heat.graph.laplacian

Module for graph-based classes

Module Contents

class Laplacian(similarity: Callable, weighted: bool = True, definition: str = 'norm_sym', mode: str = 'fully_connected', threshold_key: str = 'upper', threshold_value: float = 1.0, neighbours: int = 10)

Graph Laplacian from a dataset

Parameters:

• similarity (Callable) – Metric function that defines similarity between vertices. Should accept a data matrix \(n \times f\) as input and return an \(n\times n\) similarity matrix. Additional required parameters can be passed via a lambda function.

• definition (str) –

  Type of Laplacian

  • 'simple': Laplacian matrix for simple graphs \(L = D - A\)

  • 'norm_sym': Symmetric normalized Laplacian \(L^{sym} = I - D^{-1/2} A D^{-1/2}\)

  • 'norm_rw': Random walk normalized Laplacian \(L^{rw} = D^{-1} L = I - D^{-1}\)

• mode (str) –

  How to calculate adjacency from the similarity matrix

  • 'fully_connected' is fully-connected, so \(A = S\)

  • 'eNeighbour' is the epsilon neighbourhood, with \(A_{ji} = 0\) if \(S_{ij} > upper\) or

  \(S_{ij} < lower\); for eNeighbour an upper or lower boundary needs to be set

• threshold_key (str) – 'upper' or 'lower', defining the type of threshold for the epsilon-neighborhood

• threshold_value (float) – Boundary value for the epsilon-neighborhood

• neighbours (int) – Number of nearest neighbors to be considered for adjacency definition. Currently not implemented

similarity_metric

weighted = True

neighbours = 10

_normalized_symmetric_L(A: heat.core.dndarray.DNDarray) → heat.core.dndarray.DNDarray

Helper function to calculate the normalized symmetric Laplacian

\[L^{sym} = D^{-1/2} L D^{-1/2} = I - D^{-1/2} A D^{-1/2}\]

Parameters:

A (DNDarray) – The adjacency matrix of the graph

_simple_L(A: heat.core.dndarray.DNDarray)

Helper function to calculate the simple graph Laplacian

\[L = D - A\]

Parameters:

A (DNDarray) – The Adjacency Matrix of the graph

construct(X: heat.core.dndarray.DNDarray) → heat.core.dndarray.DNDarray

Callable to get the Laplacian matrix from the dataset X according to the specified Laplacian

Parameters:

X (DNDarray) – The data matrix, Shape = (n_samples, n_features)