# ott.geometry.graph.Graph#

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

Graph distance approximation using heat kernel .

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 , 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]) 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. apply_square_cost(arr[, axis]) Apply elementwise-square of cost matrix to array (vector or matrix). apply_transport_from_potentials(f, g, vec[, ...]) Since applying from potentials is not feasible in grids, use scalings. 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. mask(src_mask, tgt_mask[, mask_value]) Mask rows or columns of a geometry. 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(src_ixs, tgt_ixs, **kwargs) Subset rows or columns of a geometry. to_LRCGeometry([rank, tol, rng, scale]) Factorize the cost matrix using either SVD (full) or . Not implemented. 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. dtype The data type. 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 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. src_mask Mask of shape [num_a,] to compute cost_matrix statistics. tgt_mask Mask of shape [num_b,] to compute cost_matrix statistics.