ott.geometry.grid.Grid

class ott.geometry.grid.Grid(x=None, grid_size=None, cost_fns=None, num_a=None, grid_dimension=None, **kwargs)

Class describing the geometry of points taken in a cartestian product.

This class implements a geometry in which probability measures are supported on a \(d\)-dimensional cartesian grid, a cartesian product of \(d\) lists of values, each list being itself of size \(n_i\).

The transportation cost between points in the grid is assumed to be separable, namely a sum of coordinate-wise cost functions, as in

\[cost(x,y) = \sum_{i=1}^d cost_i(x_i, y_i)\]

where \(cost_i\): R x R → R.

In such a regime, and despite the fact that the total number \(n_{total}\) of points in the grid is exponential \(d\) (namely \(\prod_i n_i\)), applying a kernel in the context of regularized optimal transport can be carried out in time that is of the order of \(n_{total}^{(1+1/d)}\) using convolutions, either in the original domain or log-space domain. This class precomputes \(d\) \(n_i\) x \(n_i\) cost matrices (one per dimension) and implements these two operations by carrying out these convolutions one dimension at a time.

Parameters

• x (Optional[Sequence[ndarray]]) – list of arrays of varying sizes, describing the locations of the grid. Locations are provided as a list of jnp.ndarrays, that is \(d\) vectors of (possibly varying) size \(n_i\). The resulting grid is the Cartesian product of these vectors.

• grid_size (Optional[Sequence[int]]) – tuple of integers describing grid sizes, namely \((n_1,...,n_d)\). This will only be used if x is None. In that case the grid will be assumed to lie in the hypercube \([0,1]^d\), with the \(d\) dimensions, described as points regularly sampled in [0,1].

• cost_fns (Optional[Sequence[CostFn]]) – a sequence of \(d\) costs.CostFn’s, each being a cost taking two reals as inputs to output a real number.

• num_a (Optional[int]) – total size of grid. This parameters will be computed from other inputs and used in the flatten/unflatten functions.

• grid_dimension (Optional[int]) – dimension of grid. This parameters will be computed from other inputs and used in the flatten/unflatten functions.

• kwargs (Any) – other optional parameters to be passed on to superclass initializer, notably those related to epsilon regularization.

Methods

apply_cost(arr[, axis, fn])	Apply cost matrix to array (vector or matrix).
apply_kernel(scaling[, eps, axis])	Apply grid kernel on scaling vector.
apply_lse_kernel(f, g, eps[, vec, axis])	Apply grid kernel in log space.
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_epsilon(other)	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.
potential_from_scaling(scaling)	Compute dual potential vector from scaling vector.
prepare_divergences(*args[, static_b])	Instantiate the geometries used for a divergence computation.
rescale_cost_fn(factor)	Rescale the cost or kernel matrix using a factor.
scaling_from_potential(potential)	Compute scaling vector from dual potential.
transport_from_potentials(f, g[, axis])	Output transport matrix from potentials.
transport_from_scalings(f, g[, axis])	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.

Attributes

cost_matrices	rtype List[ndarray]
cost_matrix	Cost matrix, recomputed from kernel if only kernel was specified.
cost_rank	Output rank of cost matrix, if any was provided.
epsilon	Epsilon regularization value.
inv_scale_cost	Compute and return inverse of scaling factor for cost matrix.
is_online	Whether geometry cost/kernel should be recomputed on the fly.
is_squared_euclidean	Whether cost is computed by taking squared-Eucl.
is_symmetric	Whether geometry cost/kernel is a symmetric matrix.
kernel_matrices	rtype List[ndarray]
kernel_matrix	Kernel matrix, either provided by user or recomputed from cost.
mean_cost_matrix	Mean of cost matrix.
median_cost_matrix	Median of cost matrix.
scale_epsilon	Compute the scale of the epsilon, potentially based on data.
shape	Shape of cost or kernel matrix.