19,058 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
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