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# lecture21 - Lecture 21 SVD and Latent Semantic Indexing and...

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Lecture 21 SVD and Latent Semantic Indexing and Dimensional Reduction Shang-Hua Teng

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Singular Value Decomposition r T r r r T T v u v u v u A σ 2 1 2 2 2 1 1 1 + + + = where u 1 u r are the r orthonormal vectors that are basis of C(A) and v 1 v r are the r orthonormal vectors that are basis of C(A T )
Low Rank Approximation and Reduction T r T k k k T T k v u A v u v u v u A 1 1 1 1 2 1 2 2 2 1 1 1 σ = + + + =

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The Singular Value Decomposition · · A U V T = Σ 0 0 A U V T m x n m x r r x r r x n = Σ 0 0 m x n m x m m x n n x n
The Singular Value Reduction · · A U V T m x n m x r r x r r x n = Σ 0 0 A k U k V k T m x n m x k k x k k x n = Σ

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How Much Information Lost? T r T k k k T T k v u A v u v u v u A 1 1 1 1 2 1 2 2 2 1 1 1 σ = + + + =
Distance between Two Matrices Frobenius Norm of a matrix A. Distance between two matrices A and B ( 29 2 , 2 1 2 ∑∑ - = - = = j i ij ij F m i n j ij F B A B A A A

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How Much Information Lost?
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lecture21 - Lecture 21 SVD and Latent Semantic Indexing and...

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