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AtomicOpt.jl

AtomicOpt.jl is a Julia package for solving the following non-convex structured optimization problem:

    Find        x ∈ ℝⁿ  
    subject to  ½‖Mx-b‖² ≤ α 
    and         card(x|A) ≤ k

where M: ℝⁿ -> ℝᵐ is a linear operator, b ∈ ℝᵐ is the observation vector, A ⊆ ℝⁿ is a atomic set and card(x|A) measures the complexity of x with respect to the atomic set A. For example, when A is the set of all signed canonical vectors, i.e., A = {±e₁, ..., ±eₙ}, then card(x|A) equals to the number of nonzero entries in x. When A is the set of all normalized rank one matrices, i.e., A = { uv^T | u ∈ ℝᵐ, v ∈ ℝⁿ, ||u||₂ = ||v||₂ ≤ 1 }, then card(x|A) equals to the rank of the matrix x. Please see our paper for more detailed discussion on atomic sparsity.

Installation

To install, just call

Pkg.add("https://github.com/MPF-Optimization-Laboratory/AtomicOpt.jl.git")

Use AtomicOpt.jl to solve basis pursuit problem

using AtomicOpt
using LinearAlgebra
using Printf
import Random: seed!, randperm
m, n, k = 2^8, 2^10, 8            # No. of rows, columns, and nonzeros
M = randn(m, n)                   # Measurement operator
p = randperm(n); p = p[1:k]       # Location of k nonzeros in x
u = randn(m)/100                  # Noise
b = M*x0 + u                      # Observation
A = OneBall(n; maxrank = k)       # Atomic set
# Solve the basis pursuit problem
sol = level_set(M, b, A, α = norm(u)^2/2)
x = constructPrimal(sol)
# Report recovery error
@printf("relative difference between x0 and x: .2%f", norm(x - x0)/norm(x0))

Use AtomicOpt.jl to solve matrix completion problem

using AtomicOpt
using Printf
using LinearAlgebra
using SparseArrays
# generate a random m×n matrix with rank r
m, n, r = 100, 100, 3 
X = rand(m, n)
U, S, V = svd(X)
X0 = U[:,1:r] * Diagonal( S[1:r] ) * V[:,1:r]'
# generate a random mask with nnz ≈ m*n*p
p = 0.5
mask = sprand(Bool, m, n, p); mask = convert(SparseMatrixCSC{Float64, Int64}, mask)
# measurement
B =  mask .* X0
b =  B.nzval
# operator
Mop = MaskOP(mask)
# atomic set
A = NucBall(m, n, r)
# solve the matrix completion problem
sol = level_set(M, b, A, α = 0.0)
x = constructPrimal(sol)
X = reshape(x, m, n)
# Report recovery error
@printf("relative difference between X0 and X: .2%f", norm(X - X0)/norm(X0))

More examples

There are more examples in the folder examples.

Citing this package

If you use AtomicOpt.jl for published work, we encourage you to cite the software.

Use the following BibTeX citation:

@article{fan2022deconvolution,
  author  = {Z. Fan and H. Jeong and B. Joshi and M. P. Friedlander},
  title   = {Polar deconvolution of mixed signal},
  journal = {IEEE Transactions on Signal Processing},
  volume  = {70},
  pages   = {2713--2727},
  year    = {2022},
  month   = {May},
  doi     = {10.1109/TSP.2022.3178191}
}