66 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
Convex Tensor Decomposition via Structured Schatten Norm Regularization
We discuss structured Schatten norms for tensor decomposition that includes
two recently proposed norms ("overlapped" and "latent") for
convex-optimization-based tensor decomposition, and connect tensor
decomposition with wider literature on structured sparsity. Based on the
properties of the structured Schatten norms, we mathematically analyze the
performance of "latent" approach for tensor decomposition, which was
empirically found to perform better than the "overlapped" approach in some
settings. We show theoretically that this is indeed the case. In particular,
when the unknown true tensor is low-rank in a specific mode, this approach
performs as good as knowing the mode with the smallest rank. Along the way, we
show a novel duality result for structures Schatten norms, establish the
consistency, and discuss the identifiability of this approach. We confirm
through numerical simulations that our theoretical prediction can precisely
predict the scaling behavior of the mean squared error.Comment: 12 pages, 3 figure
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
Super-Linear Convergence of Dual Augmented-Lagrangian Algorithm for Sparsity Regularized Estimation
We analyze the convergence behaviour of a recently proposed algorithm for
regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is
based on a new interpretation of DAL as a proximal minimization algorithm. We
theoretically show under some conditions that DAL converges super-linearly in a
non-asymptotic and global sense. Due to a special modelling of sparse
estimation problems in the context of machine learning, the assumptions we make
are milder and more natural than those made in conventional analysis of
augmented Lagrangian algorithms. In addition, the new interpretation enables us
to generalize DAL to wide varieties of sparse estimation problems. We
experimentally confirm our analysis in a large scale -regularized
logistic regression problem and extensively compare the efficiency of DAL
algorithm to previously proposed algorithms on both synthetic and benchmark
datasets.Comment: 51 pages, 9 figure
Fast Convergence Rate of Multiple Kernel Learning with Elastic-net Regularization
We investigate the learning rate of multiple kernel leaning (MKL) with
elastic-net regularization, which consists 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 non-zero components of the ground truth is relatively
small, and prove that elastic-net MKL achieves the minimax learning rate on the
-mixed-norm ball. Our bound is sharper than the convergence rates ever
shown, and has a property that the smoother the truth is, the faster the
convergence rate is.Comment: 21 pages, 0 figur
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