low rank models and projections

description

• application of low rank approximation to represent data in lower-dimensional space

projecting data points:

• Given a data matrix X with m columns (each column is a data point), each column xj is projected to a k-dimensional vector hj.
• This projection is done by multiplying xj with the transpose of the product of the k largest singular values and the k right singular vectors of X.
  • Project each data point xj to a k-dimensional vector hj: $h_j=V_k^T{x}_j$

• with matrix X, use SVD to find rank-k approximation X’ = UH
  • U: matrix of k orthogonal vectors
  • H: matrix of k coefficients
• each data point (column vector)
• each feature (row vector)