247 research outputs found
Spectral norm of random tensors
We show that the spectral norm of a random tensor (or higher-order array) scales as
under some sub-Gaussian
assumption on the entries. The proof is based on a covering number argument.
Since the spectral norm is dual to the tensor nuclear norm (the tightest convex
relaxation of the set of rank one tensors), the bound implies that the convex
relaxation yields sample complexity that is linear in (the sum of) the number
of dimensions, which is much smaller than other recently proposed convex
relaxations of tensor rank that use unfolding.Comment: 5 page
Fast learning rate of multiple kernel learning: Trade-off between sparsity and smoothness
We investigate the learning rate of multiple kernel learning (MKL) with
and elastic-net regularizations. The elastic-net regularization is a
composition of an -regularizer for inducing the sparsity and an
-regularizer for controlling the smoothness. We focus on a sparse
setting where the total number of kernels is large, but the number of nonzero
components of the ground truth is relatively small, and show sharper
convergence rates than the learning rates have ever shown for both and
elastic-net regularizations. Our analysis reveals some relations between the
choice of a regularization function and the performance. If the ground truth is
smooth, we show a faster convergence rate for the elastic-net regularization
with less conditions than -regularization; otherwise, a faster
convergence rate for the -regularization is shown.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1095 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org). arXiv admin note: text overlap with
arXiv:1103.043
Sparsity-accuracy trade-off in MKL
We empirically investigate the best trade-off between sparse and
uniformly-weighted multiple kernel learning (MKL) using the elastic-net
regularization on real and simulated datasets. We find that the best trade-off
parameter depends not only on the sparsity of the true kernel-weight spectrum
but also on the linear dependence among kernels and the number of samples.Comment: 8pages, 2 figure
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