add docs

src/NGWF.jl

@@ -1,3 +1,20 @@
+"""
+    nat_spec_filter(l, D; σ = 0.25 * maximum(D), method = :regular, thres = 0.2)
+
+assemble the natural spectral graph filter centered at the l-th eigenvector via
+the distance matrix `D`.
+
+# Input Arguments
+- `l::Int64`: index of the centered eigenvector
+- `D::Matrix{Float64}`: non-trivial distance matrix of the eigenvectors
+- `σ::Float64`: Gaussian window width parameter (default: `0.25 * maximum(D)`)
+- `method::Symbol`: `:regular` or `:reduced` (default: `:regular`)
+- `thres::Float64`: cutoff threshold ∈ (0, 1).
+
+# Output Argument
+- `𝛍::Vector{Float64}`: the natural spectral graph filter
+
+"""
 function nat_spec_filter(l, D; σ = 0.25 * maximum(D), method = :regular, thres = 0.2)
     d = D[:, l]
     𝛍 = exp.(-(d ./ σ).^2)
@@ -8,6 +25,20 @@ function nat_spec_filter(l, D; σ = 0.25 * maximum(D), method = :regular, thres
     return 𝛍
 end
 
+"""
+    ngwf_all_vectors(D, 𝚽; σ = 0.2 * maximum(D))
+
+assemble the whole NGWF dictionary.
+
+# Input Arguments
+- `D::Matrix{Float64}`: non-trivial distance matrix of the eigenvectors
+- `𝚽::Matrix{Float64}`: graph Laplacian eigenvectors
+- `σ::Float64`: Gaussian window width parameter (default: `0.25 * maximum(D)`)
+
+# Output Argument
+- `𝓤::Matrix{Float64}`: the NGWF dictionary
+
+"""
 function ngwf_all_vectors(D, 𝚽; σ = 0.2 * maximum(D))
     N = size(D, 1)
     𝓤 = zeros(N, 0)
@@ -19,6 +50,23 @@ function ngwf_all_vectors(D, 𝚽; σ = 0.2 * maximum(D))
     return 𝓤
 end
 
+"""
+    rngwf_all_vectors(D, 𝚽; σ = 0.2 * maximum(D), thres = 0.2)
+
+assemble the reduced NGWF (rNGWF) dictionary.
+
+# Input Arguments
+- `D::Matrix{Float64}`: non-trivial distance matrix of the eigenvectors
+- `𝚽::Matrix{Float64}`: graph Laplacian eigenvectors
+- `σ::Float64`: Gaussian window width parameter (default: `0.25 * maximum(D)`)
+- `thres::Float64`: cutoff threshold ∈ (0, 1).
+
+# Output Argument
+- `𝓤::Matrix{Float64}`: the rNGWF dictionary
+- `dic_l2x::Dict`: a dictionary to store the filtered locations by QR at the l-th
+    centered eigenvector
+
+"""
 function rngwf_all_vectors(D, 𝚽; σ = 0.2 * maximum(D), thres = 0.2)
     N = size(D, 1)
     𝓤 = zeros(N, 0)
@@ -40,6 +88,22 @@ function ngwf_vector(D, l, x, 𝚽; σ = 0.1 * maximum(D))
     return ψ
 end
 
+"""
+    frame_approx(f, U, V; num_kept = length(f))
+
+approximate signal `f` by the frame `U`.
+
+# Input Arguments
+- `f::Vector{Float64}`: input graph signal
+- `U::Matrix{Float64}`: a frame operator (matrix or dictionary)
+- `V::Matrix{Float64}`: the dual frame operator of `U`
+- `num_kept::Int64`: number of kept coefficients (NCR)
+
+# Output Argument
+- `rel_error::Vector{Float64}`: the relative errors
+- `f_approx::Vector{Float64}`: the approximated signal
+
+"""
 function frame_approx(f, U, V; num_kept = length(f))
     g = U' * f
     ind = sortperm(g.^2; rev = true)[1:num_kept]
@@ -53,7 +117,19 @@ function frame_approx(f, U, V; num_kept = length(f))
     return rel_error, f_approx
 end
 
+"""
+    rngwf_lx(dic_l2x)
+
+find the sequential subindices of rNGWF vectors.
+
+# Input Arguments
+- `dic_l2x::Dict`: a dictionary to store the filtered locations by QR at the l-th
+    centered eigenvector
+
+# Output Argument
+- `Γ::Vector{Tuple{Int64,Int64}}`: the sequential subindices of rNGWF vectors.
 
+"""
 function rngwf_lx(dic_l2x)
     N = length(dic_l2x)
     Γ = (Tuple{Int64,Int64})[]