ott.geometry.graph.Graph

class ott.geometry.graph.Graph(laplacian, t=0.001, n_steps=100, numerical_scheme='backward_euler', tol=-1.0, **kwargs)

Graph distance approximation using heat kernel [Crane et al., 2013, Heitz et al., 2021].

Approximates the heat kernel for large n_steps, which for small t approximates the geodesic exponential kernel \(e^{\frac{-d(x, y)^2}{t}}\).

Parameters:

• laplacian (Array) – Symmetric graph Laplacian. The check for symmetry is NOT performed. See also from_graph().

• n_steps (int) – Maximum number of steps used to approximate the heat kernel.

• numerical_scheme (Literal['backward_euler', 'crank_nicolson']) – Numerical scheme used to solve the heat diffusion.

• normalize – Whether to normalize the Laplacian as \(L^{sym} = \left(D^+\right)^{\frac{1}{2}} L \left(D^+\right)^{\frac{1}{2}}\), where \(L\) is the non-normalized Laplacian and \(D\) is the degree matrix.

• tol (float) – Relative tolerance with respect to the Hilbert metric, see [Peyré and Cuturi, 2019], Remark 4.12. Used when iteratively updating scalings. If negative, this option is ignored and only n_steps is used.

• kwargs (Any) – Keyword arguments for Geometry.

• t (float)

Methods

apply_cost(arr[, axis, fn, is_linear])	Apply cost_matrix to array (vector or matrix).
apply_kernel(vec[, eps, axis])	Apply kernel_matrix on a positive vector.
apply_lse_kernel(f, g, eps[, vec, axis])	Apply kernel_matrix in log domain.
apply_square_cost(arr[, axis])	Apply elementwise-square of cost matrix to array (vector or matrix).
apply_transport_from_potentials(f, g, vec[, ...])	Not implemented.
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.
from_graph(G[, t, directed, normalize])	Construct Graph from an adjacency matrix.
marginal_from_potentials(f, g[, axis])	Not implemented.
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 2 (or 3) geometries to compute a Sinkhorn divergence.
scaling_from_potential(potential)	Compute scaling vector from dual potential.
set_scale_cost(scale_cost)	Modify how to rescale of the cost_matrix.
subset([row_ixs, col_ixs])	Subset rows or columns of a geometry.
to_LRCGeometry([rank, tol, rng, scale])	Factorize the cost matrix using either SVD (full) or [Indyk et al., 2019].
transport_from_potentials(f, g)	Not implemented.
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.

Attributes

can_LRC	Check quickly if casting geometry as LRC makes sense.
cost_matrix	Cost matrix, recomputed from kernel if only kernel was specified.
cost_rank	Output rank of cost matrix, if any was provided.
diag_cost	Diagonal of the cost matrix.
dtype	The data type.
epsilon	Epsilon regularization value.
epsilon_scheduler	Epsilon scheduler.
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_square	Whether geometry cost/kernel is a square matrix.
is_squared_euclidean	Whether cost is computed by taking squared Euclidean distance.
is_symmetric	Whether geometry cost/kernel is a symmetric matrix.
kernel_matrix	Kernel matrix.
mean_cost_matrix	Mean of the cost_matrix.
median_cost_matrix	Median of the cost_matrix.
shape	Shape of the geometry.
std_cost_matrix	Standard deviation of all values stored in cost_matrix.