:mod:`heat.graph.laplacian`
===========================
.. py:module:: heat.graph.laplacian

.. autoapi-nested-parse::

   Module for graph-based classes



Module Contents
---------------


.. py: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

   :param similarity: Metric function that defines similarity between vertices. Should accept a data matrix :math:`n \times f` as input and
                      return an :math:`n\times n` similarity matrix. Additional required parameters can be passed via a lambda function.
   :type similarity: Callable
   :param definition:
                      Type of Laplacian

                          - ``'simple'``: Laplacian matrix for simple graphs :math:`L = D - A`

                          - ``'norm_sym'``: Symmetric normalized Laplacian :math:`L^{sym} = I - D^{-1/2} A D^{-1/2}`

                          - ``'norm_rw'``: Random walk normalized Laplacian :math:`L^{rw} = D^{-1} L = I - D^{-1}`
   :type definition: str
   :param mode:
                How to calculate adjacency from the similarity matrix

                    - ``'fully_connected'`` is fully-connected, so :math:`A = S`

                    - ``'eNeighbour'`` is the epsilon neighbourhood, with :math:`A_{ji} = 0` if :math:`S_{ij} > upper` or
                    :math:`S_{ij} < lower`; for eNeighbour an upper or lower boundary needs to be set
   :type mode: str
   :param threshold_key: ``'upper'`` or ``'lower'``, defining the type of threshold for the epsilon-neighborhood
   :type threshold_key: str
   :param threshold_value: Boundary value for the epsilon-neighborhood
   :type threshold_value: float
   :param neighbours: Number of nearest neighbors to be considered for adjacency definition. Currently not implemented
   :type neighbours: int


   .. attribute:: similarity_metric
      

      

   .. attribute:: weighted
      :annotation: = True

      

   .. attribute:: neighbours
      :annotation: = 10

      

   .. role:: raw-html(raw)
      :format: html
   .. method:: _normalized_symmetric_L(A: heat.core.dndarray.DNDarray) -> heat.core.dndarray.DNDarray

      Helper function to calculate the normalized symmetric Laplacian

      .. math:: L^{sym} = D^{-1/2} L D^{-1/2} = I - D^{-1/2} A D^{-1/2}

      :param A: The adjacency matrix of the graph
      :type A: DNDarray


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

      Helper function to calculate the simple graph Laplacian

      .. math:: L = D - A

      :param A: The Adjacency Matrix of the graph
      :type A: DNDarray


   .. method:: 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

      :param X: The data matrix, Shape = (n_samples, n_features)
      :type X: DNDarray