diffusion_mapping

hypercoil.functional.connectopy.diffusion_mapping(W: Array | ndarray, k: int = 10, alpha: float = 0.5, diffusion_time: int = 0) → Tuple[Array | ndarray, Array | ndarray]

Manifold coordinates estimated using diffusion mapping.

This functionality is adapted very closely from brainspace with some minor adaptations for differentiability.

Warning

Sparse inputs are currently unsupported because an implementation of a sparse extremal eigenvalue solver does not yet exist in JAX. For sparse inputs, use the generalised connectopic functional instead – once we implement VJP rules for elementary operations on sparse matrices, anyway.

Note

The anisotropic diffusion parameter determines the kind of diffusion map produced by the algorithm.

• \(\alpha = 0\) produces Laplacian eigenmaps, corresponding to a random walk-style diffusion operator.

• \(\alpha = 0.5\) (default) corresponds to Fokker-Planck diffusion.

• \(\alpha = 1\) corresponds to Laplace-Beltrami diffusion.

Dimension:

W : \((*, N, N)\) or \((*, E)\)

* denotes any number of preceding dimensions, N denotes number of vertices, and E denotes number of edges. The shape should be \((*, N, N)\) if edge_index is not provided and \((*, E)\) if edge_index is provided.

edge_index : \((*, 2, E)\)

As above.

Q : \((*, N, k)\)

k denotes the number of diffusion maps.

L : \((*, k)\)

As above.

Parameters:

Wtensor

Edge weight tensor. This should be the graph adjacency (or affinity) matrix.

kint (default 10)

Number of eigenmaps to compute.

alphafloat \(\in [0, 1]\) (default 0.5)

Anisotropic diffusion parameter.

diffusion_timeint (default 0)

Diffusion time parameter. A value of 0 indicates that a multi-scale diffusion map should be computed, which considers all valid times (1, 2, 3, etc.).

Returns:

Qtensor

Diffusion maps.

Ltensor

Eigenvalues corresponding to diffusion maps.

See also

laplacian_eigenmaps()