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Matrix Methods - Coggle Diagram

- - - - \[ X^+ = (X^TX)^{-1}X^T\]
  - - - \(\Sigma_i^k \sigma_iu_iv_i^T \)
  - - - Uniqueness
      - leverage scores \(\mathcal{l}_{r,j} = \lVert U_k (j, :) \rVert_2 \)
        
        highest
        
        weighted random
      - Interpolatory operator
  - - - Alternating nonnegative LS
      - Alternating LS
      - Multiplicative Updates
      - Hierarchical ALS
- - - - colexicographic (stack column)/lexicographic
  - - - Higher-order SVD \(T_r\)
        
        Sequentially Truncated HoSVD
        
        quasi-optimality
    - - approximation error & compression ratio
  - - - Nonnegative CP
      - Sparse CP
      - Orthogonal decomposition
      - Symmetric CP
      - Generalized CP
        
        Other Loss
      - Expectation Maximization
    - - Border rank problem
    - - Normalized CP decomposition
- - - - Tikhonov/Ridge \( \lambda \lVert w \rVert_2^2\)
      - LASSO \( \lambda \lVert w \rVert_1 \)
    - - Mercer's theorem
      - Linear kernel
      - Polynomial kernel
      - RBF kernel
      - Sigmoid kernel
    - - log-odds
  - - - Laplace matrix \(L = D - W \)
        
        Normalized Laplacians
        
        Random Walk Laplacian \( D^{-1}L\)
        
        Symmetric Graph Laplacian \(D^{-\tfrac{1}{2}} L D^{-\tfrac{1}{2}} \)
    - - Similarity graph \(G = (V, E) \)
      - min-cut problem
        
        RatioCut
        
        N-cut
    - - Norms
      - Procedures
      - Relation to PCA
  - - - Soft Margin Classifier
      - Hyperplane \(w^Tx+b=0\)
      - linearly separable
      - Maximal Margin Classifier
        
        Quadratic Problem
        
        Sequential Minimal Optimization
        
        \[ \min \tfrac{1}{2}x^TQx+c^Tx\]
        
        feasible & bounded below
        
        Duality
        
        weak duality
        
        strong duality
        
        KKT conditions
        
        Stationarity
        
        Dual feasibility
        
        Primal feasibilty
        
        Complementary slackness
      - trivial