14,738 research outputs found
Matrix Completion via Max-Norm Constrained Optimization
Matrix completion has been well studied under the uniform sampling model and
the trace-norm regularized methods perform well both theoretically and
numerically in such a setting. However, the uniform sampling model is
unrealistic for a range of applications and the standard trace-norm relaxation
can behave very poorly when the underlying sampling scheme is non-uniform.
In this paper we propose and analyze a max-norm constrained empirical risk
minimization method for noisy matrix completion under a general sampling model.
The optimal rate of convergence is established under the Frobenius norm loss in
the context of approximately low-rank matrix reconstruction. It is shown that
the max-norm constrained method is minimax rate-optimal and yields a unified
and robust approximate recovery guarantee, with respect to the sampling
distributions. The computational effectiveness of this method is also
discussed, based on first-order algorithms for solving convex optimizations
involving max-norm regularization.Comment: 33 page
A Max-Norm Constrained Minimization Approach to 1-Bit Matrix Completion
We consider in this paper the problem of noisy 1-bit matrix completion under
a general non-uniform sampling distribution using the max-norm as a convex
relaxation for the rank. A max-norm constrained maximum likelihood estimate is
introduced and studied. The rate of convergence for the estimate is obtained.
Information-theoretical methods are used to establish a minimax lower bound
under the general sampling model. The minimax upper and lower bounds together
yield the optimal rate of convergence for the Frobenius norm loss.
Computational algorithms and numerical performance are also discussed.Comment: 33 pages, 3 figure
Low-Rank Inducing Norms with Optimality Interpretations
Optimization problems with rank constraints appear in many diverse fields
such as control, machine learning and image analysis. Since the rank constraint
is non-convex, these problems are often approximately solved via convex
relaxations. Nuclear norm regularization is the prevailing convexifying
technique for dealing with these types of problem. This paper introduces a
family of low-rank inducing norms and regularizers which includes the nuclear
norm as a special case. A posteriori guarantees on solving an underlying rank
constrained optimization problem with these convex relaxations are provided. We
evaluate the performance of the low-rank inducing norms on three matrix
completion problems. In all examples, the nuclear norm heuristic is
outperformed by convex relaxations based on other low-rank inducing norms. For
two of the problems there exist low-rank inducing norms that succeed in
recovering the partially unknown matrix, while the nuclear norm fails. These
low-rank inducing norms are shown to be representable as semi-definite
programs. Moreover, these norms have cheaply computable proximal mappings,
which makes it possible to also solve problems of large size using first-order
methods
1-Bit Matrix Completion under Exact Low-Rank Constraint
We consider the problem of noisy 1-bit matrix completion under an exact rank
constraint on the true underlying matrix . Instead of observing a subset
of the noisy continuous-valued entries of a matrix , we observe a subset
of noisy 1-bit (or binary) measurements generated according to a probabilistic
model. We consider constrained maximum likelihood estimation of , under a
constraint on the entry-wise infinity-norm of and an exact rank
constraint. This is in contrast to previous work which has used convex
relaxations for the rank. We provide an upper bound on the matrix estimation
error under this model. Compared to the existing results, our bound has faster
convergence rate with matrix dimensions when the fraction of revealed 1-bit
observations is fixed, independent of the matrix dimensions. We also propose an
iterative algorithm for solving our nonconvex optimization with a certificate
of global optimality of the limiting point. This algorithm is based on low rank
factorization of . We validate the method on synthetic and real data with
improved performance over existing methods.Comment: 6 pages, 3 figures, to appear in CISS 201
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