Laplacian Matrix for Dimensionality Reduction and Clustering
It provides foundational concepts for graph-based methods in ML, but is incremental as it reviews existing algorithms without new results.
The paper introduces the Laplacian matrix as a tool for representing graphs in machine learning, focusing on its use for dimensionality reduction and clustering to extract features and embed data in low-dimensional spaces.
Many problems in machine learning can be expressed by means of a graph with nodes representing training samples and edges representing the relationship between samples in terms of similarity, temporal proximity, or label information. Graphs can in turn be represented by matrices. A special example is the Laplacian matrix, which allows us to assign each node a value that varies only little between strongly connected nodes and more between distant nodes. Such an assignment can be used to extract a useful feature representation, find a good embedding of data in a low dimensional space, or perform clustering on the original samples. In these lecture notes we first introduce the Laplacian matrix and then present a small number of algorithms designed around it.