Convexity Estimates Research Articles

A new subclass of meromorphic functions with positive coefficients defining by linear operator

Abstract In this paper, we introduce and study a new class σ, (α,λ) of meromorphic univalent functions defined in E = {z : z ∊ ℂ and 0 < |z| < 1} = E \ {0}. We obtain coefficient inequalities, distortion theorems, extreme points, closure theorems, radius of convexity estimates and integral operators. Finally, we obtained neighbourhood result for the class σp (γ,λ).

On the Performance of Efficient Channel Estimation Strategies for Hybrid Millimeter Wave MIMO System.

Millimeter wave (mmWave) relying upon the multiple output multiple input (MIMO) is a new potential candidate for fulfilling the huge emerging bandwidth requirements. Due to the short wavelength and the complicated hardware architecture of mmWave MIMO systems, the conventional estimation strategies based on the individual exploitation of sparsity or low rank properties are no longer efficient and hence more modern and advance estimation strategies are required to recapture the targeted channel matrix. Therefore, in this paper, we proposed a novel channel estimation strategy based on the symmetrical version of alternating direction methods of multipliers (S-ADMM), which exploits the sparsity and low rank property of channel altogether in a symmetrical manner. In S-ADMM, at each iteration, the Lagrange multipliers are updated twice which results symmetrical handling of all of the available variables in optimization problem. To validate the proposed algorithm, numerous computer simulations have been carried out which straightforwardly depicts that the S-ADMM performed well in terms of convergence as compared to other benchmark algorithms and also able to provide global optimal solutions for the strictly convex mmWave joint channel estimation optimization problem.

Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms

Abstract In this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} in R n R^{n} or 𝑇 is an optimal transport map for the Monge problem with cost function d ⁢ ( x , y ) d(x,y) , the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} . Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others.

Realizing Efficient Security and Privacy in IoT Networks.

In recent times, security and privacy at the physical (PHY) layer has been a major issue of several communication technologies which comprise the internet of things (IoT) and mostly, the emerging fifth-generation (5G) cellular network. The most real-world PHY security challenge stems from the fact that the passive eavesdropper’s information is unavailable to the genuine source and destination (transmitter/receiver) nodes in the network. Without this information, it is difficult to optimize the broadcasting parameters. Therefore, in this research, we propose an efficient sequential convex estimation optimization (SCEO) algorithm to mitigate this challenge and improve the security of physical layer (PHY) in a three-node wireless communication network. The results of our experiments indicate that by using the SCEO algorithm, an optimal performance and enhanced convergence is achieved in the transmission. However, considering possible security challenges envisaged when a multiple eavesdropper is active in a network, we expanded our research to develop a swift privacy rate optimization algorithm for a multiple-input, multiple-output, multiple-eavesdropper (MIMOME) scenario as it is applicable to security in IoT and 5G technologies. The result of the investigation show that the algorithm executes significantly with minimal complexity when compared with nonoptimal parameters. We further employed the use of rate constraint together with self-interference of the full-duplex transmission at the receiving node, which makes the performance of our technique outstanding when compared with previous studies.

Restarting the accelerated coordinate descent method with a rough strong convexity estimate

We propose new restarting strategies for the accelerated coordinate descent method. Our main contribution is to show that for a well chosen sequence of restarting times, the restarted method has a nearly geometric rate of convergence. A major feature of the method is that it can take profit of the local quadratic error bound of the objective function without knowing the actual value of the error bound. We also show that under the more restrictive assumption that the objective function is strongly convex, any fixed restart period leads to a geometric rate of convergence. Finally, we illustrate the properties of the algorithm on a regularized logistic regression problem and on a Lasso problem.

Variance-stabilization-based compressive inversion under Poisson or Poisson–Gaussian noise with analytical bounds

Most existing bounds for signal reconstruction from compressive measurements make the assumption of additive signal-independent noise. However in many compressive imaging systems, the noise statistics are more accurately represented by Poisson or Poisson–Gaussian noise models. In this paper, we derive upper bounds for signal reconstruction error from compressive measurements which are corrupted by Poisson or Poisson–Gaussian noise. The features of our bounds are as follows: (1) the bounds are derived for a computationally tractable convex estimator with statistically motivated parameter selection. The estimator penalizes signal sparsity subject to a constraint that imposes a novel statistically motivated upper bound on a term based on variance stabilization transforms to approximate the Poisson or Poisson–Gaussian distributions by distributions with (nearly) constant variance. (2) The bounds are applicable to signals that are sparse as well as compressible in any orthonormal basis, and are derived for compressive systems obeying realistic constraints such as non-negativity and flux-preservation. Our bounds are motivated by several properties of the variance stabilization transforms that we develop and analyze. We present extensive numerical results for signal reconstruction under varying number of measurements and varying signal intensity levels. Ours is the first piece of work to derive bounds on compressive inversion for the Poisson–Gaussian noise model. We also use the properties of the variance stabilizer to develop a principle for selection of the regularization parameter in penalized estimators for Poisson and Poisson–Gaussian inverse problems.

A Restricted Bayes Approach to Joint Detection and Estimation Under Prior Uncertainty

The problem of joint detection and estimation under prior uncertainty is considered within the framework of restricted Bayes theory, which covers the classical Bayes and minimax frameworks as special cases. A linear combination of the average estimation risk with respect to some guessed prior probability and the maximum conditional estimation risk is minimized for generic convex estimation cost functions subject to a constraint on a linear combination of the average detection error probability with respect to the assumed prior and the maximum conditional detection error probability. Unknown random parameters common to different hypotheses as well as parameters unique to each hypothesis are assumed. The jointly optimal estimators and detector that minimize the resulting estimation performance metric under the detection error constraint are derived. With the proposed framework, it is possible to strike any desired balance between a fully minimax framework (where the worst case performance metrics appear in both the objective and the constraint functions of the optimization problem) and the standard Bayesian setting where exact knowledge of the prior distribution is assumed to be available. Unlike the previous studies in the literature, which deal with special cases of the problem studied in this work, the solution to the most general problem indicates that the optimal estimators are coupled with the optimal detector through a least favorable prior.

On a set of close-to-convex functions

In this paper, a class Ss(q) of close-to-convex functions is considered. Among the results studied for this class are its various characteristic properties such as the radius of convexity, certain bounds and coeffcient estimates. A suffcient condition for a function f to be in the class Ss(q), is also obtained.

A fully nonlinear flow for two-convex hypersurfaces in Riemannian manifolds

We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $$G_\kappa = \big ( \sum _{i < j} \frac{1}{\lambda _i+\lambda _j-2\kappa } \big )^{-1}$$ , where $$\lambda _1 \le \cdots \le \lambda _n$$ denote the curvature eigenvalues and $$\kappa $$ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that $$\lambda _1+\lambda _2>2\kappa $$ . In contrast to mean curvature flow, this flow preserves the condition $$\lambda _1+\lambda _2>2\kappa $$ in a general ambient manifold. Our main goal in this paper is to extend the surgery algorithm of Huisken–Sinestrari to this fully nonlinear flow. This is the first construction of this kind for a fully nonlinear flow. As a corollary, we show that a compact Riemannian manifold satisfying $$\overline{R}_{1313}+\overline{R}_{2323} \ge -2\kappa ^2$$ with non-empty boundary satisfying $$\lambda _1+\lambda _2 > 2\kappa $$ is diffeomorphic to a 1-handlebody. The main technical advance is the pointwise curvature derivative estimate. The proof of this estimate requires a new argument, as the existing techniques for mean curvature flow due to Huisken–Sinestrari, Haslhofer–Kleiner, and Brian White cannot be generalized to the fully nonlinear setting. To establish this estimate, we employ an induction-on-scales argument; this relies on a combination of several ingredients, including the almost convexity estimate, the inscribed radius estimate, as well as a regularity result for radial graphs. We expect that this technique will be useful in other situations as well.

Generalized characteristics and Lax–Oleinik operators: global theory

For autonomous Tonelli systems on $$\mathbb {R}^n$$ , we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics.

A general pinching principle for mean curvature flow and applications

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken (J Differ Geom 20(1):237–266, 1984), the convexity estimates of Huisken–Sinestrari (Acta Math 183(1):45–70, 1999) and the cylindrical estimate of Huisken–Sinestrari (Invent Math 175(1):137–221, 2009; see also Andrews and Langford in Anal PDE 7(5):1091–1107, 2014; Huisken and Sinestrari in J Differ Geom 101(2):267–287, 2015). Namely, we show that the curvature of the solution pinches onto the convex cone generated by the curvatures of any shrinking cylinder solutions admitted by the initial data. For example, if the initial data is \((m+1)\)-convex, then the curvature of the solution pinches onto the convex hull of the curvatures of the shrinking cylinders \(\mathbb {R}^m\times S^{n-m}_{\sqrt{2(n-m)(1-t)}}\), \(t<1\). In particular, this yields a sharp estimate for the largest principal curvature, which we use to obtain a new proof of a sharp estimate for the inscribed curvature for embedded solutions (Brendle in Invent Math 202(1):217–237, 2015; Haslhofer and Kleiner in Int Math Res Not 15:6558–6561, 2015; Langford in Proc Am Math Soc 143(12):5395–5398, 2015). Making use of a recent idea of Huisken–Sinestrari (2015), we then obtain a series of sharp estimates for ancient solutions. In particular, we obtain a convexity estimate for ancient solutions which allows us to strengthen recent characterizations of the shrinking sphere due to Huisken–Sinestrari (2015) and Haslhofer–Hershkovits (Commun Anal Geom 24(3):593–604, 2016).

Minimax Lower Bound and Optimal Estimation of Convex Functions in the Sup-Norm

Estimation of convex functions finds broad applications in science and engineering; however, the convex shape constraint complicates the asymptotic performance analysis of such estimators. This technical note is devoted to the minimax optimal estimation of univariate convex functions in a given Holder class. Particularly, a minimax lower bound in the supremum norm (or simply sup-norm) is established by constructing a novel family of piecewise quadratic convex functions in the Holder class. This result, along with a recent result on the minimax upper bound, gives rise to the optimal rate of convergence for the minimax sup-norm risk of convex functions with the Holder order between one and two. The present technical note provides the first rigorous justification of the optimal minimax risk for convex estimation on the entire interval of interest in the sup-norm.

Mean Curvature Flow of Mean Convex Hypersurfaces

In the last 15 years, White and Huisken‐Sinestrari developed a far‐reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high‐curvature regions in a mean convex flow.In the present paper, we give a new treatment of the theory of mean convex (and k‐convex) flows. This includes: (1) an estimate for derivatives of curvatures, (2) a convexity estimate, (3) a cylindrical estimate, (4) a global convergence theorem, (5) a structure theorem for ancient solutions, and (6) a partial regularity theorem.Our new proofs are both more elementary and substantially shorter than the original arguments. Our estimates are local and universal. A key ingredient in our new approach is the new noncollapsing result of Andrews [2]. Some parts are also inspired by the work of Perelman [32,33]. In a forthcoming paper [17], we will give a new construction of mean curvature flow with surgery based on the methods established in the present paper.Note added in May 2015. Since the first version of this paper was posted on arxiv in April 2013, the estimates have been used to construct mean convex flow with surgery in ℝ3 by Brendle and Huisken [5] in September 2013 and in another paper by the authors in April 2014.© 2016 Wiley Periodicals, Inc.

Convexity estimates for mean curvature flow with free boundary

We prove the estimates of [3] and [2] for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier S. Here S can be any embedded, oriented surface in Rn+1 of bounded geometry and positive inscribed radius. We also prove the estimate [4] in the case of convex flows and S=Sn, which gives an alternative proof to [6].

Sharp MSE Bounds for Proximal Denoising

Denoising has to do with estimating a signal $$\mathbf {x}_0$$x0 from its noisy observations $$\mathbf {y}=\mathbf {x}_0+\mathbf {z}$$y=x0+z. In this paper, we focus on the structured denoising where the signal $$\mathbf {x}_0$$x0 possesses a certain structure and $$\mathbf {z}$$z has independent normally distributed entries with mean zero and variance $$\sigma ^2$$ź2. We employ a structure-inducing convex function $$f(\cdot )$$f(·) and solve $$\min _\mathbf {x}\{\frac{1}{2}\Vert \mathbf {y}-\mathbf {x}\Vert _2^2+\sigma {\lambda }f(\mathbf {x})\}$$minx{12źy-xź22+źźf(x)} to estimate $$\mathbf {x}_0$$x0, for some $$\lambda >0$$ź>0. Common choices for $$f(\cdot )$$f(·) include the $$\ell _1$$l1 norm for sparse vectors, the $$\ell _1-\ell _2$$l1-l2 norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate $$\mathbf {x}^*$$xź is the normalized mean-squared error $$\text {NMSE}(\sigma )=\frac{{\mathbb {E}}\Vert \mathbf {x}^*-\mathbf {x}_0\Vert _2^2}{\sigma ^2}$$NMSE(ź)=Eźxź-x0ź22ź2. We show that NMSE is maximized as $$\sigma \rightarrow 0$$źź0 and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the $${\lambda }$$ź-scaled subdifferential $${\lambda }\partial f(\mathbf {x}_0)$$źźf(x0). When $${\lambda }$$ź is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem $$\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-\mathbf {x}\Vert _2\}$$minf(x)≤f(x0){źy-xź2}. The paper also connects these results to the generalized LASSO problem, in which one solves $$\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-{\mathbf {A}}\mathbf {x}\Vert _2\}$$minf(x)≤f(x0){źy-Axź2} to estimate $$\mathbf {x}_0$$x0 from noisy linear observations $$\mathbf {y}={\mathbf {A}}\mathbf {x}_0+\mathbf {z}$$y=Ax0+z. We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a phase transition as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.

MOTION OF HYPERSURFACES BY CURVATURE

We study the behaviour of (non-convex) solutions of a large class of fully non-linear curvature flows; specifically, we consider the evolution of closed, immersed hypersurfaces of Euclidean spaces whose pointwise normal speed is prescribed by a monotone function of their curvature which is homogeneous of degree one. It is well-known that solutions of such flows necessarily suffer finite time singularities. On the other hand, under various natural conditions, singularities are characterised by a curvature blow-up. Our first main area of study concerns the asymptotic behaviour of the curvature at a singularity. We first prove a quantitative convexity estimate for positive solutions (that is, solutions moving with inward normal speed everywhere positive) under one of the following additional assumptions: either the evolving hypersurfaces are of dimension two, or the flow speed is a convex function of the curvature. Roughly speaking, the convexity estimate states that, for positive solutions, the normalised Weingarten curvature operator is asymptotically non-negative at a singularity. We then prove a family of cylindrical estimates for flows by convex speed functions. Roughly speaking, these estimates state that, for (m+ 1)-positive solutions (that is, solutions with (m+ 1)positive Weingarten curvature), the Weingarten curvature is asymptotically m-cylindrical at a singularity unless it becomes m-positive. The convexity and cylindrical estimates yield a detailed description of the possible singularities which may form under surface flows and flows by convex speeds. Moreover, they are uniform across the class of solutions with given dimension, flow speed, and initial volume, diameter and curvature hull, which should make them useful for applications such as the development of flows with surgeries. Our second main area of study concerns the development, in the fully non-linear setting, of the recently discovered non-collapsing phenomena for the mean curvature flow; namely, we prove that embedded solutions of flows by concave speeds are interior non-collapsing, whilst embedded solutions of flows by convex or inverse-concave speeds are exterior noncollapsing. The non-collapsing results complement the above curvature estimates by ruling out certain types of asymptotic behaviour which the curvature estimates do not. (This is mainly due to the non-local nature of the non-collapsing estimates.) As a particular application, we show how non-collapsing gives rise to a particularly efficient proof of the Andrews–Huisken theorem on the convergence of convex hypersurfaces to round points under such flows.

Convexity and cylindrical estimates for mean curvature flow in the sphere

We study mean curvature flow in the sphere with the quadratic curvature condition | A | 2 ≤ 1 n − 2 H 2 + 4 K |A|^{2} \leq \frac { 1}{n-2} H^{2} + 4 K which is related to but different from two-convexity for submanifolds of the sphere. We classify type I I singularities with no further hypotheses. If H &gt; 0 H&gt; 0 , then we apply the Huisken-Sinestrari convexity estimates to this situation and show that we can classify type I I II singularities. This shows that at a singularity the surface is asymptotically convex. We then prove cylindrical estimates for the mean curvature flow and a pointwise gradient estimate which shows that near a singularity the surface is quantitatively convex.

Estimates of the attraction domain of linear systems under L 2-bounded control

Using the linear matrix inequality technique, in this paper we construct a convex estimate of the attraction domain for a linear system with L 2-bounded control in the form of linear static state feedback. A robust statement of the problem is also studied, when the control system contains structured matrix uncertainty. The results of numerical simulations are presented.

Convexity estimates for surfaces moving by curvature functions

We consider the evolution of compact surfaces by fully nonlinear, parabolic curvature flows for which the normal speed is given by a smooth, degree one homogeneous function of the principal curvatures of the evolving surface. Under no further restrictions on the speed function, we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow. This generalises the corresponding result of Huisken and Sinestrari for the mean curvature flow to the largest possible class of degree one homogeneous surface flows.

Caméras virtuelles pour l’étalonnage d’un système de réalité augmentée sur affichage semi-transparent

We present a novel extrinsic calibration method for optical see-through systems. It is primarily aimed at tablet-like systems with a semi-transparent screen, a user-tracking device and a device dedicated to the localization of the system in the environment but easily generalizable to any optical see-through setup. The proposed algorithm estimates the relative poses of those three component based on the user indicating the projections onto the screen of several reference points chosen on a known object. A convex estimation is computed through the resectioning of virtual cameras and used to initialize a global bundle adjustment. Both synthetic and real experiments show the viability of our approach.