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For the MiniBooNE data set, decay of the radial SKD during the optimisation of a random initial Nyström sample of size n = 1,000. The SGD is based on i.i.d. sampling with batch size b = 200 and stepsize γ = 2× 10 −7 , and the descent is stopped after T = 8,000 iterations; the cost is evaluated every 100 iterations.
Source publication
We study a relaxed version of the column-sampling problem for the Nyström approximation of kernel matrices, where approximations are defined from multisets of landmark points in the ambient space; such multisets are referred to as Nyström samples. We consider an unweighted variation of the radial squared-kernel discrepancy (SKD) criterion as a surr...
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Context 1
... consider a random initial Nyström sample of size n = 1,000, and optimise it through SGD with i.i.d. sampling, batch size b = 200, stepsize γ = 2 × 10 −7 ; the descent is stopped after T = 8,000 iterations. The resulting decay of the n = 100 n = 200 ini. opt. radial SKD is presented in Fig. 6 (the cost is evaluated every 100 iterations), and the trace norm of the Nyström approximation error for the initial and locally optimised samples are reported. In terms of computation time, on our machine (endowed with an 3.5 GHz Dual-Core Intel Core i7, and using a single-threaded C implementation interfaced with R), for n = 1,000, an ...
Context 2
... an 3.5 GHz Dual-Core Intel Core i7, and using a single-threaded C implementation interfaced with R), for n = 1,000, an evaluation of the radial SKD (up to the constant K 2 F ) takes 6.8 s, while an evaluation of the trace criterion K − ˆ K(S) * takes 6,600 s (the pseudoinverse of K S being computed in R); performing the optimisation reported in Fig. 6 without checking the decay of the cost takes 1,350 s. In this specific setting, the full radial-SKD optimisation process is thus roughly 5 times faster than a single evaluation of the trace ...
Citations
... The link between integral-operator approximation and potential approximation may be leveraged to design sampling strategies for low-rank approximation (where approximations are characterised by sparse finitely-supported measures). The direct minimisation of D μ under sparsity-inducing constraints is for instance considered in [10], while the possibility to locally optimised the support of approximate measures using particle-flow techniques is studied in [13]. Sequential approaches, where support points are added one-at-a-time on the basis of information provided by the directional derivatives of D μ , are investigated in [12]. ...
We describe a natural coisometry from the Hilbert space of all Hilbert-Schmidt operators on a separable reproducing kernel Hilbert space (RKHS)H and onto the RKHS G associated with the squared-modulus of the reproducing kernel of H. Through this coisometry, trace-class integral operators defined by general measures and the reproducing kernel of H are isometrically represented as potentials in G, and the quadrature approximation of these operators is equivalent to the approximation of integral functionals on G. We then discuss the extent to which the approximation of potentials in RKHSs with squared-modulus kernels can be regarded as a differentiable surrogate for the characterisation of low-rank approximation of integral operators.
