Lanczos algorithm Overview

Lanczos algorithm:

Lanczos algorithm is designed to do feature decomposition, and other similar algorithms, the acquisition of singular vectors are very close (the translation may not be the case, immediate translated quickly?) Like other algorithms, getting singular vectors is fast? This translation? ）。 The singular vector of matrix A is the eigenvector of a^t * A or a * a^t. (Should a^t be the transpose of matrix A?) Eigenvectors should be only n*n matrices, only A's transpose multiplied by a to achieve matrix A is a n*n square matrix, where A^t does not understand what it means. Lanczos algorithm uses a seed vector V (in fact, this v should be based on the previous analysis of the understanding should be that the initialization of the 13 square root 13, that is, the number of columns to open the square root), and then constantly multiplied by A to get V ' =a.times (v). (then subtract the value of the previous V ", and get an auxiliary projection matrix). In general, the matrix A is not a n*n square matrix (more generally, asymmetrical), then the constant use of v times A * a^t, the equivalent method in Mahout is a.timesquared (v), (Timessquared method is only an approximate value: change A * A^t.times (v) To sum the order, you can only pass the line of a matrix once rather than two times? What does it mean? Do not understand);

After the K-cycle, v_i=a.timessquared (V_ (i-1)), a k*k diagonal matrix (the aforementioned auxiliary projection matrix) is generated, (first of all, the singular matrix should be multiplied by the diagonal matrix and the basis matrix, Then the approximation matrix of a is obtained by multiplying the singular matrix and the A matrix, but where does the basis matrix come from? Then a matrix with a very approximate but low dimension to a matrix can be generated. What does K mean? K is the extension of the matrix a spectrum (what does this mean?) ): The preceding vectors represent a very large component of the singular vector, followed by a few vectors that represent a relatively small component. A better method is to first generate a 3*k singular vector, and then only take the previous component of the larger one-third to do the calculation (component meaning: In the previous calculation can be seen singularvector have a singular vlaue, It should be by this singularvalue to judge which is on the front, then the composition is big);

Parallel Stragegy:

The Lanczos algorithm is an awkward parallel algorithm: a matrix multiplied by a vector can only be computed on a single line, and then the total result is obtained at the end. When the true execution of A.times (v), there will be no conflict or synchronization problem (for each vector, the output is separate), and the Multi-Core method will become very fast, in Hadoop, there are some to speed up the algorithm optimization scheme. In cases where matrix A is asymmetric, a.timessquare (v) does need to be synchronized, so the time to write the hard drive is delayed until the mapper is closed. If the combiner is set, then the acceleration bottleneck is just a single node acceleration. (meaning should be said, if set the combiner, then the cluster acceleration effect should be very good, so can optimize the place is left with a single node optimization).

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