K passes over the data - where in each pass you multiply once by A and once by A' I call 2K passes over the data.
On Mon, Dec 12, 2011 at 7:48 PM, Jake Mannix <[email protected]> wrote: > On Mon, Dec 12, 2011 at 9:10 AM, Danny Bickson <[email protected] > >wrote: > > > I meant to write: twice in case of a rectangular matrix. > > By the way, if you want to have the two sides matrices [U,D,V]=svd(A) > > You will need to run Lanczos twice: once with A and another time with A'. > > So run time should be doubled. > > > > Neither of these statements are actually correct: for a rectangular matrix, > if you want the > top K singular vectors and values, you will make K passes over the data > (not 2K) each > one being an operation of (A'A)*v without ever computing A'A itself. This > operation > "timesSquared(Vector)", for a matrix with row i having d_i nonzero entries, > will scale like > sum_i(d_i^2), but still only one pass over the data. > > Also, once you have run Lanczos over A, and gotten the matrix V out, you > can recover > U in O(1) map-reduce operations, by use of the identity: U = A * V * D^-1 > > -jake > > > > > > On Mon, Dec 12, 2011 at 7:08 PM, Danny Bickson <[email protected] > > >wrote: > > > > > In each Lanczos iteration you multiply by the matrix A (in case of a > > > square matrix) > > > or twice, by the matrix A' and A. Multiplication is linear in the > number > > > of non zero edges. > > > See http://en.wikipedia.org/wiki/Lanczos_algorithm > > > Finally a decomposition of a tridiagonal matrix T for extracting the > > > eigenvalues. > > > I think it is also linear in the number of iterations (since the size > of > > T > > > is number of iterations+1). Note that this code is not distributed > since > > it > > > can be efficiently done on a single node. > > > The last step is the Ritz transformation - a product of the > intermediate > > > vectors v > > > with the eigenvectors. This step may be heavy since those matrices are > > > typically dense. > > > > > > Best, > > > > > > DB > > > > > > > > > 2011/12/12 Fernando Fernández <[email protected]> > > > > > >> Hi all, > > >> > > >> This is a question for everybody, though it may be better answered by > > Jake > > >> Mannix. Do you guys know what is the complexity of the algorithm > > >> implemented in mahout for Lancos SVD? Linear, quadratic, etc.. > > >> > > >> > > >> Thanks in advance!! > > >> Fernando. > > >> > > > > > > > > >
