春暖花开 吧：Low-rank approximations 2024-04-26 01:59:37 0 0 Low-rank approximations Give M×N matrix C and a positive integer k, we wish to find an M×N matrix Ck of rank at most k ，so as to minimize the Frobenius norm of the matrix difference X=C−Ck ，defined to be ||X||F=∑i=1M∑j=1Nx2ij−−−−−−−−⎷ .(1) Thus, the Frobenius norm of X measures the discrepancy between Ck and C ; our goal is to find a matrix Ck that minimizes this discrepancy, while constraining Ck to have rank at most k . If r is the rank of C , clearly Cr=C and the Frobenius norm of the discrepancy is zero in this case. When k isfar smallerthan r, we refer to Ck as a low-rank approximation. SVDGiven C , constuct its SVD int the form of C=UΣVT.Derive from Σ the matrix Σk formed by replacing by zeros the r−k smallest sigular values on the diagnal of Σ .Compute and output Ck=UΣkVT as the rank- k approximation to C. http://nlp.stanford.edu/IR-book/html/htmledition/low-rank-approximations-1.html 收藏（0）