Fundamentals of Matrices
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1 Maschinelles Lernen II Fundamentals of Matrices Christoph Sawade/Niels Landwehr/Blaine Nelson Tobias Scheffer
2 Matrix Examples Recap: Data Linear Model: f i x = w i T x Let X = x x n be the data matrix & W = w w k be the matrix of class models; then it follows that f x f x n W T X = f k x f k x n
3 Matrix Examples Recap: Graphs Adjacency matrix G g i, j Incidence matrix, Laplace matrix, Edge between Node i and j?
4 Matrix Examples Recap: System of Equations Elementary matrices X R m n : Basic operations of the Gaussian elimination method Addition of rows / columns Multiplication of rows / columns by a scalar Interchanging of rows / columns X is reproduced by simultaneously applying the desired operation to the identity matrix
5 Matrix Examples Recap: Linear Transformations Matrix multiplications are linear transformations: A cx + y = cax + Ay 5
6 Matrix Examples Recap: Linear Transformations Scalings A sc c = c c c 3 6
7 Matrix Examples Recap: Linear Transformations Shearing: 3 A sh c = is only scaled. c c c c c c c c c 7
8 Matrix Examples Recap: Linear Transformations Rotation z A rot θ = z-axis is invariant. cos θ sin θ 0 sin θ cos θ
9 Matrix Examples Recap: Linear Transformations Combinations: x-axis is only scaled. x A rot θ A sc c 9
10 EIGENVALUE DECOMPOSITION 0
11 Eigenvalue Problem Eigenvectors u i C m of a transformation A R m m are only scaled by A. Eigenvalues λ i C of a transformation A are the corresponding scaling factor of u i. Eigenvalue problem: Determine u i, such that Au i = λ i u i
12 Eigenvalue Problem Properties Au i = λ i u i eig A k = λ k,, λ m k, k =? m m det A = i= λ i and trace A = i= λ i det A λ i I = 0 λ i eig A If A is symmetric, then i λ i R If A 0 (positive semi-definite), then i λ i 0 If A 0 (positive definite), then i λ i > 0 Eigenvalue Decomposition: A = UΛU A = u u m λ 0 0 λ m u u m The analog for B R m n : singular values
13 Eigenvalue Problem Symmetric Matrices Au i = λ i u i It follows, if u i and u j are eigenvectors, then cu i + u j is also a solution for any c R. (Linearity) u = 0 is always a trivial solution. For symmetric matrices, the eigenvectors can be selected to be orthonormal (i.e., a basis): u i T u j = 0 i j u i T u i = i
14 Computation of Eigenvalues Power Iteration Method Iterative approach toward the largest eigenvector u of A: Initialize u 0 randomly Repeat until u k+ = Au k k u Au Rate of Convergence: u u k = O λ 2 λ where λ 2 is the 2 nd largest eigenvalue of A k
15 SINGULAR VALUE DECOMPOSITION 5
16 Singular Value Problem Suppose X is any m n matrix. The eigen-decomposition of PSD matrices C xx = XX T & K xx = X T X are C xx = UΛ m U T & K xx = VΛ n V T where U & V are orthogonal, & Λ m & Λ n are diagonal For any eigen-pair λ i, v i of K xx, there is a correspondence to C xx via Xv i C xx Xv i = XX T Xv i = XK xx v i = λ i Xv i and Xv i = λ i. The eigen-pair λ i, u i of C xx is then u i = λ i Xv i 6
17 Singular Value Problem Both matrices, C xx & K xx, have eigenvalue λ i with u i = λ 2 Xv i & v i = λ 2 X T u i The singular value decomposition (SVD) of nonsquare X is X = UΣV T where U & V are orthogonal & σ i Σ ii = The SVD is the analog of eigen-decomposition for non-square matrices X is non-singular iff all its singular values are not 0. It yields a spectral decomposition: X = i σ i u i v i T λ i 7
18 NOTES ON POSITIVE DEFINITENESS 8
19 N(0,) Positive Definiteness i λ i 0 A R m m is positive semi-definite if z R m z T Az 0 Consider the Gaussian: πσ 2 exp μ = 0 σ 2 = 2 x μ 2σ x
20 Positive Definiteness A R m m is positive semi-definite if z R m z T Az 0 Consider the Gaussian: x πσ 2 exp 2σ 2 x μ T x μ μ = 0 0 σ 2 = x
21 Positive Definiteness A R m m is positive semi-definite if z R m z T Az 0 Consider the Gaussian: 2π m/2 exp quadratic Mahalanobis distance Σ /2 2 x μ T Σ x μ x x μ = 0 0 Σ = 0 0 Covariance Matrix
22 Positive Definiteness A R m m is positive semi-definite if z R m z T Az 0 Consider the Gaussian: 2π m/2 exp quadratic Mahalanobis distance Σ /2 2 x μ T Σ x μ x x μ = 0 0 Σ = Covariance Matrix
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