Subspace clustering

• good clusters may exist in a subspace
• clustering methods may be ineffective for some datasets in the normal space
• linear combination of dimension or a subset of current dimensions
• In the case where data is not axis aliged, then use linear combination

Axis aligned clusters

• subset of original d dimensions
• O(2^d) choices
• Build dimension enumeration tree (all possible combinations of dimensions)
• project onto each new dimensionality and evaluate clustering

Clique method

• grid over data
• Every dimension is split into b bins/divisions
• O(d^b)
• Density = # of points (can have many definitions)
• Keep track of dense cells in each dimension
• tries to find axis aligned cells
• Clique will join each cell that share a face
• \(\frac{x_{range}}{x} \cdot \frac{y_{range}}{Y}\)
• Bad in high dimensions
• sensitive to bin boundaries
• fast (can use itemset mining), but misses a lot of data points

Random projections

• randomized to select "good" subspace

For i=1 maxiters:
   pick a random sead point xp
   for j=1 to max sample
      pick a sample of points
        score is the size of the cluster * dimenisonality (higher dimensions is better)

General subspaces

• projective k-means
• Generalization of PCA to subspaces
• Projetedive kmeans/subspace PCA
• Find direction of most variance/ find k subspace of varying dimensionalities
• true clusters are orthogonal

algorithm

• initialization
• iterative refinement (Expectation maximation)
  • repeat until convergence
  • for each partition D_i do a PCA(D_i ) and determine the dimenionality
  • D_i <- dim(PCA(D_i ))
  • map point onto subspace, determine distance between projection and point, smallest distance wins