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Matrix Inversions (Approximate Matrices) and Log Determinant (Jacobians)

  • Reduced Order Matrix
    • Gram, Covariance
    • Data Independent - Fourier Features
    • Data Dependent - Nyström
  • Sparse Matrix
  • Structured Matrix - Diagonal, TriDiagonal, Kronecker, Block Diagonal, Triangular
  • Log Determinant - Stochastic, Hutchins
  • SVD/PCA/EOFs/POD
    • Randomized Sketching + Deterministic Post-Processing
    • Order of Matrix Multiplications
  • Vector Jacobian Products
  • Linear Solve
    • Cholesky
    • Iterative - Conjugate Gradient

Applications

  • randomized SVD
    • Gap-Filling with EOFs
    • Eigenmaps - Laplacian, Schrodinger, Locality Projections
  • Log Determinants - GPs, Fokker-Planck PDE
  • Vector Jacobian Products - 3DVar, 4DVar
  • Linear Solvers
    • Regression - Linear, Ridge, Lasso, Elastic Net, Kernel, Fourier Features, Nystrom
    • State Estimation - DeNoise, DeBlurr, Gap Filling (SVD), 3DVarNet, SC 4DVarNet
    • Poisson Solver - Periodic, Dirichlet, Neumann, Staggered Grid