Optimal Transport with Linear Programming

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This numerical tours details how to solve the discrete optimal transport problem (in the case of measures that are sums of Diracs) using linear programming.

In [2]:
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('solutions/optimaltransp_1_linprog')

Optimal Transport of Discrete Distribution

We consider two dicretes distributions $$ \forall k=0,1, \quad \mu_k = \sum_{i=1}^{n_k} p_{k,i} \de_{x_{k,i}} $$ where $n_0,n_1$ are the number of points, $\de_x$ is the Dirac at location $x \in \RR^d$, and $ X_k = ( x_{k,i} )_{i=1}^{n_i} \subset \RR^d$ for $k=0,1$ are two point clouds.

We define the set of couplings between $\mu_0,\mu_1$ as $$ \Pp = \enscond{ (\ga_{i,j})_{i,j} \in (\RR^+)^{n_0 \times n_1} }{ \forall i, \sum_j \ga_{i,j} = p_{0,i}, \: \forall j, \sum_i \ga_{i,j} = p_{1,j} } $$

The Kantorovitch formulation of the optimal transport reads $$ \ga^\star \in \uargmin{\ga \in \Pp} \sum_{i,j} \ga_{i,j} C_{i,j} $$ where $C_{i,j} \geq 0$ is the cost of moving some mass from $x_{0,i}$ to $x_{1,j}$.

The optimal coupling $\ga^\star$ can be shown to be a sparse matrix with less than $n_0+n_1-1$ non zero entries. An entry $\ga_{i,j}^\star \neq 0$ should be understood as a link between $x_{0,i}$ and $x_{1,j}$ where an amount of mass equal to $\ga_{i,j}^\star$ is transfered.

In the following, we concentrate on the $L^2$ Wasserstein distance. $$ C_{i,j}=\norm{x_{0,i}-x_{1,j}}^2. $$

The $L^2$ Wasserstein distance is then defined as $$ W_2(\mu_0,\mu_1)^2 = \sum_{i,j} \ga_{i,j}^\star C_{i,j}. $$

The coupling constraint $$ \forall i, \sum_j \ga_{i,j} = p_{0,i}, \: \forall j, \sum_i \ga_{i,j} = p_{1,j}$ $$ can be expressed in matrix form as $$ \Sigma(n_0,n_1) \ga = [p_0;p_1] $$ where $ \Sigma(n_0,n_1) \in \RR^{ (n_0+n_1) \times (n_0 n_1) } $.

In [3]:
flat = @(x)x(:);
Cols = @(n0,n1)sparse( flat(repmat(1:n1, [n0 1])), ...
             flat(reshape(1:n0*n1,n0,n1) ), ...
             ones(n0*n1,1) );
Rows = @(n0,n1)sparse( flat(repmat(1:n0, [n1 1])), ...
             flat(reshape(1:n0*n1,n0,n1)' ), ...
             ones(n0*n1,1) );
Sigma = @(n0,n1)[Rows(n0,n1);Cols(n0,n1)];

We use a simplex algorithm to compute the optimal transport coupling $\ga^\star$.

In [4]:
maxit = 1e4; tol = 1e-9;
otransp = @(C,p0,p1)reshape( perform_linprog( ...
        Sigma(length(p0),length(p1)), ...
        [p0(:);p1(:)], C(:), 0, maxit, tol), [length(p0) length(p1)] );

Dimensions $n_0, n_1$ of the clouds.

In [5]:
n0 = 60;
n1 = 80;

Compute a first point cloud $X_0$ that is Gaussian. and a second point cloud $X_1$ that is Gaussian mixture.

In [6]:
gauss = @(q,a,c)a*randn(2,q)+repmat(c(:), [1 q]);
X0 = randn(2,n0)*.3;
X1 = [gauss(n1/2,.5, [0 1.6]) gauss(n1/4,.3, [-1 -1]) gauss(n1/4,.3, [1 -1])];

Density weights $p_0, p_1$.

In [7]:
normalize = @(a)a/sum(a(:));
p0 = normalize(rand(n0,1));
p1 = normalize(rand(n1,1));

Shortcut for display.

In [8]:
myplot = @(x,y,ms,col)plot(x,y, 'o', 'MarkerSize', ms, 'MarkerEdgeColor', 'k', 'MarkerFaceColor', col, 'LineWidth', 2);

Display the point clouds. The size of each dot is proportional to its probability density weight.

In [9]:
clf; hold on;
for i=1:length(p0)
    myplot(X0(1,i), X0(2,i), p0(i)*length(p0)*10, 'b');
end
for i=1:length(p1)
    myplot(X1(1,i), X1(2,i), p1(i)*length(p1)*10, 'r');
end
axis([min(X1(1,:)) max(X1(1,:)) min(X1(2,:)) max(X1(2,:))]); axis off;

Compute the weight matrix $ (C_{i,j})_{i,j}. $

In [10]:
C = repmat( sum(X0.^2)', [1 n1] ) + ...
    repmat( sum(X1.^2), [n0 1] ) - 2*X0'*X1;

Compute the optimal transport plan.

In [11]:
gamma = otransp(C,p0,p1);

Check that the number of non-zero entries in $\ga^\star$ is $n_0+n_1-1$.

In [12]:
fprintf('Number of non-zero: %d (n0+n1-1=%d)\n', full(sum(gamma(:)~=0)), n0+n1-1);
Number of non-zero: 139 (n0+n1-1=139)

Check that the solution satifies the constraints $\ga \in \Cc$.

In [13]:
fprintf('Constraints deviation (should be 0): %.2e, %.2e.\n', norm(sum(gamma,2)-p0(:)),  norm(sum(gamma,1)'-p1(:)));
Constraints deviation (should be 0): 5.38e-16, 1.01e-17.

Displacement Interpolation

For any $t \in [0,1]$, one can define a distribution $\mu_t$ such that $t \mapsto \mu_t$ defines a geodesic for the Wasserstein metric.

Since the $W_2$ distance is a geodesic distance, this geodesic path solves the following variational problem $$ \mu_t = \uargmin{\mu} (1-t)W_2(\mu_0,\mu)^2 + t W_2(\mu_1,\mu)^2. $$ This can be understood as a generalization of the usual Euclidean barycenter to barycenter of distribution. Indeed, in the case that $\mu_k = \de_{x_k}$, one has $\mu_t=\de_{x_t}$ where $ x_t = (1-t)x_0+t x_1 $.

Once the optimal coupling $\ga^\star$ has been computed, the interpolated distribution is obtained as $$ \mu_t = \sum_{i,j} \ga^\star_{i,j} \de_{(1-t)x_{0,i} + t x_{1,j}}. $$

Find the $i,j$ with non-zero $\ga_{i,j}^\star$.

In [14]:
[I,J,gammaij] = find(gamma);

Display the evolution of $\mu_t$ for a varying value of $t \in [0,1]$.

In [15]:
clf;
tlist = linspace(0,1,6);
for i=1:length(tlist)
    t=tlist(i);
    Xt = (1-t)*X0(:,I) + t*X1(:,J);
    subplot(2,3,i);
    hold on;
    for i=1:length(gammaij)
        myplot(Xt(1,i), Xt(2,i), gammaij(i)*length(gammaij)*6, [t 0 1-t]);
    end
    title(['t=' num2str(t,2)]);
    axis([min(X1(1,:)) max(X1(1,:)) min(X1(2,:)) max(X1(2,:))]); axis off;
end

Optimal Assignement

In the case where the weights $p_{0,i}=1/n, p_{1,i}=1/n$ (where $n_0=n_1=n$) are constants, one can show that the optimal transport coupling is actually a permutation matrix. This properties comes from the fact that the extremal point of the polytope $\Cc$ are permutation matrices.

This means that there exists an optimal permutation $ \si^\star \in \Sigma_n $ such that $$ \ga^\star_{i,j} = \choice{ 1 \qifq j=\si^\star(i), \\ 0 \quad\text{otherwise}. } $$ where $\Si_n$ is the set of permutation (bijections) of $\{1,\ldots,n\}$.

This permutation thus solves the so-called optimal assignement problem $$ \si^\star \in \uargmin{\si \in \Sigma_n}$ \sum_{i} C_{i,\si(j)}. $$

Same number of points.

In [16]:
n0 = 40;
n1 = n0;

Compute points clouds.

In [17]:
X0 = randn(2,n0)*.3;
X1 = [gauss(n1/2,.5, [0 1.6]) gauss(n1/4,.3, [-1 -1]) gauss(n1/4,.3, [1 -1])];

Constant distributions.

In [18]:
p0 = ones(n0,1)/n0;
p1 = ones(n1,1)/n1;

Compute the weight matrix $ (C_{i,j})_{i,j}. $

In [19]:
C = repmat( sum(X0.^2)', [1 n1] ) + ...
    repmat( sum(X1.^2), [n0 1] ) - 2*X0'*X1;

Display the coulds.

In [20]:
clf; hold on;
myplot(X0(1,:), X0(2,:), 10, 'b');
myplot(X1(1,:), X1(2,:), 10, 'r');
axis equal; axis off;

Solve the optimal transport.

In [21]:
gamma = otransp(C,p0,p1);

Show that $\ga$ is a binary permutation matrix.

In [22]:
clf;
imageplot(gamma);

Display the optimal assignement.

In [23]:
clf; hold on;
[I,J,~] = find(gamma);
for k=1:length(I)
    h = plot( [X0(1,I(k)) X1(1,J(k))], [X0(2,I(k)) X1(2,J(k))], 'k' );
    set(h, 'LineWidth', 2);
end
myplot(X0(1,:), X0(2,:), 10, 'b');
myplot(X1(1,:), X1(2,:), 10, 'r');
axis equal; axis off;