Compression results

Following plots show the Frobenius norm error \(||{A-\tilde A}||_{\text{Frob}}\) of compressing matrices with pMMF vs. other sketching methods, as a function of the dimension of the compressed core. In each figure, the error is normalized by \(||{A-A_k}||_\text{\Frob}\), where \(A_k\) is the best rank \(k\) approximation to \(A\). The four datasets are "HEPph", "AstroPh", "CondMat" and "Gisette", with \(k\le 100\) in the first three and \(k\le12\) in "Gisette".

HEPph AstroPh
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CondMat Gisette
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Preconditioning results

Following figures show the residual as a function of the iteration number of the conjugate gradient algorithm when solving \(Ax= b\), where \(b\) is a random vector. The times indicated in the legend of each plot are the wall clock times to convergence on an 8 core 2.6 GHz machine. The per-iteration time of pMMF preconditioning is usually \(2-10\) times faster than of other methods. The pMMF, incomplete Cholesky, and SSOR experiments are shown in orange, yellow and magenta, respectively. The blue curve is the solution without using any precondition.

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