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Outline. Singular Value Decomposition Example of PCA: Eigenfaces. Singular Value Decomposition. A sample set of M N -dimensional points can be written as a matrix each row of which represents a sample point. PCA of the sample is then equivalent to solving the SVD
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Outline • Singular Value Decomposition • Example of PCA: Eigenfaces
Singular Value Decomposition • A sample set of MN-dimensional points can be written as a matrix each row of which represents a sample point. • PCA of the sample is then equivalent to solving the SVD problem for this matrix; that is, finding the decomposition • W is diagonal matrix, V is orthogonal square matrix, columns of U are orthogonal, columns of V are orthogonal : Also
SVD-remarks • Meaning: columns of V represent the KL transform axes, ordered by respective values in W (singular values), which are amount of variation, in descending order. • The new axes (columns of V) are also eigenvectors of XXT ifX is square matrix • From the orthogonal basis vectors given as columns of V we omit those to which correspond small values in W. • SVD provides unique decomposition for the given data. • Taking the first m<N eigenvectors (rows in VT) we get the optimal approximation in the sense of L2 norm.
SVD-remarks • When M<N the singular values wj for j=M+1,…,N are zero.
