# ot.weak

Weak optimal ransport solvers

## Functions

ot.weak.weak_optimal_transport(Xa, Xb, a=None, b=None, verbose=False, log=False, G0=None, **kwargs)[source]

Solves the weak optimal transport problem between two empirical distributions

\begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align}

where :

• $$X^a$$ and $$X^b$$ are the sample matrices.

• $$\mathbf{a}$$ and $$\mathbf{b}$$ are the sample weights

Note

This function is backend-compatible and will work on arrays from all compatible backends. But the algorithm uses the C++ CPU backend which can lead to copy overhead on GPU arrays.

Uses the conditional gradient algorithm to solve the problem proposed in [39].

Parameters:
• Xa ((ns,d) array-like, float) – Source samples

• Xb ((nt,d) array-like, float) – Target samples

• a ((ns,) array-like, float) – Source histogram (uniform weight if empty list)

• b ((nt,) array-like, float) – Target histogram (uniform weight if empty list))

• G0 ((ns,nt) array-like, float) – initial guess (default is indep joint density)

• numItermax (int, optional) – Max number of iterations

• numItermaxEmd (int, optional) – Max number of iterations for emd

• stopThr (float, optional) – Stop threshold on the relative variation (>0)

• stopThr2 (float, optional) – Stop threshold on the absolute variation (>0)

• verbose (bool, optional) – Print information along iterations

• log (bool, optional) – record log if True

Returns:

• gamma (array-like, shape (ns, nt)) – Optimal transportation matrix for the given parameters

• log (dict, optional) – If input log is true, a dictionary containing the cost and dual variables and exit status

References

ot.bregman.sinkhorn

Entropic regularized OT

ot.optim.cg

General regularized OT

ot.weak.weak_optimal_transport(Xa, Xb, a=None, b=None, verbose=False, log=False, G0=None, **kwargs)[source]

Solves the weak optimal transport problem between two empirical distributions

\begin{align}\begin{aligned}\gamma = \mathop{\arg \min}_\gamma \quad \sum_i \mathbf{a}_i \left(\mathbf{X^a}_i - \frac{1}{\mathbf{a}_i} \sum_j \gamma_{ij} \mathbf{X^b}_j \right)^2\\s.t. \ \gamma \mathbf{1} = \mathbf{a}\\ \gamma^T \mathbf{1} = \mathbf{b}\\ \gamma \geq 0\end{aligned}\end{align}

where :

• $$X^a$$ and $$X^b$$ are the sample matrices.

• $$\mathbf{a}$$ and $$\mathbf{b}$$ are the sample weights

Note

This function is backend-compatible and will work on arrays from all compatible backends. But the algorithm uses the C++ CPU backend which can lead to copy overhead on GPU arrays.

Uses the conditional gradient algorithm to solve the problem proposed in [39].

Parameters:
• Xa ((ns,d) array-like, float) – Source samples

• Xb ((nt,d) array-like, float) – Target samples

• a ((ns,) array-like, float) – Source histogram (uniform weight if empty list)

• b ((nt,) array-like, float) – Target histogram (uniform weight if empty list))

• G0 ((ns,nt) array-like, float) – initial guess (default is indep joint density)

• numItermax (int, optional) – Max number of iterations

• numItermaxEmd (int, optional) – Max number of iterations for emd

• stopThr (float, optional) – Stop threshold on the relative variation (>0)

• stopThr2 (float, optional) – Stop threshold on the absolute variation (>0)

• verbose (bool, optional) – Print information along iterations

• log (bool, optional) – record log if True

Returns:

• gamma (array-like, shape (ns, nt)) – Optimal transportation matrix for the given parameters

• log (dict, optional) – If input log is true, a dictionary containing the cost and dual variables and exit status

References

ot.bregman.sinkhorn
ot.optim.cg