Matrix Factorization and Latent Semantic Indexing Low-Rank Approximation Approximation error How good (bad) is this approximation? It's the best possible, measured by the frobenius norm of the error. minA-xl=4-Alk=12∑ X: rank(X)=k i=k+1 where the o, are ordered such that o 20i+1 Suggests why frobenius error drops as k increased 22Matrix Factorization and Latent Semantic Indexing 22 Approximation error ▪ How good (bad) is this approximation? ▪ It’s the best possible, measured by the Frobenius norm of the error: where the i are ordered such that i  i+1. Suggests why Frobenius error drops as k increased.  = = + − = − = r i k F k F i X rank X k A X A A 1 2 : min ( )  Low-Rank Approximation