Laplace matrix/Mapping/Clustering

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Laplace matrix is a very ingenious thing, it is a kind of matrix of description graph, it is widely used in the field of machine learning such as dimensionality reduction, classification and clustering. what is a laplace matrix Laplace Matrix

First of all, what is the Laplace matrix, the English name Laplacian matrix, the specific form of the first from the diagram, assuming that there is an image as shown below,
　　
Each of its points have corresponding edge connection, we use an indicator (this place can be arbitrarily selected, such as Euclidean distance, geodesic distance, or Gaussian similarity, etc.) to measure the similarity of two points, expressed as W=∑wij, no edge connection of its similarity is naturally zero, W is a symmetric matrix The sum of the similarity of a point to all points is expressed as D=dig (d);d =rowsum (W); D is a diagonal array; our Laplace matrix is l=d−w.

properties of Laplace matrices

Properties:
(1) L is a semi-positive definite matrix.
(2) L the minimum special value is 0, the corresponding special direction is a full 1-column vector.
(3) The LF=ΛDF has m non-negative real eigenvalues, 0