Sort Of Approximation Research Articles

Approximate sorting and its applications in I/O model

Sorting is a fundamental component in many different applications including web indexing engines, geographic information systems, data mining and database systems. However, the existing optimal algorithms still cause huge costs when the input data becomes very large. Thus, the approximate sorting for big data is considered in this paper. The goal of approximate sorting for big data is to generate an approximate sorted result, but using less CPU and I/O cost. For big data, we consider the approximate sorting and applications for approximate sorted results in I/O model.The quality of approximate sorting results is usually measured by the distance metrics on permutation space. However, the existing metrics on permutation space are not available for external approximate sorting algorithms. Thus, we propose a new kind of metric named External metric, which ignores the errors and dislocations that happened in each I/O block. The External metric of Spearman's footrule metric is named as External Spearman's footrule metric (short as ESP metric), which is analyzed in this paper.Furthermore, to facilitate a better evaluation of the approximate sorted result, we propose a new metric, named as errors, which directly states the number of dislocation of the elements. The External errors are also considered in this paper.Then, according to the rate-distortion relationship under these two metrics, the lower bound of these two metrics on external approximate sorting problem with t I/O operations is proved. We propose a k-passes external approximate sorting algorithm, named as EASORT, and prove that EASORT is asymptotically optimal under ESP metric.Finally, we consider the applications on approximate sorting results. An index for the approximate sorted result is proposed. The single and range query on approximate sorted result using this index are also analyzed. Further, the sort-merge join on two relations, where one of the relations is approximate sorted or both relations are approximate sorted, are all discussed in this paper.

On the modelling of energetic multi-jet QCD events

Physics beyond the Standard Model (BSM) may be unveiled by studying events with a high number of outgoing jets, produced at the LHC with energies above the TeV scale (energetic multi-jet events). Such events are dominated by QCD processes, where the calculations rely on some sort of approximation. Therefore, it is important to develop a robust approach for modeling such events that could probe the existence of BSM signals. In this note, jet spatial distributions in energetic multi-jet processes were compared using several state-of-the-art MC event generators. Slight differences were found, indicating modelling limitations. Therefore, a data-driven technique for the estimation of processes with a final state that contains a large number of jets is proposed. This procedure can predict jet multiplicities up to a precision of 25% in energetic multi-jet events.

Efficient spatio-temporal Gaussian regression via Kalman filtering

We study the non-parametric reconstruction of spatio-temporal dynamical processes via Gaussian Processes (GPs) regression from sparse and noisy data. GPs have been mainly applied to spatial regression where they represent one of the most powerful estimation approaches also thanks to their universal representing properties. Their extension to dynamical processes has been instead elusive so far since classical implementations lead to unscalable algorithms or require some sort of approximation. We propose a novel procedure to address this problem by coupling GPs regression and Kalman filtering. In particular, assuming space/time separability of the covariance (kernel) of the process and rational time spectrum, we build a finite-dimensional discrete-time state-space process representation amenable to Kalman filtering. With sampling over a finite set of fixed spatial locations, our major finding is that the current Kalman filter state represents a sufficient statistic to compute the minimum variance estimate of the process at any future time over the entire spatial domain. In machine learning, a representer theorem states that an important class of infinite-dimensional variational problems admits a computable and finite-dimensional exact solution. In view of this, our result can be interpreted as a novel Dynamic Representer Theorem for GPs. We then extend the study to situations where the spatial input locations set varies over time. The proposed algorithms are tested on both synthetic and real field data, providing comparisons with standard GP and truncated GP regression techniques.

D-Matrix: A Novel Ranking Procedure for Prioritizing Data Items

In this article, we propose a ranking method based on a matrix, called D-matrix, with the special identical diagonal values. This ranking system has five properties: (1) it can provide both biased and bias-free ranking, and except for that, the working matrix can be built in two ways: results merging and results separating for both biased and bias-free matrices. (2) it can perform the webpage ranking with a sparse matrix to generate ratings for pages instead of constructing complicated, irreducible, and stochastic matrices as the Google PageRank matrix does, thereby accelerating the computation speed. (3) this D-matrix has a solution no matter how much data is selected. If there are no comparisons among items, then all the items end up with the same equal ratings. (4) the ranking system has the least effects on data variation. If one item changes, only those connecting to it get different ratings, those without connection retain the same ratings. (5) this D-matrix has a $\ddot {R}$ support matrix with a delicate diagonal value which may or may not appear crucial. These five features are illustrated with five different and comprehensive examples. Besides that, a 2017 game of the National Football League data is tested where the D-matrix generates a reasonable result. Furthermore, we introduce a new approximate non-dominated sorting method based on D-matrix and thereby put forth a new algorithm for solving the multi-objective optimization problems. Experimental results indicate that our algorithm can maintain a better spread of solutions on many standard test functions.

Missing values and dragonfly operations in fuzzy relational compositions

Three-valued logics were found by logicians as an important topic focusing on dealing with truth-values different from the standard True and False values. The variety of such values, including “Irrelevant”, “Non-applicable”, “Indeterminable”, “Incosistent”, “Graded truth” or “Unknown”, generated a wide variety of distinct three-valued logics, each focusing on a distinct type of the third value and the consequent aspects of the related logic. Indeed, there is no single approach that would correctly model all the motivating situations and serve perfectly to all practical problems. Furthermore, one has to keep in mind that these logical or even only purely algebraic approaches serve as a sort of approximation of the modeled real situation. Indeed, some of them might deserve very complex approaches using several other techniques and scientific fields related to the uncertainty theories.However, the logical/algebraic approaches may serve as very appropriate, comprehensible, elegant and efficient way to treat such truth values that are neither True, nor False. Following some of the previous works, we will call such values by the word “undefined” and make a short revision of the three-valued logics dealing with such undefined values. Secondly, we will review some extensions of these three-valued logics to many-valued logics, i.e., in particular partial fuzzy logics, which extend typical, usually [0,1]-valued fuzzy logics by a dummy value ⋆ in order to represent the undefined truth value. Furthermore, we recall that none of them is primarily proposed in order to deal with the missing values in fuzzy relational compositions and thus, the first attempts to deal with such values in fuzzy relational compositions was built on a combination of two algebras for partial fuzzy logics, namely Bochvar and Sobociński. However, it is clear that this combination of two algebras in the definition of fuzzy relational compositions is a sort of higher-level construction of a rather heuristic origin. Therefore, in this paper, we go back one level lower and design a new set of operations for the purpose of dealing with missing values. This algebra employs the lower estimation approach and it is designed in order to preserve as many properties from the residuated lattices as possible. Further properties of the proposed operations are provided and formally proved. Finally, the application potential is demonstrated on a real example of the taxonomical classification of dragonflies. Based on the primary application, we call the proposed algebra of operations as Dragonfly algebra or simply Dragonfly operations.

Nonperturbative Quantization à La Heisenberg: Modified Gravities, Wheeler-DeWitt Equations, and Monopoles in QCD

For field theories in which no small parameter is available, we use Heisenberg’s quantization procedure to propose a definition of nonperturbative quantum states in terms of the complete set of Green functions. We present the corresponding quantization schemes in the case of Einstein gravity and gauge theories. To illustrate the procedure of quantization, we show that: (1) modified theories of gravity appear as an effective approximation of nonperturbative quantum gravity; (2) the Wheeler-DeWitt equations appear as a sort of approximation of the quantization procedure a la Heisenberg, and (3) it is possible to carry out explicit nonperturbative calculations in quantum chromodynamics, and we obtain the energy spectrum of a quantum monopole and some thermodynamic quantities for a gas of noninteracting quantum monopoles.

Assessment of algorithms for computing moist available potential energy

Atmospheric moist available potential energy (MAPE) has traditionally been defined as the potential energy of a moist atmosphere relative to that of the adiabatically sorted reference state defining a global potential energy minimum. Although the Munkres algorithm can in principle find such a reference state exactly, its computational cost has prompted much interest in developing heuristic methods for computing MAPE in practice. Comparisons of the accuracy of such approximate algorithms have so far been limited to a small number of test cases; this work provides an assessment of the performance of the algorithms across a wide range of atmospheric soundings, in two different locations. We determine that the divide‐and‐conquer algorithm is the best suited to practical application, but suffers from the previously unexplored shortcoming that it can produce a reference state with higher potential energy than the actual state, resulting in a negative value of MAPE. Additionally, we show that it is possible to construct an algorithm exploiting a previously derived theoretical expression linking MAPE to Convective Available Potential Energy (CAPE). This approach has a similar accuracy to existing approximate sorting algorithms, whilst providing greater insight into the physical source of MAPE. In light of these results, we discuss possible ways to improve on the construction of Available Potential Energy (APE) theory for a moist atmosphere.

Tales from the prehistory of Quantum Gravity

The main purpose of this paper is to analyse the earliest work of L\'eon Rosenfeld, one of the pioneers in the search of Quantum Gravity, the supposed theory unifying quantum theory and general relativity. We describe how and why Rosenfeld tried to face this problem in 1927, analysing the role of his mentors: Oskar Klein, Louis de Broglie and Th\'eophile De Donder. Rosenfeld asked himself how quantum mechanics should \textit{concretely} modify general relativity. In the context of a five-dimensional theory, Rosenfeld tried to construct a unifying framework for the gravitational and electromagnetic interaction and wave mechanics. Using a sort of "general relativistic quantum mechanics" Rosenfeld introduced a wave equation on a curved background. He investigated the metric created by what he called `quantum phenomena', represented by wave functions. Rosenfeld integrated Einstein equations in the weak field limit, with wave functions as source of the gravitational field. The author performed a sort of semi-classical approximation obtaining at the first order the Reissner-Nordstr\"om metric. We analyse how Rosenfeld's work is part of the history of Quantum Mechanics, because in his investigation Rosenfeld was guided by Bohr's correspondence principle. Finally we briefly discuss how his contribution is connected with the task of finding out which metric can be generated by a quantum field, a problem that quantum field theory on curved backgrounds will start to address 35 years later.

, <Br> Legal Proof, Mathematical Proof and Scientific Explanation

תקציר בעברית: חיבור זה סובב סביב פשרה ומהותה של ההוכחה המשפטית הנוגעת בקביעת עובדות . אפתח במספר הערות מקדימות: ראשית, הנושא שלפנינו אינו פשוט, אלא מורכב למדיי, ומצריך עיון בלתי שטחי בתופעת ההוכחה וההסבר בשטחים אינטלקטואליים אחרים. שנית, אודות נושא זה לא נכתב אלא מעט . כאן חשוב להדגיש. במהלך כ-שלושים השנים האחרונים החלה להתפתח ספרות תיאורטית ענפה, אינטרדיספלינארית באופייה, העוסקת במימד האפיסטמולוגי של דיני הראיות, בפרט בשיטה האנגלו-אמריקאית . כך לדוגמא קיים דיון רחב היקף סביב השאלה בדבר מהותה של הסתברות שראויה לשמש, אם בכלל, קובעי עובדות, בפרט שופטים במלאכת הסקת מסקנות עובדתיות . כמו כן קיים ניתוח על אודות קדם ההנחות ההכרתיות של המסורת שבתוכה מתפקדים דיני הראיות . כן מוצעים ומוצגים מודלים אנליטיים ונורמטיביים להבנה ,פירוש או שינוי ועדכון של דיני הראיות הנוהגים . כל זאת תוך שימוש בתובנות מתחומי תורת ההחלטות סטטיסטיקה ותורת המשחקים, סוציולוגיה, פסיכולוגיה קוגניטיבית ,פילוסופיה של ההכרה ותחומים נוספים. כל אלו סובבים סביב העניין שלפנינו אך אינם מציבים אותו כנושא הישיר של בחינתם. עניינו כאן בלוגיקה להבדיל מן האפיסטמולוגיה. עניינו בתכניו המופשטים של מושג התקפות בדיני ההוכחה להבדיל מהצגה של אמצעים ההכרה ורכישת הידע השיפוטיים . נקודת מבטינו מוסבת לעבר בצורות או האופנים המופשטים ביותר של המעבר מראיות לעובדות להבדיל מאופן הטיפול בראיות מסוימות או בסוגים מיוחדים של היסקים עובדתיים (לפיכך לא נטפל במישרין באבחנות אפשריות שבין הסקה נסיבתית להסקה על יסוד עדויות ישירות ,חזקות או הודאת בעל הדין או הנאשם)שלישית, ככלל אתמקד כפי שכותרתו של המאמר מרמזת בהשוואת דפוס ההוכחה המשפטית עם זה המתמטי-עיוני וזה הנוהג במדעי הטבע, (ובמדעי ההתנהגות והחברה ככל שהם מציבים את דגם ההוכחה של המדעים המדויקים כמופת של הוכחה ). רק כבדרך אגב אעיר כמה הערות באשר לרעיונות בדבר השוואת ההיסק המשפטי עם דגם ההוכחה וההסבר במדעי הרוח ובשטחים אינטלקטואליים אחרים העוסקים בפרשנות ובהבניה של אפיזודות אנושיות.ורביעית מה שייכתב כאן אינו יותר מהערות תיאורטיות ראשוניות שאינן בוודאי עולות לכדי עמדה שיטתית ומלוטשת באשר לנושא. מכל מקום עיקר חשיבות הדברים, היא לעורר מחשבה ולהפנות את הדעת לסוגיה שטרם זכתה לעיון שלעניות דעתי לו היא ראויה. נפנה אם כך לגוף העניין עצמו In this article I will seek to discuss the essence of legal proof in relation to fact-finding; in particular, a facet thereof that would seem to dovetail with scientific research. I refer, in this regard, to the manner in which the law establishes generalizations (i.e., legal revelations) regarding reality, which it then uses for fact-finding in a given instance; as well as the manner in which the law uses such generalizations for specific factual deduction. The principle intuition that our article seeks to contend with is how the law uses a sort of approximation of the scientific process within the constraints of legal practicality. Essentially, a sort of “layman’s science” that attempts to accomplish the same goals as science - only, not as well, and less methodically. The conclusion we will seek to show in this article is that the manner in which the law derives generalizations about reality, and the manner in which it uses same, fundamentally differs from the manner in which science is conducted. Theories in natural sciences (and largely with respect to social sciences) are framed as generally as possible. Legal proof, on the other hand, is specific . In fact, the most common form of induction used is the cognitive analogy, wherein the discretion employed is far more similar in its characteristics to clinical discretion, such as that used in the field of medicine, then to those typically employed in the fields of natural sciences.

Anomaly Detection and Classification for Streaming Data using PDEs

Nondominated sorting, also called Pareto Depth Analysis (PDA), is widely used in multi-objective optimization and has recently found important applications in multi-criteria anomaly detection. Recently, a partial differential equation (PDE) continuum limit was discovered for nondominated sorting leading to a very fast approximate sorting algorithm called PDE-based ranking. We propose in this paper a fast real-time streaming version of the PDA algorithm for anomaly detection that exploits the computational advantages of PDE continuum limits. Furthermore, we derive new PDE continuum limits for sorting points within their nondominated layers and show how the new PDEs can be used to classify anomalies based on which criterion was more significantly violated. We also prove statistical convergence rates for PDE-based ranking, and present the results of numerical experiments with both synthetic and real data.

Electronic structures of elements according to ionization energies.

The electronic structures of elements in the periodic table were analyzed using available experimental ionization energies. Two new parameters were defined to carry out the study. The first parameter-apparent nuclear charge (ANC)-quantified the overall charge of the nucleus and inner electrons observed by an outer electron during the ionization process. This parameter was utilized to define a second parameter, which presented the shielding ability of an electron against the nuclear charge. This second parameter-electron shielding effect (ESE)-provided an insight into the electronic structure of atoms. This article avoids any sort of approximation, interpolation or extrapolation. First experimental ionization energies were used to obtain the two aforementioned parameters. The second parameter (ESE) was then graphed against the electron number of each element, and was used to read the corresponding electronic structure. The ESE showed spikes/peaks at the end of each electronic shell, providing insight into when an electronic shell closes and a new one starts. The electronic structures of elements in the periodic table were mapped using this methodology. These graphs did not show complete agreement with the previously known "Aufbau" filling rule. A new filling rule was suggested based on the present observations. Finally, a new way to organize elements in the periodic table is suggested. Two earlier topics of effective nuclear charge, and shielding factor were also briefly discussed and compared numerically to demonstrate the capability of the new approach.

Optimizing Graph Processing on GPUs

Distributed vertex-centric model has been recently proposed for large-scale graph processing. Due to the simple but efficient programming abstraction, similar graph computing frameworks based on GPUs are gaining more and more attention. However, prior works of GPU-based graph processing suffer from load imbalance and irregular memory access because of the inherent characteristics of graph applications. In this paper, we propose a generalized graph computing framework for GPUs to simplify existing models but with higher performance. In particular, two novel algorithmic optimizations, lightweight approximate sorting and data layout transformation, are proposed to tackle the performance issues of current systems. With extensive experimental evaluation under a wide range of real world and synthetic workloads, we show that our system can achieve 1.6 $\times$ to 4.5 $\times$ speedups over the state-of-the-art.

Redundant Network Traffic Elimination with GPU Accelerated Rabin Fingerprinting

Recently, redundant network traffic elimination has attracted a lot of attention from both the academia and the industry. A core challenge and enabling technique in implementing redundancy elimination is to perform content-based chunking, which typically involves the computationally heavy Rabin fingerprinting algorithm. In this paper, we propose a GPU-based implementation of Rabin fingerprinting to address this issue. To maximize performance gains, a diverse set of optimization strategies, such as efficient buffer management, GPU memory hierarchy optimization, and balanced load distribution, is proposed by either exploiting the intrinsic hardware features or addressing domain-specific challenges. Extensive evaluations on both the overall and microscopic performance reveal the effectiveness of the GPU-accelerated Rabin fingerprinting algorithm, and we can achieve up to 40 Gpbs throughput on a GTX 780 card. The throughput shows 1.87 $\times$ speedup against the state-of-the-art using comparable hardware. In addition, although some optimization designs are specific for the problem, techniques proposed in this work including the indexed compact buffer scheme and approximate sorting would also be beneficial and applicable to other network applications leveraging GPU acceleration.

Approximate non-dominated sorting for evolutionary many-objective optimization

Non-dominated sorting has widely been adopted and shown to be very effective in dominance based evolutionary multi-objective optimization where the number of objectives is two or three. In dealing with many-objective optimization problems, where the number of objectives is more than three, however, the effectiveness of dominance based evolutionary algorithms will seriously degrade and the computational complexity of non-dominated sorting will considerably increase. In this paper, we propose an approximate non-dominated sorting algorithm for many-objective optimization, where the dominance relationship between two solutions is determined by a maximum of three objective comparisons on top of a sorted population according to one of the objectives. The time complexity of the proposed approximate sorting is independent of the number of objectives, which significantly enhances the time efficiency, in particular when the number of objectives is large. To examine the influence of inaccurate sorting on the search performance, the proposed approximate sorting method is embedded in three popular dominance based evolutionary many-objective optimization algorithms. Our experimental results on 16 widely used many-objective optimization benchmark problems suggest that the approximate non-dominated sorting method is not only computationally much more efficient than the state-of-the-art accurate non-dominated sorting methods, but also leads to better search performance on most test problems used in this study.

A Bernoulli mean estimate with known relative error distribution

AbstractSuppose that are independent identically distributed Bernoulli random variables with mean p, so and . Any estimate of p has relative error . This paper builds a new estimate of p with the remarkable property that the relative error of the estimate does not depend in any way on the value of p. This allows the easy construction of exact confidence intervals for p of any desired level without needing any sort of limit or approximation. In addition, is unbiased. For ∊ and δ in (0, 1), to obtain an estimate where , the new algorithm takes on average at most samples. It is also shown that any such algorithm that applies whenever requires at least samples on average. The same algorithm can also be applied to estimate the mean of any random variable that falls in . The used here employs randomness external to the sample, and has a small (but nonzero) chance of being above 1. It is shown that any nontrivial where the relative error is independent of p must also have these properties. Applications of this methodology include finding exact p‐values and randomized approximation algorithms for # P complete problems. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 173–182, 2017

Bounds for Permutation Rate-Distortion

We study the rate-distortion relationship in the set of permutations endowed with the Kendall $\tau $ -metric and the Chebyshev metric (the $\ell _\infty $ -metric). This paper is motivated by the application of permutation rate-distortion to the average-case and worst-case distortion analysis of algorithms for ranking with incomplete information and approximate sorting algorithms. For the Kendall $\tau $ -metric, we provide bounds for various distortion regimes, while for the Chebyshev metric, we present bounds that are valid for all distortions and are especially accurate for small distortions. In addition, for the Chebyshev metric, we provide a construction for covering codes.

Approximate sorting of data streams with limited storage

We consider the problem of approximate sorting of a data stream (in one pass) with limited internal storage where the goal is not to rearrange data but to output a permutation that reflects the ord...

Casimir torque between nanostructured plates

We investigate in detail the Casimir torque induced by quantum vacuum fluctuations between two nanostructured plates. Our calculations are based on the scattering approach and take into account the coupling between different modes induced by the shape of the surface which are neglected in any sort of proximity approximation or effective medium approach. We then present an experimental setup aiming at measuring this torque.

Modification of Near Sets Theory

Most real life situations need some sort of approximation to fit mathematical models. The beauty of using topology in approximation is achieved via obtaining approximation for qualitative subsets without coding or using assumption. The aim of this paper is to introduce different approaches to near sets by using general relations and special neighborhoods. Some fundamental properties and characterizations are given. We obtain a comparison between these new approximations and traditional approximations introduced by Peters [23].

Robust Control of Electromagnetic Levitation System

Nonlinear \({H}_{\infty }\) control is a promising way of attenuating negative effects of external disturbances in nonlinear systems and is robust against uncertainties of the plant. However, partial differential equation (PDE) obtained from this method rarely can be solved analytically. Thus, some sort of approximation is required. Furthermore, produced controllers by this method using Taylor series approximation (especially higher order controllers) are complicated and hard to implement. In this paper, a robust and nonlinear controller is introduced which combines linear \({H}_{\infty }\) controller and feedback linearization technique. First, a nonlinear state transformation is applied to the nonlinear system such that the resulting system is linear. Next, a linear \({H}_{\infty }\) controller is designed for the feedback linearized system. By this method, approximation is not required in designing \({H}_{\infty }\) controller because for linear systems, resulting PDE is simplified to a Riccati equation. Proposed approach has been applied to a Magnetic Levitation (Maglev) system, and its performance is compared with third-order nonlinear \({H}_{\infty }\) controller to verify the effectiveness and robustness of new method. Simulation results show that proposed controller attenuates external disturbance much better than nonlinear \({H}_{\infty }\), and simultaneously has less control effort.