class ott.geometry.geometry.Geometry(cost_matrix=None, kernel_matrix=None, epsilon=None, relative_epsilon=None, scale_epsilon=None, src_mask=None, tgt_mask=None, scale_cost=1.0, **kwargs)[source]#

Base class to define ground costs/kernels used in optimal transport.

Optimal transport problems are intrinsically geometric: they compute an optimal way to transport mass from one configuration onto another. To define what is meant by optimality of a transport requires defining a cost, of moving mass from one among several sources, towards one out of multiple targets. These sources and targets can be provided as points in vectors spaces, grids, or more generally exclusively described through a (dissimilarity) cost matrix, or almost equivalently, a (similarity) kernel matrix.

Once that cost or kernel matrix is set, the Geometry class provides a basic operations to be run with the Sinkhorn algorithm.

  • cost_matrix (Optional[ndarray]) – jnp.ndarray<float>[num_a, num_b]: a cost matrix storing n x m costs.

  • kernel_matrix (Optional[ndarray]) – jnp.ndarray<float>[num_a, num_b]: a kernel matrix storing n x m kernel values.

  • epsilon (Union[Epsilon, float, None]) – a regularization parameter. If a Epsilon scheduler is passed, other parameters below are ignored in practice. If the parameter is a float, then this is understood to be the regularization that is needed, unless relative_epsilon below is True, in which case epsilon is understood as a normalized quantity, to be scaled by the mean value of the cost_matrix.

  • relative_epsilon (Optional[bool]) – whether epsilon is passed relative to scale of problem, here understood as mean value of cost_matrix.

  • scale_epsilon (Optional[float]) – the scale multiplier for epsilon.

  • scale_cost (Union[bool, int, float, Literal[‘mean’, ‘max_cost’, ‘median’]]) – option to rescale the cost matrix. Implemented scalings are ‘median’, ‘mean’ and ‘max_cost’. Alternatively, a float factor can be given to rescale the cost such that cost_matrix /= scale_cost. If True, use ‘mean’.

  • tgt_mask (Optional[ndarray]) – Mask specifying valid rows when computing some statistics of cost_matrix, see src_mask.

  • tgt_mask – Mask specifying valid columns when computing some statistics of cost_matrix, see tgt_mask.

  • kwargs (Any) – additional kwargs to epsilon scheduler.

  • src_mask (Optional[jax._src.numpy.ndarray.ndarray]) –


When defining a Geometry through a cost_matrix, it is important to select an epsilon regularization parameter that is meaningful. That parameter can be provided by the user, or assigned a default value through a simple rule, using the mean_cost_matrix.


apply_cost(arr[, axis, fn])

Apply cost_matrix to array (vector or matrix).

apply_kernel(scaling[, eps, axis])

Apply kernel_matrix on positive scaling vector.

apply_lse_kernel(f, g, eps[, vec, axis])

Apply kernel_matrix in log domain on a pair of dual potential variables.

apply_square_cost(arr[, axis])

Apply elementwise-square of cost matrix to array (vector or matrix).

apply_transport_from_potentials(f, g, vec[, ...])

Apply transport matrix computed from potentials to a (batched) vec.

apply_transport_from_scalings(u, v, vec[, axis])

Apply transport matrix computed from scalings to a (batched) vec.


Copy the epsilon parameters from another geometry.

marginal_from_potentials(f, g[, axis])

Output marginal of transportation matrix from potentials.

marginal_from_scalings(u, v[, axis])

Output marginal of transportation matrix from scalings.

mask(src_mask, tgt_mask[, mask_value])

Mask rows or columns of a geometry.


Compute dual potential vector from scaling vector.

prepare_divergences(*args[, static_b])

Instantiate 2 (or 3) geometries to compute a Sinkhorn divergence.


Compute scaling vector from dual potential.

subset(src_ixs, tgt_ixs, **kwargs)

Subset rows or columns of a geometry.

to_LRCGeometry(rank[, tol, seed])

Factorize the cost matrix in sublinear time [Indyk et al., 2019].

transport_from_potentials(f, g)

Output transport matrix from potentials.

transport_from_scalings(u, v)

Output transport matrix from pair of scalings.

update_potential(f, g, log_marginal[, ...])

Carry out one Sinkhorn update for potentials, i.e. in log space.

update_scaling(scaling, marginal[, ...])

Carry out one Sinkhorn update for scalings, using kernel directly.



Cost matrix, recomputed from kernel if only kernel was specified.


Output rank of cost matrix, if any was provided.


Epsilon regularization value.


Compute and return inverse of scaling factor for cost matrix.


Whether geometry cost/kernel should be recomputed on the fly.


Whether cost is computed by taking squared-Eucl.


Whether geometry cost/kernel is a symmetric matrix.


Kernel matrix, either provided by user or recomputed from cost_matrix.


Mean of the cost_matrix.


Median of the cost_matrix.


Compute the scale of the epsilon, potentially based on data.


Shape of the geometry.


Mask of shape [num_a,] to compute cost_matrix statistics.


Mask of shape [num_b,] to compute cost_matrix statistics.