class ott.core.sinkhorn_lr.LRSinkhorn(rank, gamma=10.0, gamma_rescale=True, epsilon=0.0, initializer='random', lse_mode=True, inner_iterations=10, use_danskin=True, implicit_diff=False, kwargs_dys=None, kwargs_init=None, **kwargs)[source]#

A Low-Rank Sinkhorn solver for linear reg-OT problems.

The algorithm is described in [Scetbon et al., 2021] and the implementation contained here is adapted from LOT.

The algorithm minimizes a non-convex problem. It therefore requires special care to initialization and convergence. Convergence is evaluated on successive evaluations of the objective. The algorithm is only provided for the balanced case.

  • rank (int) – the rank constraint on the coupling to minimize the linear OT problem

  • gamma (float) – the (inverse of) gradient step size used by mirror descent.

  • gamma_rescale (bool) – Whether to rescale \(\gamma\) every iteration as described in [Scetbon and Cuturi, 2022].

  • epsilon (float) – entropic regularization added on top of low-rank problem.

  • initializer (Union[Literal[‘random’, ‘rank2’, ‘k-means’, ‘generalized-k-means’], LRInitializer, None]) –

    How to initialize the \(Q\), \(R\) and \(g\) factors. Valid options are:

    If None, KMeansInitializer is used when the linear problem’s geometry is PointCloud or LRCGeometry. Otherwise, use RandomInitializer.

  • lse_mode (bool) – whether to run computations in lse or kernel mode. At the moment, only lse_mode = True is implemented.

  • inner_iterations (int) – number of inner iterations used by the algorithm before re-evaluating progress.

  • use_danskin (bool) – use Danskin theorem to evaluate gradient of objective w.r.t. input parameters. Only True handled at this moment.

  • implicit_diff (bool) – Whether to use implicit differentiation. Currently, only implicit_diff = False is implemented.

  • kwargs_dys (Optional[Mapping[str, Any]]) – keyword arguments passed to dykstra_update().

  • kwargs_init (Optional[Mapping[str, Any]]) – keyword arguments for LRInitializer.

  • kwargs (Any) – Keyword arguments for Sinkhorn.



Create a low-rank Sinkhorn initializer.

dykstra_update(c_q, c_r, h, gamma, ot_prob)


Tuple[ndarray, ndarray, ndarray]

init_state(ot_prob, init)

Return the initial state of the loop.

kernel_step(ot_prob, state, iteration)

Not implemented.

lse_step(ot_prob, state, iteration)

LR Sinkhorn LSE update.

one_iteration(ot_prob, state, iteration, ...)

Carries out one LR sinkhorn iteration.

output_from_state(ot_prob, state)

Create an output from a loop state.



Whether entropy regularization is used.


Powers used to compute the p-norm between marginal/target.


Upper bound on number of times inner_iterations are carried out.