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Concave function
About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.
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TL;DR: For the Borell-Brascamp-Lieb inequalities, when near equality is realized, the involved functions must be L 1-close to be p-concave and to coincide up to homotheties of their graphs as mentioned in this paper.
Abstract: We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be L1-close to be p-concave and to coincide up to homotheties of their graphs.
13 citations
TL;DR: In this article, the reverse Rogers-Shephard inequality for log-concave functions was shown to hold for convex bodies with opposite barycenters under the condition that their polar bodies have opposite locations.
Abstract: We will prove a reverse Rogers–Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of $$\ell _p$$
-diferences of convex bodies under the condition that their polar bodies have opposite barycenters.
13 citations
24 Jun 2007
TL;DR: Convexity/concavity properties of symbol error rates (SER) of the maximum likelihood detector operating in the AWGN channel (non-fading and fading) are studied and universal bounds for the SER 1st and 2nd derivatives are obtained.
Abstract: Convexity/concavity properties of symbol error rates (SER) of the maximum likelihood detector operating in the AWGN channel (non-fading and fading) are studied. Generic conditions are identified under which the SER is a convex/concave function of the SNR. Universal bounds for the SER 1st and 2nd derivatives are obtained, which hold for arbitrary constellations and are tight for some of them. Applications of the results are discussed, which include optimum power allocation in spatial multiplexing systems, optimum power/time sharing to decrease or increase (jamming problem) error rate, and implication for fading channels.
13 citations
TL;DR: From characterizations, it is obtained that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order n ≥ 3 if and only if it is 2-matrix monotones (matrix convex) and that Conjecture 4.2 proposed in Chen does not hold in general.
Abstract: We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order n ? 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order n ? 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363---385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.
13 citations
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TL;DR: In this article, the authors developed a theory of matrix completion for the extreme case of noisy 1-bit observations, where instead of observing a subset of the real-valued entries of a matrix M, they 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 is whether or not it is possible to obtain an accurate estimate of M from this data.
Abstract: 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.
13 citations