Minimax Estimation Research Articles

Visual object tracking with multi-scale superpixels and color-feature guided kernelized correlation filters

While discriminative correlation filter based tracking algorithms have achieved competitive results and demonstrated successfully, there still remain some challenges, such as handling the scenarios of scale variation, fast motion, etc. A region of interest with a fixed size, which is usually used by discriminative correlation filter based tracking algorithms to train correlation filter and track object, makes the trackers hard to deal with the challenges of fast motion and scale variation. It also restricts the use of object structure information. In this paper, we propose a Multi-Scale Superpixels and Color Feature Guided Kernelized Correlation Filters (MSSCF-KCF) to deal with the problems mentioned above. Firstly, we treat tracking procedure as optimizing the combination of components of an object, and propose a multi-scale superpixel method to segment object image based on a proposed global confidence mask which determines the center and size of object patches automatically. Then, KCF is embedded into Bayesian filter framework, in which the proposed color guided confidence map is viewed as an observation model, to expand the tracking range and improve the performance when handling fast motion. Finally, we propose a center distance matrix of each object patch to use structure information of object. A novel min–max criterion as well as drop-out strategy is used to search the optimal combination and estimate the state of object, which also helps the tracker to cope with scale variation. Besides, monitoring update strategy is proposed to monitor tracking operation and adjust parameters dynamically. The proposed algorithm is tested on a self-made dataset and some challenging sequences with a novel evaluation criterion. The results demonstrate the feasibility and effectiveness compared with the state-of-the-art methods.

Distributed multiple‐input–multiple‐output radar waveform and mismatched filter design with expanded mainlobe

Distributed multiple-input–multiple-output radar usually needs to transmit nearly orthogonal waveforms, but the constant-modulus requirement from engineering applications often makes sidelobe levels difficult to suppress. In this study, the authors resort to two approaches to further suppress the sidelobe levels, i.e. orthogonal phase-coded waveforms with expanded mainlobe (OPCWEM) and its mismatched filters design. Specifically, they first propose a minimisation criterion to design nearly OPCWEM, which is a non-convex optimisation problem and solved by a least-pth minimax algorithm. Next, they design mismatched filters with given OPCWEM, which is formulated as a convex optimisation problem with min–max criterion and solved by a convex optimisation algorithm. Numerical results validate the effectiveness of the two approaches and the Doppler sensitivity is also analysed.

Multichannel High Resolution Wide Swath SAR Imaging for Hypersonic Air Vehicle with Curved Trajectory.

Synthetic aperture radar (SAR) equipped on the hypersonic air vehicle in near space has many advantages over the conventional airborne SAR. However, its high-speed maneuvering characteristics with curved trajectory result in serious range migration, and exacerbate the contradiction between the high resolution and wide swath. To solve this problem, this paper establishes the imaging geometrical model matched with the flight trajectory of the hypersonic platform and the multichannel azimuth sampling model based on the displaced phase center antenna (DPCA) technology. Furthermore, based on the multichannel signal reconstruction theory, a more efficient spectrum reconstruction model using discrete Fourier transform is proposed to obtain the azimuth uniform sampling data. Due to the high complexity of the slant range model, it is difficult to deduce the processing algorithm for SAR imaging. Thus, an approximate range model is derived based on the minimax criterion, and the optimal second-order approximate coefficients of cosine function are obtained using the two-population coevolutionary algorithm. On this basis, aiming at the problem that the traditional Omega-K algorithm cannot compensate the residual phase with the difficulty of Stolt mapping along the range frequency axis, this paper proposes an Exact Transfer Function (ETF) algorithm for SAR imaging, and presents a method of range division to achieve wide swath imaging. Simulation results verify the effectiveness of the ETF imaging algorithm.

LARGE-SCALE MIMO MC-CDMA SYSTEM USING COMBINED MULTIPLE BEAMFORMING AND SPATIAL MULTIPLEXING

This paper proposes a novel Large-scale (massive) Multi-input Multi-output Multi-carrier Code division multiple access (LS MIMO MC-CDMA) model and application to Fifth-Generation Mobile Communication Systems (5G). This system uses combined cylindrical array antenna multiple beamforming architecture with spatial multiplexing. The model is optimized by Min-Max criteria in order to minimize side lobes and maximize compression of propagation loss. The Monte Carlo simulation results unify with the analytical solution for system performance.

Two‐stage minimax stochastic unit commitment

The study proposes a stochastic optimisation approach based on discrete scenarios and the minimax criterion to deal with demand uncertainty in the two-stage unit commitment problem. To avoid making an over-conservative decision, the approach is designed to assign a probability-based weight to each demand scenario and provide a unit commitment schedule that minimises the maximum possible-weighted generation cost. Unit commitment is determined in the first stage and the economic dispatch under the scenario that corresponds to the maximum possible-weighted generation cost is determined in the second stage. The problem is formulated as a type of min–max–min mixed integer programming model. By introducing an auxiliary variable, the model is further transformed into a minimisation problem. A Benders decomposition algorithm is developed to solve the problem. The Benders master problem determines the unit commitment decision, while the Benders subproblem determines the dispatch decision. Multiple Benders feasibility cuts are constructed when the Benders subproblem is infeasible. Valid inequalities are derived to improve the lower bound provided by the Benders master problem. Numerical results show the performance of the algorithm.

Secrecy Analysis of Generalized Space-Shift Keying Aided Visible Light Communication

This paper investigates the physical layer security problem of visible light communication (VLC) systems relying on generalized space-shift keying (GSSK) termed as GSSK-VLC. The GSSK-VLC system considered is assumed to be comprised of three nodes: a transmitter equipped with multiple light-emitting diodes, a legitimate receiver, and a passive eavesdropper. Each of them is equipped with a single photo-detector. Specifically, the average mutual information (AMI) of a GSSK-VLC system is derived. We also obtain both a lower bound and an accurate closed-form expression of the approximate AMI, which can be employed for efficiently estimating the achievable secrecy rate of GSSK-VLC systems. Furthermore, the pairwise error probability and bit error rate of GSSK-VLC systems are analyzed, and again some closed-form expressions are obtained. Additionally, in order to enhance the secrecy performance of the GSSK-VLC system, an optimal LED pattern selection algorithm is proposed under the minimax criterion. We show that the proposed LED pattern selection algorithm is capable of enhancing both the AMI between the transmitter and legitimate user and the achievable secrecy rate of the GSSK-VLC system.

Модифицированный алгоритм построения областей достижимости линейных дискретных динамических систем

The paper considered the modified general recursion algebraic method for the linear discrete-time dynamic systems reachable sets computation. The algorithms of application programs for reachable sets constructing is described. The comparative analysis of original (Shorikov A.F. [Minimax Estimation and Control in Discrete-Time Dynamic Systems]. Yekaterinburg. Ural State University Publ., 1997. 242 p. (in Russ.)) and suggest algorithms is provided by giving of numerical example.

Robust Balanced Optimization

An instance of a balanced optimization problem with vector costs consists of a ground set X, a cost vector for every element of X, and a system of feasible subsets over X. The goal is to find a feasible subset that minimizes the so-called imbalance of values in every coordinate of the underlying vector costs. Balanced optimization problems with vector costs are equivalent to the robust optimization version of balanced optimization problems under the min-max criterion. We regard these problems as a family of optimization problems; one particular member of this family is the known balanced assignment problem. We investigate the complexity and approximability of robust balanced optimization problems in a fairly general setting. We identify a large family of problems that admit a 2-approximation in polynomial time, and we show that for many problems in this family this approximation factor 2 is best-possible (unless P=NP). We pay special attention to the balanced assignment problem with vector costs and show that this problem is NP-hard even in the highly restricted case of sum costs. We conclude by performing computational experiments for this problem.

Исследование корректирующей способности синхрокодов для модели декодера с согласованной обработкой

Increase of the operation reliability of the frame synchronization schemes of communication systems affected by interferences might be achieved by using sync sequences optimal according to the minimax criterion and having the best ratio of the maximum side-lobe level to the main peak of the non-periodic autocorrelation function. Obviously, the longer the minimax sequence, the higher its correction ability. However, with the increase in the length of the sync code, a conflict arises between the corrective ability and the speed of data transfer. This is caused by the decrease in the duration of the information section of the frame. At the same time, as the length of the code increases, the complexity of the technical implementation of the decoder synchronization scheme increases as well. This means that when developing an information transmission system, it is necessary to find a compromise between the complexity of the technical implementation of the decoder and the required noise immunity. To solve this problem, it is necessary to study the correcting ability of binary sync codes. The purpose of this work was to study the correcting ability of binary sequences with good non-periodic auto-correlation properties using a reception scheme with matched processing. This article discusses the possibility of using Barker sequences of lengths 7, 11, and 13 and M-sequences as corrective sync codes. The authors carry out an analysis of the correcting ability of the decoder model with matched processing for Barker codes of different lengths under the conditions of a one, two, and threefold error. Analytical expressions have been obtained to determine the decoding error probability of a binary minimax code depending on the length of the sync code. The dependence of the corrective ability of the code and the maximum level of the side lobe on the length of the sync code in the range from 4 to 128 with the best nonperiodic autocorrelation functions (NPAF) is obtained. It has been established that there are length ranges for minimax sequences within which their correction ability remains constant, that is, a sync code of shorter length has the same correction ability as a longer code. Thus, the research results allow us to optimize the complexity of the technical implementation of a sync signal decoder depending on the requirements for general noise immunity and information transfer rate in a communication system. The proposed approach can be successfully used to modify the existing synchronization schemes of communication systems with code division of channels.

Project risk management in the construction of high-rise buildings

This paper shows the project risk management methods, which allow to better identify risks in the construction of high-rise buildings and to manage them throughout the life cycle of the project. One of the project risk management processes is a quantitative analysis of risks. The quantitative analysis usually includes the assessment of the potential impact of project risks and their probabilities. This paper shows the most popular methods of risk probability assessment and tries to indicate the advantages of the robust approach over the traditional methods. Within the framework of the project risk management model a robust approach of P. Huber is applied and expanded for the tasks of regression analysis of project data. The suggested algorithms used to assess the parameters in statistical models allow to obtain reliable estimates. A review of the theoretical problems of the development of robust models built on the methodology of the minimax estimates was done and the algorithm for the situation of asymmetric “contamination” was developed.

Resource Optimization of Wireless Information and Energy Supply Control Systems With Massive MIMO

We study a wireless information and energy supply control system with frequency-division operation. In the considered system, a control station (CS) with a massive antenna array transmits energy and control information to multiple control terminals (CTs), and the CTs simultaneously transmit sensing information to the CS using harvested energy. The objective of this letter is to minimize the maximum control cost among all the CTs through resource allocation, while satisfying the transmission rate requirements of both the sensing and control information. The transmission rates are first derived, and then, a relationship between control cost and power allocation is established. Finally, we prove that all the CTs should have the same control cost to meet the min–max criterion, based on which a resource allocation scheme is proposed for the min–max problem. Numerical results indicate that the proposed resource allocation scheme is able to significantly reduce the system control cost.

Asymptotic Optimality of One-Group Shrinkage Priors in Sparse High-dimensional Problems

We study asymptotic optimality of inference in a high-dimensional sparse normal means model using a broad class of one-group shrinkage priors. Assuming that the proportion of non-zero means is known, we show that the corresponding Bayes estimates asymptotically attain the minimax risk (up to a multiplicative constant) for estimation with squared error loss. The constant is shown to be 1 for the important sub-class of “horseshoe-type” priors proving exact asymptotic minimaxity property for these priors, a result hitherto unknown in the literature. An empirical Bayes version of the estimator is shown to achieve the minimax rate in case the level of sparsity is unknown. We prove that the resulting posterior distributions contract around the true mean vector at the minimax optimal rate and provide important insight about the possible rate of posterior contraction around the corresponding Bayes estimator. Our work shows that for rate optimality, a heavy tailed prior with sufficient mass around zero is enough, a pole at zero like the horseshoe prior is not necessary. This part of the work is inspired by Pas et al. (2014). We come up with novel unifying arguments to extend their results over the general class of priors. Next we focus on simultaneous hypothesis testing for the means under the additive 0−1 loss where the means are modeled through a two-groups mixture distribution. We study asymptotic risk properties of certain multiple testing procedures induced by the class of one-group priors under study, when applied in this set-up. Our key results show that the tests based on the “horseshoe-type” priors asymptotically achieve the risk of the optimal solution in this two-groups framework up to the correct constant and are thus asymptotically Bayes optimal under sparsity (ABOS). This is the first result showing that in a sparse problem a class of one-group priors can exactly mimic the performance of an optimal two-groups solution asymptotically. Our work shows an intrinsic technical connection between the theories of minimax estimation and simultaneous hypothesis testing for such one-group priors.

Accuracy assessment for high-dimensional linear regression

This paper considers point and interval estimation of the $\ell_q$ loss of an estimator in high-dimensional linear regression with random design. We establish the minimax rate for estimating the $\ell_{q}$ loss and the minimax expected length of confidence intervals for the $\ell_{q}$ loss of rate-optimal estimators of the regression vector, including commonly used estimators such as Lasso, scaled Lasso, square-root Lasso and Dantzig Selector. Adaptivity of the confidence intervals for the $\ell_{q}$ loss is also studied. Both the setting of known identity design covariance matrix and known noise level and the setting of unknown design covariance matrix and unknown noise level are studied. The results reveal interesting and significant differences between estimating the $\ell_2$ loss and $\ell_q$ loss with $1\le q <2$ as well as between the two settings. New technical tools are developed to establish rate sharp lower bounds for the minimax estimation error and the expected length of minimax and adaptive confidence intervals for the $\ell_q$ loss. A significant difference between loss estimation and the traditional parameter estimation is that for loss estimation the constraint is on the performance of the estimator of the regression vector, but the lower bounds are on the difficulty of estimating its $\ell_q$ loss. The technical tools developed in this paper can also be of independent interest.

A Practical, Robust Methodology for Acquiring New Observation Data Using Computationally Expensive Groundwater Models

AbstractRegional groundwater flow models play an important role in decision making regarding water resources; however, the uncertainty embedded in model parameters and model assumptions can significantly hinder the reliability of model predictions. One way to reduce this uncertainty is to collect new observation data from the field. However, determining where and when to obtain such data is not straightforward. There exist a number of data‐worth and experimental design strategies developed for this purpose. However, these studies often ignore issues related to real‐world groundwater models such as computational expense, existing observation data, high‐parameter dimension, etc. In this study, we propose a methodology, based on existing methods and software, to efficiently conduct such analyses for large‐scale, complex regional groundwater flow systems for which there is a wealth of available observation data. The method utilizes the well‐established d‐optimality criterion, and the minimax criterion for robust sampling strategies. The so‐called Null‐Space Monte Carlo method is used to reduce the computational burden associated with uncertainty quantification. And, a heuristic methodology, based on the concept of the greedy algorithm, is proposed for developing robust designs with subsets of the posterior parameter samples. The proposed methodology is tested on a synthetic regional groundwater model, and subsequently applied to an existing, complex, regional groundwater system in the Perth region of Western Australia. The results indicate that robust designs can be obtained efficiently, within reasonable computational resources, for making regional decisions regarding groundwater level sampling.

NOMA-Based Multi-Pair Two-Way Relay Networks With Rate Splitting and Group Decoding

In this paper, we develop a non-orthogonal multiple access (NOMA)-based multi-pair two-way relay (TWR) network, in which a rate splitting scheme and a successive group decoding strategy are employed. By exploiting the interference signals received from neighbor users with the leverage of the full-duplex technique, we enhance the decoding ability of each user and further achieve an effective multiuser interference management for the network. We propose different decoding strategies for different types of nodes by processing the received signals with only local incoming channel state information in different manners. Moreover, under the limited group decoding size, each individual node decodes its own desired messages along with a fraction of the interference successively. We further investigate the joint uplink and downlink fair rate allocation problem based on the max–min criterion, and the solution to which also contains the optimal group decoding schedule. Simulation results in terms of ergodic rate and outrage probability corroborate the superiority of our NOMA-based multi-pair TWR network over the OMA-based counterpart.

Sparse CCA: Adaptive estimation and computational barriers

Canonical correlation analysis is a classical technique for exploring the relationship between two sets of variables. It has important applications in analyzing high dimensional datasets originated from genomics, imaging and other fields. This paper considers adaptive minimax and computationally tractable estimation of leading sparse canonical coefficient vectors in high dimensions. First, we establish separate minimax estimation rates for canonical coefficient vectors of each set of random variables under no structural assumption on marginal covariance matrices. Second, we propose a computationally feasible estimator to attain the optimal rates adaptively under an additional sample size condition. Finally, we show that a sample size condition of this kind is needed for any randomized polynomial-time estimator to be consistent, assuming hardness of certain instances of the Planted Clique detection problem. The result is faithful to the Gaussian models used in the paper. As a byproduct, we obtain the first computational lower bounds for sparse PCA under the Gaussian single spiked covariance model.

Hazard rate estimation for left truncated and right censored data

Left truncation and right censoring (LTRC) presents a unique challenge for nonparametric estimation of the hazard rate of a continuous lifetime because consistent estimation over the support of the lifetime is impossible. To understand the problem and make practical recommendations, the paper explores how the LTRC affects a minimal (called sharp) constant of a minimax MISE convergence over a fixed interval. The corresponding theory of sharp minimax estimation of the hazard rate is presented, and it shows how right censoring, left truncation and interval of estimation affect the MISE. Obtained results are also new for classical cases of censoring or truncation and some even for the case of direct observations of the lifetime of interest. The theory allows us to propose a relatively simple data-driven estimator for small samples as well as the methodology of choosing an interval of estimation. The estimation methodology is tested numerically and on real data.

Utilitarian preference for redistribution: a concern with max-min

This paper studies an optimal linear tax or basic income regime with an extensive margin in the supply of labor and a utililitarian welfare criterion. The presence the extensive margin in labor supply breaks the connection between allocational efficiency and aggregate utility maximization. In a general framework, some redistribution maximizes aggregate utility, but a lesser degree than what Rawls’ max-min criterion would require. Rawls’ concept of the Original Position is useful, but the rationality of the max-min criterion is questionable.

Minimax signal detection under weak noise assumptions

We consider minimax signal detection in the sequence model. Working with certain ellipsoids in the space of square-summable sequences of real numbers, with a ball of positive radius removed, we obtain upper and lower bounds for the minimax separation radius in the non-asymptotic framework, i.e., for a fixed value of the involved noise level. We use very weak assumptions on the noise (i.e., fourth moments are assumed to be uniformly bounded). In particular, we do not use any kind of Gaussian distribution or independence assumption on the noise. It is shown that the established minimax separation rates are not faster than the ones obtained in the classical sequence model (i.e., independent standard Gaussian noise) but, surprisingly, are of the same order as the minimax estimation rates in the classical setting. Under an additional condition on the noise, the classical minimax separation rates are also retrieved in benchmark well-posed and ill-posed inverse problems.

Minimax theory of estimation of linear functionals of the deconvolution density with or without sparsity

The present paper considers the problem of estimating a linear functional $\Phi=\int_{-\infty}^{\infty}\varphi(x)f(x)\,dx$ of an unknown deconvolution density $f$ on the basis of $n$ i.i.d. observations, $Y_{1},\ldots,Y_{n}$ of $Y=\theta+\xi$, where $\xi$ has a known pdf $g$, and $f$ is the pdf of $\theta$. The objective of the present paper is to develop the a general minimax theory of estimating $\Phi$, and to relate this problem to estimation of functionals $\Phi_{n}=n^{-1}\sum_{i=1}^{n}\varphi(\theta_{i})$ in indirect observations. In particular, we offer a general, Fourier transform based approach to estimation of $\Phi$ (and $\Phi_{n}$) and derive upper and minimax lower bounds for the risk for an arbitrary square integrable function $\varphi$. Furthermore, using technique of inversion formulas, we extend the theory to a number of situations when the Fourier transform of $\varphi$ does not exist, but $\Phi$ can be presented as a functional of the Fourier transform of $f$ and its derivatives. The latter enables us to construct minimax estimators of the functionals that have never been handled before such as the odd absolute moments and the generalized moments of the deconvolution density. Finally, we generalize our results to the situation when the vector $\mathbf{{\theta}}$ is sparse and the objective is estimating $\Phi$ (or $\Phi_{n}$) over the nonzero components only. As a direct application of the proposed theory, we automatically recover multiple recent results and obtain a variety of new ones such as, for example, estimation of the mixing probability density function with classical and Berkson errors and estimation of the $(2M+1)$-th absolute moment of the deconvolution density.