Matrix Factorization and Latent Semantic Indexing Low-Rank Approximation SVD Low-rank approximation Whereas the term-doc matrix a may have m=50000 N=10 million(and rank close to 50000) We can construct an approximation Aloo with rank 100. of all rank 100 matrices it would have the lowest Frobenius error Great. but why would we?? Answer Latent Semantic Indexing C. Eckart, G. Young, The approximation of a matrix by another of lower rank Psychometrika, 1, 211-218, 1936Matrix Factorization and Latent Semantic Indexing 23 SVD Low-rank approximation ▪ Whereas the term-doc matrix A may have M=50000, N=10 million (and rank close to 50000) ▪ We can construct an approximation A100 with rank 100. ▪ Of all rank 100 matrices, it would have the lowest Frobenius error. ▪ Great … but why would we?? ▪ Answer: Latent Semantic Indexing C. Eckart, G. Young, The approximation of a matrix by another of lower rank. Psychometrika, 1, 211-218, 1936. Low-Rank Approximation