Arbitrary Precision Research Articles

Correspondenceless scan-to-map-scan matching of homoriented 2D scans for mobile robot localisation

The objective of this study is improving the location estimate of a mobile robot capable of motion on a plane and mounted with a conventional 2D LIDAR sensor, given an initial guess for its location on a 2D map of its surroundings. Documented herein is the theoretical reasoning behind solving a matching problem between two homoriented 2D scans, one derived from the robot’s physical sensor and one derived by simulating its operation within the map, in a manner that does not require the establishing of correspondences between their constituting rays. Two results are proved and subsequently shown through experiments. The first is that the true position of the sensor can be recovered with arbitrary precision when the physical sensor reports faultless measurements and there is no discrepancy between the environment the robot operates in and its perception of it by the robot. The second is that when either is affected by disturbance, the location estimate is bound in a neighbourhood of the true location whose radius is proportional to the affecting disturbance.

Nonatomic aggregative games with infinitely many types

We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of player types. These equilibria are characterized through an infinite-dimensional variational inequality. We show, under monotonicity conditions, a convergence theorem which enables to compute such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of player types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different player types, with actions subject to coupling constraints.

Real-Time mmWave Channel Sounding Through Switched Beamforming With 3-D Dual-Polarized Phased-Array Antennas.

We propose a 28.5-GHz channel sounder that switches through all antennas of multiple dual-polarized 8 × 8 phased arrays at the transmitter and receiver and performs beamforming in postprocessing through digital weights to synthesize a sweepable beam. To our knowledge, we are the first to implement-what we refer to as-switched beamforming with phased arrays for millimeter-wave channel sounding, realized through highly stable Rubidium clocks and local oscillators coupled with precision over-the-air calibration techniques developed in house. By circumventing the time-consuming programming of analog weights that is associated with analog beamforming-what phased arrays are designed for-we can sweep a 3-D double-omnidirectional dual-polarized channel in just 1.3 ms, for real-time sounding. By in turn circumventing the coarse precision of analog weights, we can synthesize ideal beam patterns thanks to the effectively infinite precision of digital weights, enabling fine weight calibration for the nonidealities of the system hardware and fine weight tapering for sidelobe suppression. This translates to average estimation errors of 0.47° in 3-D double-directional angle, 0.48 dB in co-polarized path gain, and 0.18 ns in delay, as substantiated by field measurements.

Bose-Einstein statistics for a finite number of particles

This article presents a study of the grand canonical Bose-Einstein (BE) statistics for a finite number of particles in an arbitrary quantum system. The thermodynamical quantities that identify BE condensation -- namely, the fraction of particles in the ground state and the specific heat -- are calculated here exactly in terms of temperature and fugacity. These calculations are complemented by a numerical calculation of fugacity in terms of the number of particles, without taking the thermodynamic limit. The main advantage of this approach is that it does not rely on approximations made in the vicinity of the usually defined critical temperature, rather it makes calculations with arbitrary precision possible, irrespective of temperature. Graphs for the calculated thermodynamical quantities are presented in comparison to the results previously obtained in the thermodynamic limit. In particular, it is observed that for the gas trapped in a 3-dimensional box the derivative of specific heat reaches smaller values than what was expected in the thermodynamic limit -- here, this result is also verified with analytical calculations. This is an important result for understanding the role of the thermodynamic limit in phase transitions and makes possible to further study BE statistics without relying neither on the thermodynamic limit nor on approximations near critical temperature.

The Stochastic Augmented Lagrangian method for domain adaptation

Among various topics explored in the transfer learning community, domain adaptation (DA) has been of primary interest and successfully applied in diverse fields. However, theoretical understanding of learning convergence in DA has not been sufficiently explored. To address such an issue, this paper presents the Stochastic Augmented Lagrangian method (SALM) to solve the optimization problem associated with domain adaptation. In contrast to previous works, the SALM is able to find the optimal Lagrangian multipliers, as opposed to manually selecting the multipliers which could result in significantly suboptimal solutions. Additionally, the SALM is the first algorithm which can find a feasible point with arbitrary precision for domain adaptation problems with bounded penalty parameters. We also observe that with unbounded penalty parameters, the proposed algorithm is able to find an approximate stationary point of infeasibility. We validate our theoretical analysis with several experimental results using benchmark data sets including MNIST, SYNTH, SVHN, and USPS.

Finite variation sensitivity analysis for discrete topology optimization of continuum structures

This paper proposes two novel approaches to perform more suitable sensitivity analyses for discrete topology optimization methods. To properly support them, we introduce a more formal description of the Bi-directional Evolutionary Structural Optimization (BESO) method, in which the sensitivity analysis is based on finite variations of the objective function. The proposed approaches are compared to a naive strategy; to the conventional strategy, referred to as First-Order Continuous Interpolation (FOCI) approach; and to a strategy previously developed by other researchers, referred to as High-Order Continuous Interpolation (HOCI) approach. The novel Woodbury approach provides exact sensitivity values and is a better alternative to HOCI. Although HOCI and Woodbury approaches may be computationally prohibitive, they provide useful expressions for a better understanding of the problem. The novel Conjugate Gradient Method (CGM) approach provides sensitivity values with arbitrary precision and is computationally viable for a small number of steps. The CGM approach is a better alternative to FOCI since, for appropriate initial conditions, it is always more accurate than the conventional strategy. The standard compliance minimization problem with volume constraint is considered to illustrate the methodology. Numerical examples are presented together with a broad discussion about BESO-type methods.

Computing Puiseux series: a fast divide and conquer algorithm

Let F∈𝕂[X,Y] be a polynomial of total degree D defined over a perfect field 𝕂 of characteristic zero or greater than D. Assuming F separable with respect to Y, we provide an algorithm that computes all singular parts of Puiseux series of F above X=0 in an expected Ø ˜(Dδ ) operations in 𝕂, where δ is the valuation of the resultant of F and its partial derivative with respect to Y. To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in 𝕂[[X]][Y] up to an arbitrary precision X N with Ø ˜(D(δ +N)) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with Ø ˜(D 3 ) arithmetic operations and, if 𝕂=ℚ, with Ø ˜((h+1)D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F.

Fast and exact quantification of motif occurrences in biological sequences

BackgroundIdentification of motifs and quantification of their occurrences are important for the study of genetic diseases, gene evolution, transcription sites, and other biological mechanisms. Exact formulae for estimating count distributions of motifs under Markovian assumptions have high computational complexity and are impractical to be used on large motif sets. Approximated formulae, e.g. based on compound Poisson, are faster, but reliable p value calculation remains challenging. Here, we introduce ‘motif_prob’, a fast implementation of an exact formula for motif count distribution through progressive approximation with arbitrary precision. Our implementation speeds up the exact calculation, usually impractical, making it feasible and posit to substitute currently employed heuristics.ResultsWe implement motif_prob in both Perl and C+ + languages, using an efficient error-bound iterative process for the exact formula, providing comparison with state-of-the-art tools (e.g. MoSDi) in terms of precision, run time benchmarks, along with a real-world use case on bacterial motif characterization. Our software is able to process a million of motifs (13–31 bases) over genome lengths of 5 million bases within the minute on a regular laptop, and the run times for both the Perl and C+ + code are several orders of magnitude smaller (50–1000× faster) than MoSDi, even when using their fast compound Poisson approximation (60–120× faster). In the real-world use cases, we first show the consistency of motif_prob with MoSDi, and then how the p-value quantification is crucial for enrichment quantification when bacteria have different GC content, using motifs found in antimicrobial resistance genes. The software and the code sources are available under the MIT license at https://github.com/DataIntellSystLab/motif_prob.ConclusionsThe motif_prob software is a multi-platform and efficient open source solution for calculating exact frequency distributions of motifs. It can be integrated with motif discovery/characterization tools for quantifying enrichment and deviation from expected frequency ranges with exact p values, without loss in data processing efficiency.

De-biasing strategic communication

This paper studies the effect of disclosing conflicts of interest on strategic communication when the sender has lying costs. I present a simple economic mechanism under which such disclosure often leads to more informative and, at the same time, also to more biased messages. This benefits rational receivers but exerts a negative externality from them on naive or delegating receivers. Disclosure is thus not a Pareto-improvement among receivers. I identify general conditions of the information structure under which this effect manifests and show that whenever it does, full disclosure is socially inefficient. These results hold independently of the degree of the receivers' risk-aversion and for an arbitrary precision of the disclosure statement.

Exponential Error Suppression for Near-Term Quantum Devices

As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level $E$ at a cost in qubit-count $n$ that is merely poly-logarithmic in $1/E$. However in the NISQ era, the complexity and scale required to adopt even the smallest QEC is prohibitive. Instead, error mitigation techniques have been employed; typically these do not require an increase in qubit-count but cannot provide exponential error suppression. Here we show that, for the crucial case of estimating expectation values of observables (key to almost all NISQ algorithms) one can indeed achieve an effective exponential suppression. We introduce the Error Suppression by Derangement (ESD) approach: by increasing the qubit count by a factor of $n\geq 2$, the error is suppressed exponentially as $Q^n$ where $Q<1$ is a suppression factor that depends on the entropy of the errors. The ESD approach takes $n$ independently-prepared circuit outputs and applies a controlled derangement operator to create a state whose symmetries prevent erroneous states from contributing to expected values. The approach is therefore `NISQ-friendly' as it is modular in the main computation and requires only a shallow circuit that bridges the $n$ copies immediately prior to measurement. Imperfections in our derangement circuit do degrade performance and therefore we propose an approach to mitigate this effect to arbitrary precision due to the remarkable properties of derangements. a) they decompose into a linear number of elementary gates -- limiting the impact of noise b) they are highly resilient to noise and the effect of imperfections on them is (almost) trivial. In numerical simulations validating our approach we confirm error suppression below $10^{-6}$ for circuits consisting of several hundred noisy gates (two-qubit gate error $0.5\%$) using no more than $n=4$ circuit copies.

Deep neural network for detecting arbitrary precision peptide features through attention based segmentation

A promising technique of discovering disease biomarkers is to measure the relative protein abundance in multiple biofluid samples through liquid chromatography with tandem mass spectrometry (LC-MS/MS) based quantitative proteomics. The key step involves peptide feature detection in the LC-MS map, along with its charge and intensity. Existing heuristic algorithms suffer from inaccurate parameters and human errors. As a solution, we propose PointIso, the first point cloud based arbitrary-precision deep learning network to address this problem. It consists of attention based scanning step for segmenting the multi-isotopic pattern of 3D peptide features along with the charge, and a sequence classification step for grouping those isotopes into potential peptide features. PointIso achieves 98% detection of high-quality MS/MS identified peptide features in a benchmark dataset. Next, the model is adapted for handling the additional ‘ion mobility’ dimension and achieves 4% higher detection than existing algorithms on the human proteome dataset. Besides contributing to the proteomics study, our novel segmentation technique should serve the general object detection domain as well.

Implementation of high-precision computation capabilities into the open-source dynamic simulation framework YADE

This paper deals with the implementation of arbitrary precision calculations into the open-source discrete element framework YADE published under the GPL-2+ free software license. This new capability paves the way for the simulation framework to be used in many new fields such as quantum mechanics. The implementation details and associated gains in the accuracy of the results are discussed. Besides the “standard” double (64 bits) type, support for the following high-precision types is added: long double (80 bits), float128 (128 bits), mpfr_float_backend (arbitrary precision) and cpp_bin_float (arbitrary precision). Benchmarks are performed to quantify the additional computational cost involved with the new supported precisions. Finally, a simple calculation of a chaotic triple pendulum is performed to demonstrate the new capabilities and the effect of different precisions on the simulation result.

Interpreting uninterpretable predictors: kernel methods, Shtarkov solutions, and random forests

Many of the best predictors for complex problems are typically regarded as hard to interpret physically. These include kernel methods, Shtarkov solutions, and random forests. We show that, despite the inability to interpret these three predictors to infinite precision, they can be asymptotically approximated and admit conceptual interpretations in terms of their mathematical/statistical properties. The resulting expressions can be in terms of polynomials, basis elements, or other functions that an analyst may regard as interpretable.

Current calculation of irreversible electrode reaction mechanismsin linear sweep voltammetry

Application of exponential infinite series gives highly accurate analytical solution contributing to the theory of linear sweep voltammetry for single scan experiments. We have calculated theoretical dimensionless current function (usually denoted as π1/2χ(bt)) at relevant potentials for irreversible charge transfer without a coupled chemical reaction. For this purpose several transformation techniques were used, which convert the derived infinite series into summable sequences. Since infinite series of further electrochemical mechanisms with irreversible electrode reaction have similar features (particularly those comprising preceding and catalytic chemical reaction), the same approach can be successfully applied also for further electrochemical mechanisms. The respective infinite series are divergent in the most important potential region at and after voltammetric peak therefore their transformation by Epsilon and Levin transform techniques was used. Necessary arbitrary precision arithmetic (APA) was implemented by UBASIC. The results were compared to the customary solution of Nicholson and Shain, who computed the current-potential curves by means of numerical solution of the integral equations but with a much lower precision. Our results were obtained in a broad potential range including the potential regions where the series are divergent. Obtained current functions are precise to 12 valid decimal numbers, which is utilizable for evaluation of the results achieved by various faster but less precise digital simulation techniques.

On the Numerical Solution of the Near Field Refractor Problem

A numerical scheme is presented to solve the one source near field refractor problem to arbitrary precision and it is proved that for a given error, the scheme terminates in a finite number of iterations. The convergence of the algorithm depends upon proving appropriate Lipschitz estimates for the refractor measure. The algorithm is presented in general terms and has independent interest.

Invariance properties of exact solutions of the radiative transfer equation

In this work, special invariance properties of a class of exact solutions of the radiative transfer equation (RTE) pertaining to a uniform Lambertian illumination of any non-absorbing homogeneous and inhomogeneous volume are presented and discussed. This class of solutions of the RTE traces a reference ground under which light propagation can be studied in a special simplified regime. Despite the difficulties to obtain general solutions of the radiative transfer equation, the condition of Lambertian illumination determines a unique regime of photon transport where quite easy and simple invariant solutions can be obtained in all generality for homogeneous and inhomogeneous geometries. These solutions are invariant both with respect to the geometry (size and shape of the volume) and with respect to the scattering properties, i.e. scattering coefficient, scattering function and homogeneity of the considered domain. Another evident advantage of these solutions is that they are exact solutions known with arbitrary precision and can thus be used as reference standard for photon migration studies.

Privacy Preserving Classification of EEG Data Using Machine Learning and Homomorphic Encryption

Data privacy is a major concern when accessing and processing sensitive medical data. A promising approach among privacy-preserving techniques is homomorphic encryption (HE), which allows for computations to be performed on encrypted data. Currently, HE still faces practical limitations related to high computational complexity, noise accumulation, and sole applicability the at bit or small integer values level. We propose herein an encoding method that enables typical HE schemes to operate on real-valued numbers of arbitrary precision and size. The approach is evaluated on two real-world scenarios relying on EEG signals: seizure detection and prediction of predisposition to alcoholism. A supervised machine learning-based approach is formulated, and training is performed using a direct (non-iterative) fitting method that requires a fixed and deterministic number of steps. Experiments on synthetic data of varying size and complexity are performed to determine the impact on runtime and error accumulation. The computational time for training the models increases but remains manageable, while the inference time remains in the order of milliseconds. The prediction performance of the models operating on encoded and encrypted data is comparable to that of standard models operating on plaintext data.

Complete-active-space extended Koopmans theorem method

The complete-active-space (CAS) extended Koopmans theorem (EKT) method is defined as a special case of the EKT in which the reference state is a CAS configuration interaction (CI) expansion and the electron removal operator acts only on the active orbitals. With these restrictions, the EKT is equivalent to the CI procedure involving all hole-state configurations derived from the active space of the reference wavefunction and has properties analogous to those of the original Koopmans theorem. The equivalence is used to demonstrate in a transparent manner that the first ionization energy predicted by the EKT is in general not exact, i.e., not equal to the difference between the full CI energies of the neutral and the ion, but can approach the full CI result with arbitrary precision even within a finite basis set. The findings also reconcile various statements about the EKT found in the literature.

Analytic integration of the Newton potential over cuboids and an application to fast multipole methods

Abstract We present simplified formulae for the analytic integration of the Newton potential of polynomials over boxes in two- and three-dimensional space. These are implemented in an easy-to-use C++ library that allows computations in arbitrary precision arithmetic which is also documented here. We describe how these results can be combined with fast multipole methods to evaluate the Newton potential of more general, non-polynomial densities.

Approximation Properties of Solutions to Multipoint Boundary-Value Problems

We consider a wide class of linear boundary-value problems for systems of $m$ ordinary differential equations of order $r$, known as general boundary-value problems. Their solutions $y:[a,b]\to \mathbb{C}^{m}$ belong to the Sobolev space $(W_1^{r})^m$, and the boundary conditions are given in the form $By=q$ where $B:(C^{(r-1)})^{m}\to\mathbb{C}^{rm}$ is an arbitrary continuous linear operator. We prove that a solution to such a problem can be approximated with an arbitrary precision in $(W_1^{r})^m$ by solutions to multipoint boundary-value problems with the same right-hand sides. These multipoint problems are built explicitly and do not depend on the right-hand sides of the general boundary-value problem. For these problems, we obtain estimates of errors of solutions in the normed spaces $(W_1^{r})^m$ and $(C^{(r-1)})^{m}$.