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Krylov subspace methods to accelerate kernel machines on graphs

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  • In classical frameworks as the Euclidean space, positive definite kernels as well as their analytic properties are explicitly available and can be incorporated directly in kernel-based learning algorithms. This is different if the underlying domain is a discrete irregular graph. In this case, respective kernels have to be computed in a preliminary step in order to apply them inside a kernel machine. Typically, such a kernel is given as a matrix function of the graph Laplacian. Its direct calculation leads to a high computational burden if the size of the graph is very large. In this work, we investigate five different block Krylov subspace methods to obtain cheaper iterative approximations of these kernels. We will investigate convergence properties of these Krylov subspace methods and study to what extent these methods are able to preserve the symmetry and positive definiteness of the original kernels they are approximating. We will further discuss the computational complexity and the memory requirements of these methods, as well as possible implications for the kernel predictors in machine learning.

    Mathematics Subject Classification: Primary: 65F60, 65F50; Secondary: 65D15.

    Citation:

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  • Figure 1.  Comparison between Lanczos and Chebyshev Krylov subspace approximation of a diffusion and a variation spline kernel on the path graph $ G_1 $. The blue line indicates the support of the approximant, the error with respect to the exact kernel column is measured in the uniform norm

    Figure 2.  Kernel interpolant on the bunny graph using $ N = 20 $ samples and the variational spline kernel with parameters $ s = 2 $ and $ \epsilon = 0.05 $

    Figure 3.  Uniform error $ \|y - y^{(\mathrm{kr})}\|_\infty $ for the five block Krylov methods $ \mathrm{kr} \in \{\mathrm{cbl}, \mathrm{gbl}, \mathrm{sbl}, \mathrm{cheb}, \mathrm{cheb}^2\} $ in terms of the iteration numbers $ m $

    Figure 4.  Eigenvalues of $ \mathrm{E}_W^* p_{\phi,5}^{(\mathrm{kr})}( \mathbf{L}) \mathrm{E}_W $ for the five methods $ \mathrm{kr} \in \{\mathrm{cbl}, \mathrm{gbl}, \mathrm{sbl}, \mathrm{cheb}, \mathrm{cheb}^2\} $, with kernel $ \phi( \mathbf{L}) = e^{-t \mathbf{L}} $, $ t = 20 $

    Table 1.  Required operations to calculate $ p_{\phi,m-1}^{(\mathrm{kr})}( \mathbf{L}) \mathrm{E}_W $ for the Krylov space methods $ \mathrm{kr} \in \{\mathrm{cbl},\mathrm{gbl},\mathrm{sbl},\mathrm{cheb}\} $

    Operations $ \rm cbl $ $ \rm gbl $ $ \rm sbl $ $ \rm cheb $
    MVs $ m N $ $ m N $ $ m N $ $ m N $
    DOTs $ \mathcal{O}(m N^2) $ $ \mathcal{O}(m N) $ $ \mathcal{O}(m N) $ -
    AXPYs $ \mathcal{O}(m N^2) $ $ \mathcal{O}(m N) $ $ \mathcal{O}(m N) $ $ \mathcal{O}(m N) $
    $ \phi(\mathbf{H}_m) / c_k(\phi) $ $ \mathcal{O}(m N^3) + \mathcal{O}(m^2 N^2) $ $ \mathcal{O}(m^2) $ $ \mathcal{O}(m^2 N) $ $ \mathcal{O}(m \log m) $
     | Show Table
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    Table 2.  Memory requirements for the calculation of the matrix polynomial $ p_{\phi,m-1}^{(\mathrm{kr})}( \mathbf{L}) \mathrm{E}_W $ for the Krylov space methods $ \mathrm{kr} \in \{\mathrm{cbl},\mathrm{gbl},\mathrm{sbl},\mathrm{cheb}\} $

    Storage $ \rm cbl $ $ \rm gbl $ $ \rm sbl $ $ \rm cheb $
    $ \mathrm{Q}_k/ T_k( \mathbf{L}) \mathrm{E}_W $
    $ \mathbf{H}_m / c_k(\phi) $
    $ m n N $
    $ \mathcal{O}(m N^2) $
    $ m n N $
    $ \mathcal{O}(m) $
    $ m n $
    $ \mathcal{O}(m) $
    $ 2n $
    $ m $
     | Show Table
    DownLoad: CSV
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