As we see, spectral clustering allows us to look at more complex cluster assigments based on more than just proximity.


University of Michigan - Ann Arbor

Spectral Clustering is a clustering algorithm that formulates the building of clusters as a graph partitioning problem and uses the eigenvectors of a similarity matrix to break the data into clusters.

Spectral Clustering

https://towardsdatascience.com/spectral-clustering-82d3cff3d3b7

Image Reference : Spectral vs K means clustering

Resources referred to : 
1. [towardsdatascience.com/spectral-clustering-82d3cff3d3b7](https://towardsdatascience.com/spectral-clustering-82d3cff3d3b7)
2. [en.wikipedia.org/wiki/Spectral_clustering](https://en.wikipedia.org/wiki/Spectral_clustering) 

Reference : spectral clustering

Algorithm : 
Input : Similarity Matrix and number of clusters k.
1. Represent all data points as nodes in a graph.
2. Build the similarity graph given a similarity matrix. Calculate it's weighted adjacency matrix W.  Let these have the dimentionality n x n.
3. Build the unnormalized Laplacian L (n x n) 
4.  If we wish to cluster the data into k points (user defined), then we consider the first k eigenvectors of the matrix L.
5.  Build an n x k matrix with each of the first k eigenvectors as it's columns. Let us call this matrix M.
6.  Interpret each **ROW** of the matrix M as a separate data point $a_i$ . 
7.  Run the k-mean clustering algorithm to put these distinct $a_i$ data points into clusters.

Spectral Clustering Algorithm

Papers and class slides referred to : 
1. [www.cs.cmu.edu/~aarti/Class/10701/readings/Luxburg06_TR.pdf](https://www.cs.cmu.edu/~aarti/Class/10701/readings/Luxburg06_TR.pdf)
2. [www.cs.cmu.edu/~aarti/Class/10701/slides/Lecture21_2.pdf](https://www.cs.cmu.edu/~aarti/Class/10701/slides/Lecture21_2.pdf)
3. [people.cs.pitt.edu/~milos/courses/cs3750-Fall2011/lectures/class17.pdf](https://people.cs.pitt.edu/~milos/courses/cs3750-Fall2011/lectures/class17.pdf)
4. [projecteuclid.org/download/pdfview_1/euclid.aos/1431695635](https://projecteuclid.org/download/pdfview_1/euclid.aos/1431695635) 

Paper References : spectral clustering

1. Doesn't make assumptions about the underlying structure of clusters and is better at finding irregularly shaped clusters
2. Very efficient for sparse data sets.

Advantages of Spectral Clustering

1. Computationally intensive
2. Not immediately apparent how to choose similarity functions or assigning the number of clusters without knowing something about the underlying data structure.
3. Uses K means, so the cluster assignments may not always be stable/the same.

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