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Contents1 Singular Value Decomposition (SVD) Singular Vectors .. Singular Value Decomposition (SVD) .. Best RankkApproximations .. Power Method for Computing the Singular Value Decomposition .. Applications of Singular Value Decomposition .. Component Analysis .. a Mixture of Spherical Gaussians .. Application of SVD to a Discrete Optimization Problem .. as a Compression Algorithm .. Decomposition .. Vectors and ranking documents .. Bibliographic Notes .. Exercises .. 2811 Singular Value Decomposition (SVD)The singular value decomposition of a matrixAis the factorization ofAinto theproduct of three matricesA=UDVT where the columns ofUandVare orthonormal andthe matrixDis diagonal with positive real entries. The SVD is useful in many tasks. Herewe mention some examples. First, in many applications, the data matrixAis close to amatrix of low rank and it is useful to find a low rank matrix which is a good approximationto the data matrix.
Contents 1 Singular Value Decomposition (SVD) 2 ... 1 is perpendicular to W, any unit vector in W will do as w 2. If not, choose w 2 to be the unit vector in Wperpendicular to the projection of v 1 onto W:Since v 1 was chosen to maximize jAv 1j2, it follows that jAw 1j2 jAv 1j2.
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