40,838 research outputs found
Estimation in high dimensions: a geometric perspective
This tutorial provides an exposition of a flexible geometric framework for
high dimensional estimation problems with constraints. The tutorial develops
geometric intuition about high dimensional sets, justifies it with some results
of asymptotic convex geometry, and demonstrates connections between geometric
results and estimation problems. The theory is illustrated with applications to
sparse recovery, matrix completion, quantization, linear and logistic
regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change
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
1-Bit Matrix Completion
In this paper we develop a theory of matrix completion for the extreme case
of noisy 1-bit observations. Instead of observing a subset of the real-valued
entries of a matrix M, we obtain a small number of binary (1-bit) measurements
generated according to a probability distribution determined by the real-valued
entries of M. The central question we ask is whether or not it is possible to
obtain an accurate estimate of M from this data. In general this would seem
impossible, but we show that the maximum likelihood estimate under a suitable
constraint returns an accurate estimate of M when ||M||_{\infty} <= \alpha, and
rank(M) <= r. If the log-likelihood is a concave function (e.g., the logistic
or probit observation models), then we can obtain this maximum likelihood
estimate by optimizing a convex program. In addition, we also show that if
instead of recovering M we simply wish to obtain an estimate of the
distribution generating the 1-bit measurements, then we can eliminate the
requirement that ||M||_{\infty} <= \alpha. For both cases, we provide lower
bounds showing that these estimates are near-optimal. We conclude with a suite
of experiments that both verify the implications of our theorems as well as
illustrate some of the practical applications of 1-bit matrix completion. In
particular, we compare our program to standard matrix completion methods on
movie rating data in which users submit ratings from 1 to 5. In order to use
our program, we quantize this data to a single bit, but we allow the standard
matrix completion program to have access to the original ratings (from 1 to 5).
Surprisingly, the approach based on binary data performs significantly better
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