*X) + lambda*norm1(X);
    end
    %% fista solution 
    opts.pos    = true;   % change to false for unconstrained problems
    opts.lambda = lambda;
    X_fista     = lasso_fista(Y, D, [], opts);
    %% spams solution 
    param.lambda     = lambda;
    param.lambda2    = 0;
    param.numThreads = 1;
    param.mode       = 2;
    param.pos        = opts.pos;
    X_spams          = mexLasso(Y, D, param);
    %% compare costs 
    cost_spams = calc_F(X_spams);
    cost_fista = calc_F(X_fista);
    fprintf('cost_fista = %.5s\n', cost_fista);
    fprintf('cost_spams = %.5s\n', cost_spams);
end

will generate an output like this:

cost_fista = 8.39552e+00
cost_spams = 8.39552e+00

2. Weighted l1 minimization (lambda is a vector or a matrix)

See demo_lasso_weighted.m

3. Full test

Run fista_lasso_fulltest.m to see the full test.

Results should look like this:

A toy example:
Data dimension                : 300
No. of samples                : 70
No. of atoms in the dictionary: 100
=====================================================
Lasso FISTA solution vs SPAMS solution,
 both of the following values should be close to 0.
1. average(norm1(X_fista - X_spams)) = 0.000028
2. costfista - cost_spams            = 0.000003
SPAMS provides a better cost.
=====================================================
Lasso Weighted FISTA solution vs SPAMS solution,
 both of the following values should be close to 0.
1. average(norm1(X_fista - X_spams)) = 0.000015
2. costfista - cost_spams            = -0.000004
FISTA provides a better cost.
================Positive Constraint===================
Lasso FISTA solution vs SPAMS solution,
 both of the following values should be close to 0.
1. average(norm1(X_fista - X_spams)) = 0.000025
2. costfista - cost_spams            = 0.003537
SPAMS provides a better cost.
================Positive Constraint===================
Lasso Weighted FISTA solution vs SPAMS solution,
 both of the following values should be close to 0.
1. average(norm1(X_fista - X_spams)) = 0.000016
2. costfista - cost_spams            = -0.000005
FISTA provides a better cost.

Elastic net problems

Optimization problem: This function solves the Elastic Net problem:

if lambda is a scalar, or :

if lambda is a matrix. In case lambda is a vector, it will be convert to a matrix with same columns and its # of columns = # of columns of X.

MATLAB function:

See fista_enet.m

function X = fista_enet(Y, D, Xinit, opts)
    opts    = initOpts(opts);
    lambda  = opts.lambda;
    lambda2 = opts.lambda2;

    if numel(lambda) > 1 && size(lambda, 2)  == 1
        lambda = repmat(opts.lambda, 1, size(Y, 2));
    end
    if numel(Xinit) == 0
        Xinit = zeros(size(D,2), size(Y,2));
    end
    %% cost f
    function cost = calc_f(X)
        cost = 1/2 *normF2(Y - D*X) + lambda2/2*normF2(X);
    end 
    %% cost function 
    function cost = calc_F(X)
        if numel(lambda) == 1 % scalar 
            cost = calc_f(X) + lambda*norm1(X);
        elseif numel(lambda) == numel(X)
            cost = calc_f(X) + norm1(lambda.*X);
        end
    end 
    %% gradient
    DtD = D'*D + lambda2*eye(size(D, 2));
    DtY = D'*Y;
    function res = grad(X) 
        res = DtD*X - DtY;
    end 
    %% Checking gradient 
    if opts.check_grad
        check_grad(@calc_f, @grad, Xinit);
    end 
    %% Lipschitz constant 
    L = max(eig(DtD));
    %% Use fista 
    opts.max_iter = 500;
    [X, ~, ~] = fista_general(@grad, @proj_l1, Xinit, L, opts, @calc_F);
end

Example:

See demo_enet.m

Row sparsity problems

Optimization problem:

where 'm' is number of rows and is the i-th row of .

Matlab function:

function X = fista_row_sparsity(Y, D, Xinit, opts)
    opts = initOpts(opts);
    lambda = opts.lambda;

    if numel(lambda) > 1 && size(lambda, 2)  == 1
        lambda = repmat(opts.lambda, 1, size(Y, 2));
    end
    if numel(Xinit) == 0
        Xinit = zeros(size(D,2), size(Y,2));
    end
    %% cost f
    function cost = calc_f(X)
        cost = 1/2 *normF2(Y - D*X);
    end 
    %% cost function 
    function cost = calc_F(X)
        cost = calc_f(X) + lambda*norm12(X);
    end 
    %% gradient
    DtD = D'*D;
    DtY = D'*Y;
    function res = grad(X) 
        res = DtD*X - DtY;
    end 
    %% Checking gradient 
    if opts.check_grad
        check_grad(@calc_f, @grad, Xinit);
    end 
    %% Lipschitz constant 
    L = max(eig(DtD));
    %% Use fista 
    opts.max_iter = 500;
    [X, ~, ~] = fista_general(@grad, @proj_l12, Xinit, L, opts, @calc_F);

Example:

function test_row_sparsity()
    clc
    d      = 30;    % data dimension
    N      = 70;    % number of samples 
    k      = 50;    % dictionary size 
    lambda = 0.01;
    Y      = normc(rand(d, N));
    D      = normc(rand(d, k));
    %% cost function 
    function c = calc_F(X)
        c = 0.5*normF2(Y - D*X) + lambda*norm12(X);
    end
    %% fista solution 
    opts.pos = true;
    opts.lambda = lambda;
    opts.check_grad = 0;
    X_fista = fista_row_sparsity(Y, D, [], opts);
    cost_fista = calc_F(X_fista);
    fprintf('cost_fista = %.5s\n', cost_fista);
end

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README.md

README.md

Update 11/06/17: FISTA with backtracking is tested with lasso, lasso_weighted, and Elastic net.

A simple implementation of FISTA

Table of content

General Optimization problem

Algorithms

If `L(f)` is easy to calculate,

In case `L(f)` is hard to find,

Usage

`fista_general.m`

`fista_backtracking`

Examples

Lasso (and weighted) problems

Elastic net problems

Row sparsity problems

Group sparsity problems (implement later)

Files

README.md

Latest commit

History

README.md

File metadata and controls

Update 11/06/17: FISTA with backtracking is tested with lasso, lasso_weighted, and Elastic net.

A simple implementation of FISTA

Table of content

General Optimization problem

Algorithms

If L(f) is easy to calculate,

In case L(f) is hard to find,

Usage

fista_general.m

fista_backtracking

Examples

Lasso (and weighted) problems

Elastic net problems

Row sparsity problems

Group sparsity problems (implement later)

If `L(f)` is easy to calculate,

In case `L(f)` is hard to find,

`fista_general.m`

`fista_backtracking`