CAOL(x, λ, h0; p=0, maxiters=2000, tol=1e-13, trace=false)

Learn convolutional analysis operators, i.e., sparsifying filters, for signals x with sparsity regularization λ and initial filters h0. Preserve the first p filters in h0, i.e., consider them handcrafted.

x can be either:

• a vector of training signals each a D-dimensional array
• a (D+1)-dimnensional array with training sample i being slice x[:,...,:,i]

h0 can be either:

• a vector of filters each a D-dimensional array
• a tuple (H0,R) where each column of the matrix H0 is a vectorized filter and where R gives the size/shape of the filter.

The filters must be orthogonal to one another and each filter must be normalized to have norm 1/sqrt(filter length), i.e., H0'H0 ≈ (1/size(H0,1))*I.

When trace=false, CAOL returns just the learned filters (in the same form as h0). When trace=true, CAOL also returns

• the iterates (only the learned filters since handcrafted filters do not change)
• the objective function values (evaluated on only the learned filters)
• the iterate differences norm(H[t]-H[t-1])/norm(H[t]) used for the stopping criterion

Examples

Passing in a vector of ten 100x50 training signals and a vector of 3x3 DCT filters

julia> x = [randn(100,50) for _ in 1:10];
julia> h0 = generatefilters(:DCT,(3,3));
julia> h = CAOL(x,1e-3,h0,maxiters=10);

Passing the ten 100x50 training signals as an array

julia> X = randn(100,50,10);
julia> h0 = generatefilters(:DCT,(3,3));
julia> h = CAOL(X,1e-3,h0,maxiters=10);

Passing the DCT filters in matrix form

julia> x = [randn(100,50) for _ in 1:10];
julia> H0 = generatefilters(:DCT,(3,3),form=:matrix);
julia> H = CAOL(x,1e-3,(H0,(3,3)),maxiters=10);

Getting the trace

julia> x = [randn(100,50) for _ in 1:10];
julia> H0 = generatefilters(:DCT,(3,3),form=:matrix);
julia> H, Htrace, objtrace, Hdifftrace = CAOL(x,1e-3,(H0,(3,3)),maxiters=10,trace=true);