G. Golub and C. Reinsch, Handbook for automatic computation II, linear algebra, Springer-Verlag, New York, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Delta Delta oe r 1 = Delta Delta Delta oe q = 0 and dene the truncated SVD approximation A k of A as A k = k X i=1 u i Delta oe i Delta v T i = U k Sigma k V T k ; 3. 2) then min rank(B) k kA Gamma Bk 2 F = kA Gamma A k k 2 F = q X i=k 1 oe 2 i Proof: see [54] 2 Theorem 3.1.3 states that the best rank k approximation of A with respect to the Frobenius norm is A k as dened by (3.2) We can even say that [55] A k is the best rank k approximation of A with respect to any unitarily invariant norm and hence min rank(B) k kA Gamma Bk 2 = kA Gamma A k k ....

G. Golub and C. Reinsch, Handbook for automatic computation II, linear algebra, Springer-Verlag, New York, 1971.

....document space, and thus partially resolves the word choice (synonyms) problem in information retrieval, and redundant semantic relationships in text categorization. Mathematically, LSI with a truncated SVD is the best approximation of X in the reduced k dim subspace (Eckart Young Theorem, see [Golub and Reinsch, 1971]) However, the improved results in information retrieval and ltering indicates the LSI goes beyond mathematical approximations. From statistical point of view, LSI amounts to an e ective dimensionality reduction, similar to principal component analysis in statistics. Dimensions with small ....

G. Golub and C. Reinsch. Handbook for Automatic Computation II, Linear Algebra. Springer-Verlag, New York, 1971.

....Theorem 2.2 [Eckart and Young] Let the SVD of A be given by Equation (1) with r = rank(A) p=min(m;n) and define A k = k X i=1 u i Delta oe i Delta v T i ; 2) then min rank(B) k kA Gamma Bk 2 F =kA Gamma A k k 2 F =oe 2 k 1 Delta Delta Delta oe 2 p : Proof 2. 2 See [16]. In other words, A k , which is constructed from the k largest singular triplets of A, is the closest rank k matrix to A [15] In fact, A k is the best approximation to A for any unitarily invariant norm [22] Hence, min rank(B) k kA Gamma Bk 2 =kA Gamma A k k 2 =oe k 1 : 3) 2.1 Latent ....

G. GOLUB AND C. REINSCH, Handbook for automatic computation II, linear algebra, Springer-Verlag, New York, 1971.

....the U and V matrices and the first (largest) k singular values of A are used to construct a rank k approximation to A via A k = U k Sigma k V T k . The columns of U and V are orthogonal, such that U T U = V T V = I r , where r is the rank of the matrix A. A theorem due to Eckart and Young [15] suggests that A k , constructed from the k largest singular triplets 1 of A, is the closest rank k approximation (in the least squares sense) to A [5] With regard to LSI, A k is the closest k dimensional approximation to the original term document space represented by the incidence matrix A. ....

G. Golub and C. Reinsch. Handbook for Automatic Computation II, Linear Algebra. Springer-Verlag, New York, 1971.

....[14] Theorem 2.2. Eckart and Young] Let the SVD of A be given by Equation (1) with r = rank(A) p = min(m;n) and define Ak = k X i=1 u i Delta oe i Delta v T i ; 2) then min rank(B) k kA Gamma Bk 2 F = kA Gamma Akk 2 F = oe 2 k 1 Delta Delta Delta oe 2 p : Proof. See [15]. In other words, Ak , which is constructed from the k largest singular triplets of A, is the closest rank k matrix to A [14] In fact, Ak is the best approximation to A for any unitarily invariant norm [21] Hence, min rank(B) k kA Gamma Bk2 = kA Gamma Akk2 = oe k 1 : 3) 2.1. Latent ....

G. Golub and C. Reinsch, Handbook for automatic computation II, linear algebra, SpringerVerlag, New York, 1971.

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