Optimal recovery of multivariate contractions

Because of the irregular sampling pattern of raw altimeter data, many oceanographic applications rely on information from sea surface height (SSH) products gridded on regular grids where gaps have been filled with interpolation. Today, the operational SSH products are created using the simple, but robust, optimal interpolation (OI) method. If well tuned, the OI becomes computationally cheap and provides accurate results at low resolution. However, OI is not adapted to produce high-resolution and high-frequency maps of SSH. To improve the interpolation of SSH satellite observations, a data-driven approach (i.e., constructing a dynamical forecast model from the data) was recently proposed: analog data assimilation (AnDA). AnDA adaptively chooses analog situations from a catalog of SSH scenes—originating from numerical simulations or a large database of observations—which allow the temporal propagation of physical features at different scales, while each observation is assimilated. In this article, we review the AnDA and OI algorithms and compare their skills in numerical experiments. The experiments are observing system simulation experiments (OSSE) on the Lorenz-63 system and on an SSH reconstruction problem in the Gulf of Mexico. The results show that AnDA, with no necessary tuning, produces comparable reconstructions as does OI with tuned parameters. Moreover, AnDA manages to reconstruct the signals at higher frequencies than OI. Finally, an important additional feature for any interpolation method is to be able to assess the quality of its reconstruction. This study shows that the standard deviation estimated by AnDA is flow dependent, hence more informative on the reconstruction quality, than the one estimated by OI.

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

Interpolation is responsible for digital signal resampling and can significantly degrade the original signal quality if not done properly. For many years, optimal interpolation algorithms were sought within constrained classes of interpolation kernel functions. We derive a new family of unconstrained, finitely supported L 2 -optimal interpolation kernels H L (x), and compare their properties to the previously known results. Our research demonstrates that L 2-optimal kernels provide superior interpolation quality, and can be efficiently applied to any digital signal, of arbitrary nature, bandwidth, and dimensionality.

In this paper, we consider two distance-based relaxed variants of the maximum clique problem (Max Clique), named Maxd-Clique and Maxd-Club for positive integers d. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of $$n^{1-\varepsilon }$$ for any real $$\varepsilon > 0$$ unless $${{{\mathcal {P}}}} = {{\mathcal {NP}}}$$ , since they are identical to Max Clique (Hastad in Acta Math 182(1):105–142, 1999; Zuckerman in Theory Comput 3:103–128, 2007). In addition, it is $${{\mathcal {NP}}}$$ -hard to approximate Maxd-Clique and Maxd-Club to within a factor of $$n^{1/2 - \varepsilon }$$ for any fixed integer $$d\ge 2$$ and any real $$\varepsilon > 0$$ (Asahiro et al. in Approximating maximum diameter-bounded subgraphs. In: Proc of LATIN 2010, Springer, pp 615–626, 2010; Asahiro et al. in Optimal approximation algorithms for maximum distance-bounded subgraph problems. In: Proc of COCOA, Springer, pp 586–600, 2015). As for approximability of Maxd-Clique and Maxd-Club, a polynomial-time algorithm, called ReFindStar $$_d$$ , that achieves an optimal approximation ratio of $$O(n^{1/2})$$ for Maxd-Clique and Maxd-Club was designed for any integer $$d\ge 2$$ in Asahiro et al. (2015, Algorithmica 80(6):1834–1856, 2018). Moreover, a simpler algorithm, called ByFindStar $$_d$$ , was proposed and it was shown in Asahiro et al. (2010, 2018) that although the approximation ratio of ByFindStar $$_d$$ is much worse for any odd $$d\ge 3$$ , its time complexity is better than ReFindStar $$_d$$ . In this paper, we implement those approximation algorithms and evaluate their quality empirically for random graphs. The experimental results show that (1) ReFindStar $$_d$$ can find larger d-clubs (d-cliques) than ByFindStar $$_d$$ for odd d, (2) the size of d-clubs (d-cliques) output by ByFindStar $$_d$$ is the same as ones by ReFindStar $$_d$$ for even d, and (3) ByFindStar $$_d$$ can find the same size of d-clubs (d-cliques) much faster than ReFindStar $$_d$$ . Furthermore, we propose and implement two new heuristics, Hclub $$_d$$ for Maxd-Club and Hclique $$_d$$ for Maxd-Clique. Then, we present the experimental evaluation of the solution size of ReFindStar $$_d$$ , Hclub $$_d$$ , Hclique $$_d$$ and previously known heuristic algorithms for random graphs and Erdős collaboration graphs.

Optimal router node placement (RNP) is an effective method for improving the performance of wireless mesh networks (WMN). However, solving the RNP problem in WMN is difficult because it is NP-hard. As a result, this problem can only be solved using approximate optimization algorithms such as heuristics and meta-heuristics. In this study, we propose a new and effective method for solving the RNP problem. The idea behind this method is to solve the RNP problem in two stages using an optimal algorithm with fewer variables than the original RNP problem. In stage 1, we build an RNP sub problem using 15% to 20% of the number of routers, with the objective function of minimizing coverage overlap between routers to form a core network. Stage 2 is built into another RNP sub problem with the remaining number of routers, and the objective function is to maximize the network connectivity. Each sub problem was solved using an approximate optimal algorithm. The experimental results demonstrate that, in terms of client coverage and network connectivity, our proposed method outperforms widely used RNP problem-solving methods.

We consider a game of persuasion with evidence between a sender and a receiver. The sender has private information. By presenting evidence on the information, the sender wishes to persuade the receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender’s utility depends solely on whether or not the receiver takes the action. The receiver’s utility depends on both the action as well as the sender’s private information. We study three natural variations. First, we consider sequential equilibria of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and then the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and then the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, as well as polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm for a natural problem in algorithmic economics.

In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the order algorithms produces polytopes for which the accuracy in Hausdorff metric is inversely proportional to the number of vertices (faces) in the degree of 2/(d-1). Numerical approximation algorithms can be adaptive (active) when the vertices or faces are constructed successively, depending on the information obtained in the process of approximation, and non-adaptive (passive) when parameters of algorithms are defined on the basis of a priory information available. Approximation algorithms differ in the use of operations applied to the approximated body. Most common are indicator, support and distance (Minkowski) functions calculations. Some optimal active algorithms for arbitrary bodies approximation are known using support or distance function calculation operation. Optimal passive algorithms for smooth bodies approximation are known using support function calculation operation and extremal curvature information. It is known that there are no optimal non-adaptive algorithms for arbitrary bodies approximation using support function calculation operation. We consider optimal non-adaptive algorithms for arbitrary bodies approximation using projection function calculation operation.

Effective data assimilation algorithms require the estimation and specification of second-moment forecast error statistics. The imposition of multivariate constraints has been found to be a particularly effective way of extracting the maximum amount of information from the observations. In the optimal interpolation (OI) algorithm, geostrophic constraints were imposed, and these have been extended to the global case in the newer three-dimensional variational (3DVAR) algorithm by the application of the linear balance equation or a Rossby–Hough expansion. This study shows that the imposition of the linear balance equation (or the Rossby–Hough expansion) in the Tropics (although mathematically attractive) is not correct and may have deleterious effects on the assimilated products. A procedure is developed, based on the singular-value decomposition (svd) of the linear balance equation, for generating global forecast error covariances in which the multivariate coupling between wind and mass is completely user controlled. Thus, two modal spaces are defined: a spectral (spherical harmonic) space for controlling the redness of the spectrum and a balance (svd) space for controlling the coupling. In this way it is possible for the rotational wind and mass error covariances to be closely coupled in line extratropics and essentially univariate at low latitudes.

Based on the optimal interpolation (OI) algorithm, a daily fusion product of high-resolution global ocean columnar atmospheric water vapor with a resolution of 0.25° was generated in this study from multisource remote sensing observations. The product covers the period from 2003 to 2018, and the data represent a fusion of microwave radiometer observations, including those from the Special Sensor Microwave Imager Sounder (SSMIS), WindSat, Advanced Microwave Scanning Radiometer for Earth Observing System sensor (AMSR-E), Advanced Microwave Scanning Radiometer 2 (AMSR2), and HY-2A microwave radiometer (MR). The accuracy of this water vapor fusion product was validated using radiosonde water vapor observations. The comparative results show that the overall mean deviation (Bias) is smaller than 0.6 mm; the root mean square error (RMSE) and standard deviation (SD) are better than 3 mm, and the mean absolute deviation (MAD) and correlation coefficient (R) are better than 2 mm and 0.98, respectively.

Informative path planning is an important and challenging problem in robotics that remains to be solved in a manner that allows for wide-spread implementation and real-world practical adoption. Among various reasons for this, one is the lack of approaches that allow for informative path planning in high-dimensional spaces and non-trivial sensor constraints. In this work we present a sampling-based approach that allows us to tackle the challenges of large and high-dimensional search spaces. This is done by performing informed sampling in the high-dimensional continuous space and incorporating potential information gain along edges in the reward estimation. This method rapidly generates a global path that maximizes information gain for the given path budget constraints. We discuss the details of our implementation for an example use case of searching for multiple objects of interest in a large search space using a fixed-wing UAV with a forward-facing camera. We compare our approach to a sampling-based planner baseline and demonstrate how our contributions allow our approach to consistently out-perform the baseline by 18.0%. With this we thus present a practical and generalizable informative path planning framework that can be used for very large environments, limited budgets, and high dimensional search spaces, such as robots with motion constraints or high-dimensional configuration spaces. [Code] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> Codebase: https://github.com/castacks/tigris [Video] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sup> Video: https://youtu.be/bMw5nUGL5GQ

The use of effective distance functions has been explored for many data mining problems including clustering, nearest neighbor search, and indexing. Recent research results show that if the Pearson variation of the distance distribution converges to zero with increasing dimensionality, the distance function will become unstable (or meaningless) in high dimensional space even with the commonly used Lp metric on the Euclidean space. This result has spawned many subsequent studies. We first comment that although the prior work provided the sufficient condition for the unstability of a distance function, the corresponding proof has some defects. Also, the necessary condition for unstability (i.e., the negation of the sufficient condition for the stability) of a distance function, which is required for function design, remains unknown. Consequently, we first provide in this paper a general proof for the sufficient condition of unstability. More importantly, we go further to prove that the rapid degradation of Pearson variation for a distance distribution is in fact a necessary condition of the resulting unstability. With the result, we will then have the necessary and the sufficient conditions for unstability, which in turn imply the sufficient and necessary conditions for stability. This theoretical result derived leads to a powerful means to design a meaningful distance function. Explicitly, in light of our results, we design in this paper a meaningful distance function, called Shrinkage-Divergence Proximity (abbreviated as SDP), based on a given distance function. It is empirically shown that the SDP significantly outperforms prior measures for its being stable in high dimensional data space and robust to noise, and is thus deemed more suitable for distance-based clustering applications than the priorly

Many approximation algorithms have been presented in the last decades for ${\cal NP}$-hard search problems. The focus of this paper is on cryptographic applications, where it is desirable to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms—efficient approximation algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, Feigenbaum et al. [ACM Trans. Algorithms, 2 (2006), pp. 435–472] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to relations is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms; applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. We argue that our privacy requirement is natural and quite minimal. We show that, even under this minimal definition of privacy, for well-studied problems such as vertex cover and max exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similarly to Halevi et al. [in Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 550–559], we define a relaxed notion of approximation algorithms that leak (a little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for max exact 3SAT that leak a little information.

An adaptive denoising method based on 2D empirical mode decomposition (EMD) is proposed to improve the infrared image quality of inspection UNMANNED aerial vehicles (UAVs) and provide guarantee for improving the inspection level of distribution network. Through rapid adaptive two-dimensional empirical mode decomposition algorithm decomposition of a UAV collected for distribution network inspection original noise of infrared image, get more than the IMF component and the residual amount, a forecast noise dominated the IMF component parameters such as threshold value and the variance of noise, using the estimated parameters in combination with the optimal linear interpolation algorithm of noise threshold function of leading the IMF component implementation of threshold denoising. After the denoised IMF component is obtained, the denoised infrared image is obtained after reconstruction with the signal-dominated IMF component, and the adaptive denoising of the infrared image of the distribution network inspection UAV is realized. The experimental results show that the method in this paper can maintain the details of the image, improve the definition, significantly improve the visual effect, the overall denoising performance is stable and feasible, and ensure the quality inspection UAV to collect infrared images.

Global optimisation problems in high-dimensional and infinite-dimensional spaces arise in various real-world applications such as engineering, economics and finance, geophysics, biology, machine learning, optimal control, etc. Among stochastic approaches to global optimisation, biology-inspired methods are currently very popular in the literature. Bio-inspired approaches imitate natural ecological and evolutionary processes and are reported to be efficient in a large number of practical study cases. However, many bio-inspired methods can possess some vital drawbacks. For example, due to their semi-empirical nature, convergence to the globally optimal solution cannot always be guaranteed. Another major obstacle is that the existing methods often struggle with higher dimensionality of the space of parameters, which results in a slow convergence. Moreover, it is often difficult to adjust the dimensionality of the space of parameters in the corresponding computer code for a practical realisation of the optimisation method. Here, we present a bio-inspired global stochastic optimisation method applicable in Hilbert function spaces. The proposed method is an evolutionary algorithm inspired by the Darwin’s famous idea of survival of the fittest and is, therefore, referred to as the ‘Survival of the Fittest Algorithm’ (SoFA). Mathematically, the convergence of SoFA is a consequence of the fundamental property of localisation of probabilistic measure in a Hilbert space and we rigorously prove the convergence of the introduced algorithm for a generic class of functionals. The approach is simple in terms of practical coding. As an insightful, real-world problem, we apply our method to find the globally optimal trajectory for the daily vertical migration of zooplankton in the ocean and lakes, this phenomenon is considered to be the largest synchronised movement of biomass on Earth. We maximise fitness in a function space derived from a von-Foerster stage-structured population model with biologically realistic parameters. We show that for problems of fitness maximisation in high-dimensional spaces, SoFA provides better performance as compared to some other stochastic global optimisation algorithms. We highlight the links between the new optimisation algorithm and natural selection processes in ecosystems occurring within a population via gradual exclusion of competitive con-specific strains.

The purpose of reoptimization using approximation methods—application of knowledge about the solution of the initial instance I, provided to achieve a better quality of approximation (approximation ratio) of an algorithm for determining optimal or close to it solutions of some “minor” changes of instance I. To solve the problem Ins-Max-EkCSP-P (reoptimization of Max-EkCSP-P with the addition of one constraint) with approximation resistant predicate P exists a polynomial threshold (optimal) -approximation algorithm, where the threshold “random” approximation ratio of P). When the unique games conjecture (UGC) is hold there exists a polynomial threshold (optimal) -approximation algorithm (where and the integrality gap of semidefinite relaxation of Max-EkCSP-P problem Z) to solve the problem Ins-Max-EkCSP-P.