Implementing Wasserstein Barycenter problem using JuMP in Julia.¶
The goal of this notebook is to implement a model to calculate the Wasserstein barycenter in Julia with JuMP and then solve it using MosekTools. For additional info about the data used, theoretical explanation of the calculation of barycenters, references and for more insight in construction of the model, please consult the corresponding Python notebook. Data files can be found in http://yann.lecun.com/exdb/mnist/.
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using LinearAlgebra
using Plots
pyplot()
using JuMP
using MosekTools
In [2]:
#Define the number of images for the barycenter calculation.
n = 20
#Read the images from the file.
function read_idx(filename)
f = open(filename,"r")
data_layout = zeros(UInt8,4)
readbytes!(f,data_layout,4)
data_zero = reinterpret(UInt16,data_layout[1:2])
data_type,data_dimensions = reinterpret(UInt8,data_layout[3:4])
data_shape = Int32[]
for i = 1:data_layout[4]
s = zeros(UInt8,4)
readbytes!(f,s,4)
s = map(hton,reinterpret(Int32,s))
push!(data_shape,s[1])
end
idx_data = zeros(UInt8,cumprod(data_shape)[length(data_shape)])
read!(f,idx_data)
idx_data = reshape(idx_data,Tuple(reverse(data_shape)))
return(idx_data)
end
data = read_idx("train-images-idx3-ubyte")
labels = read_idx("train-labels-idx1-ubyte")
#Select the images.
mask = labels .== 1
train_ones = data[:,:,mask]
train = train_ones[:,:,1:n]
x = [i for i=1:28]
y = reverse(x)
f,ax = PyPlot.plt.subplots(2,5,sharey=true,sharex=true,figsize=(10,5))
PyPlot.plt.xticks([5,10,15,20,25])
for i = 1:10
rand_pick = rand(1:size(train_ones)[3])
ax[i].pcolormesh(x,y,transpose(train_ones[:,:,rand_pick]))
end
Barycenters using JuMP¶
In [3]:
function single_pmf(data)
#Takes a list of images and extracts the probability mass function.
v = vec(data[:,:,1])
v = v./cumsum(v)[length(v)]
for im_k in 2:size(data)[3]
image = data[:,:,im_k]
arr = vec(image)
v_size = size(arr)[1]
v = hcat(v, arr./cumsum(arr)[length(arr)])
end
return v,size(v)[1]
end
function ms_distance(m,n)
#Squared Euclidean distance calculation between the pixels.
d = ones(m,m)
coor_I = []
for c_i in 1:n
append!(coor_I,ones(Int,n).*c_i)
end
coor_J = repeat(1:n,n)
coor = hcat(coor_I,coor_J)
for i in 1:m
for j in 1:m
d[i,j] = norm(coor[i,:]-coor[j,:]).^2
end
end
return d
end
function wasserstein_barycenter(data)
M= direct_model(Mosek.Optimizer())
if length(size(data))==3
K = size(data)[3]
else
K = 1
end
v,N = single_pmf(data)
d = ms_distance(N,size(data)[2])
#Define indices
M_i = 1:N
M_j = 1:N
M_k = 1:K
#Adding variables
M_pi = @variable(M, M_pi[i = M_i, j = M_j, k = M_k] >= 0.0)
M_mu = @variable(M, M_mu[i = M_i] >= 0.0)
#Adding constraints
@constraint(M, c3_expr[k = M_k, j = M_j], sum(M_pi[:,j,k]) == v[j,k])
@constraint(M, c2_expr[k = M_k, i = M_i], sum(M_pi[i,:,k]) == M_mu[i])
#Objective
W_obj = @objective(M, Min, sum(d[i,j]*M_pi[i,j,k] for i=M_i,j=M_j,k=M_k)/K)
return M,M_mu
end
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In [4]:
function run_model(data)
@time begin
M,M_mu = wasserstein_barycenter(data)
optimize!(M)
end
println("Solution status = ",termination_status(M))
println("Primal objective value = ",objective_value(M))
mu_level = value.(M_mu)
return mu_level
end
function show_barycenter(bary_center)
bary_center = reshape(bary_center,(28,28))
x = [i for i=1:28]
y = reverse(x)
PyPlot.plt.pcolormesh(x,y,transpose(bary_center))
PyPlot.plt.title("Non-regularized Wasserstein Barycenter")
PyPlot.plt.show()
end
Out[4]:
In [5]:
bary_center = run_model(train)
println("******")
In [6]:
show_barycenter(bary_center)
This work is licensed under a Creative Commons Attribution 4.0 International License. The MOSEK logo and name are trademarks of Mosek ApS. The code is provided as-is. Compatibility with future release of MOSEK or the Fusion API are not guaranteed. For more information contact our support.
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