remove additional file

Author: zsteve

src/OptimalTransport.jl

@@ -534,4 +534,138 @@ function ot_reg_plan(mu, nu, C, eps; reg_func="L2", method="lorenz", kwargs...)
     end
 end
 
+
+"""
+    quadreg(mu, nu, C, ϵ; θ = 0.1, tol = 1e-5,maxiter = 50,κ = 0.5,δ = 1e-5)
+
+Computes the optimal transport plan of histograms `mu` and `nu` with cost matrix `C` and quadratic regularization parameter `ϵ`, 
+using the semismooth Newton algorithm [Lorenz 2016].
+
+This implementation makes use of IterativeSolvers.jl and SparseArrays.jl.
+
+Parameters:\n
+θ: starting Armijo parameter.\n
+tol: tolerance of marginal error.\n
+maxiter: maximum interation number.\n
+κ: control parameter of Armijo.\n
+δ: small constant for the numerical stability of conjugate gradient iterative solver.\n
+
+Tips:
+If the algorithm does not converge, try some different values of θ.
+
+Reference:
+Lorenz, D.A., Manns, P. and Meyer, C., 2019. Quadratically regularized optimal transport. arXiv preprint arXiv:1903.01112v4.
+"""
+function quadreg(mu, nu, C, ϵ; θ=0.1, tol=1e-5, maxiter=50, κ=0.5, δ=1e-5)
+    if !(sum(mu) ≈ sum(nu))
+        throw(ArgumentError("Error: mu and nu must lie in the simplex"))
+    end
+
+    N = length(mu)
+    M = length(nu)
+
+    # initialize dual potentials as uniforms
+    a = ones(M) ./ M
+    b = ones(N) ./ N
+    γ = spzeros(M, N)
+
+    da = spzeros(M)
+    db = spzeros(N)
+
+    converged = false
+
+    function DualObjective(a, b)
+        A = a .* ones(N)' + ones(M) .* b' - C'
+
+        return 0.5 * norm(A[A .> 0], 2)^2 - ϵ * (dot(nu, a) + dot(mu, b))
+    end
+
+    # computes minimizing directions, update γ
+    function search_dir!(a, b, da, db)
+        P = a * ones(N)' .+ ones(M) * b' .- C'
+
+        σ = 1.0 * sparse(P .>= 0)
+        γ = sparse(max.(P, 0) ./ ϵ)
+
+        G = vcat(
+            hcat(spdiagm(0 => σ * ones(N)), σ),
+            hcat(sparse(σ'), spdiagm(0 => sparse(σ') * ones(M))),
+        )
+
+        h = vcat(γ * ones(N) - nu, sparse(γ') * ones(M) - mu)
+
+        x = cg(G + δ * I, -ϵ * h)
+
+        da = x[1:M]
+        return db = x[(M + 1):end]
+    end
+
+    function search_dir(a, b)
+        P = a * ones(N)' .+ ones(M) * b' .- C'
+
+        σ = 1.0 * sparse(P .>= 0)
+        γ = sparse(max.(P, 0) ./ ϵ)
+
+        G = vcat(
+            hcat(spdiagm(0 => σ * ones(N)), σ),
+            hcat(sparse(σ'), spdiagm(0 => sparse(σ') * ones(M))),
+        )
+
+        h = vcat(γ * ones(N) - nu, sparse(γ') * ones(M) - mu)
+
+        x = cg(G + δ * I, -ϵ * h)
+
+        return x[1:M], x[(M + 1):end]
+    end
+
+    # computes optimal maginitude in the minimizing directions
+    function search_t(a, b, da, db, θ)
+        d = ϵ * dot(γ, (da .* ones(N)' .+ ones(M) .* db')) - ϵ * (dot(da, nu) + dot(db, mu))
+
+        ϕ₀ = DualObjective(a, b)
+        t = 1
+
+        while DualObjective(a + t * da, b + t * db) >= ϕ₀ + t * θ * d
+            t *= κ
+
+            if t < 1e-15
+                # @warn "@ i = $i, t = $t , armijo did not converge"
+                break
+            end
+        end
+        return t
+    end
+
+    for i in 1:maxiter
+
+        # search_dir!(a, b, da, db)
+        da, db = search_dir(a, b)
+
+        t = search_t(a, b, da, db, θ)
+
+        a += t * da
+        b += t * db
+
+        err1 = norm(γ * ones(N) - nu, Inf)
+        err2 = norm(sparse(γ') * ones(M) - mu, Inf)
+
+        if err1 <= tol && err2 <= tol
+            converged = true
+            @warn "Converged @ i = $i with marginal errors: \n err1 = $err1, err2 = $err2 \n"
+            break
+        elseif i == maxiter
+            @warn "Not Converged with errors:\n err1 = $err1, err2 = $err2 \n"
+        end
+
+        @debug " t = $t"
+        @debug "marginal @ i = $i: err1 = $err1, err2 = $err2 "
+    end
+
+    if !converged
+        @warn "SemiSmooth Newton algorithm did not converge"
+    end
+
+    return sparse(γ')
+end
+
 end