Explicit Error Research Articles

An effective path planning approach for robot welding considering redundant kinematics

Robot welding with redundant kinematics demands precise coordination of the welding path and redundant axis motion for stable operation. This paper introduces an effective G3 continuous path planning approach for a 6-DOF friction stir welding robot, integrating redundant axis motion to enhance welding stability and quality. The method encompasses a two-pronged strategy: a five-axis path corner smoothing technique and a motion synchronization strategy. Initially, a robust toolpath preprocessing algorithm is presented to mitigate curvature extremities of the inserted smoothing curves within predefined error limits. Subsequently, leveraging G3 continuity criteria, a B-spline-based smoothing method is advanced for refining tool position and orientation paths, with an analytical determination of spline control points facilitated by an explicit smoothing error expression. An effective motion synchronization technique is then proposed, which formulates the tool position, tool orientation, and redundant axis path as explicit functions of the position path arc length. The effectiveness of the proposed method is demonstrated through simulations and experiments on a robot welding platform, with the integration of a jerk-continuous feedrate profile for smooth motion execution. The findings indicate a significant improvement in robotic welding quality by integrating the proposed path planning method with existing workpiece posture optimization techniques.

Approximation of optimal feedback controls for stochastic reaction-diffusion equations

In this paper we present a method to approximate optimal feedback controls for stochastic reaction-diffusion equations. We derive two approximation results providing the theoretical foundation of our approach and allowing for explicit error estimates. The approximation of optimal feedback controls by neural networks is discussed as an explicit application of our method. We illustrate our findings in the case of a linear quadratic control problem with a numerical example.

Error estimates of effective boundary conditions for the heat equation with optimally aligned coatings

We are interested in the validity of effective boundary conditions for a heat equation on a coated body as the thickness of the coating shrinks to zero. The coating is optimally aligned in the sense that the normal vector in the coating is an eigenvector of the thermal tensor. If the heat equation satisfies Neumann boundary condition on the outer boundary of the coating, Chen et al. (Arch. Ration. Mech. Anal. 206 (2012) 911-951) derived the complete list of effective boundary conditions satisfied by the limiting model. In this paper we provide explicit error estimates between the full model and the effective model. Moreover, our error estimates are independent of time, which shows that the maximal time interval in which the effective boundary conditions remain valid are infinite. The proof is based on H2 estimates for solutions of the full model, characterization of large time behaviors for solutions of the effective model, and interaction estimates between the two models.

An 88 dB SNDR 100 kHz BW Sturdy MASH Delta-Sigma Modulator Using Self-Cascoded Floating Inverter Amplifiers

Battery-powered Internet-of-Things applications require high-resolution, energy-efficient analog-to-digital converters (ADCs). There are still limited works on sub-MHz-bandwidth ADC designs. This paper presents a sturdy multi-stage shaping (SMASH) discrete-time (DT) delta-sigma modulator (DSM) structure using a self-cascoded floating-inverter-based dynamic amplifier (FIA). The proposed structure removes the explicit quantization error extraction of the first loop and all the feedback DACs in the cascaded loop, decreasing the design complexity of the circuit. This enables the proposed DT DSM to operate at a higher speed, which is suitable for achieving high-order noise at a low oversampling ratio (OSR). The proposed self-cascoded FIA is more power-efficient and can acquire more than 45 dB DC gain under a 1.2 V supply. The DT DSM implemented in a piece of 55 nm CMOS technology measures an 88.0 dB peak signal-to-noise-and-distortion ratio (SNDR) in a 100 kHz bandwidth (BW) and an 85.3 dB dynamic range (DR), consuming 249.1 μW from a 1.2 V supply at 10 MS/s. The obtained 174.0 dB SNDR-based Schreier figure-of-merit (FoMs) is competitive within state-of-art high-resolution (SNDR > 85 dB) and general-purpose (sub-MHz-bandwidth) ΔΣ ADCs.

Optimally truncated WKB approximation for the 1D stationary Schrödinger equation in the highly oscillatory regime

This paper is dedicated to the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the highly oscillatory regime. We compute an approximate solution based on the well-known WKB-ansatz, which relies on an asymptotic expansion w.r.t. the small parameter ɛ. Assuming that the coefficient in the equation is analytic, we derive an explicit error estimate for the truncated WKB series, in terms of ɛ and the truncation order N. For any fixed ɛ, this allows to determine the optimal truncation order Nopt which turns out to be proportional to ɛ−1. When chosen this way, the resulting error of the optimally truncated WKB series behaves like O(exp(−r/ɛ)), with some parameter r>0. The theoretical results established in this paper are confirmed by several numerical examples.

On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields

Deep Learning is having a remarkable impact on the design of Reduced Order Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited as a powerful tool for tackling complex problems for which classical methods might fail. In this respect, deep autoencoders play a fundamental role, as they provide an extremely flexible tool for reducing the dimensionality of a given problem by leveraging on the nonlinear capabilities of neural networks. Indeed, starting from this paradigm, several successful approaches have already been developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs). Nevertheless, when it comes to stochastic problems parameterized by random fields, the current understanding of DL-ROMs is mostly based on empirical evidence: in fact, their theoretical analysis is currently limited to the case of PDEs depending on a finite number of (deterministic) parameters. The purpose of this work is to extend the existing literature by providing some theoretical insights about the use of DL-ROMs in the presence of stochasticity generated by random fields. In particular, we derive explicit error bounds that can guide domain practitioners when choosing the latent dimension of deep autoencoders. We evaluate the practical usefulness of our theory by means of numerical experiments, showing how our analysis can significantly impact the performance of DL-ROMs.

Interplay between depth and width for interpolation in neural ODEs

Neural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width p and the number of transitions between layers L (corresponding to a depth of L+1). Specifically, we construct explicit controls interpolating either a finite dataset D, comprising N pairs of points in Rd, or two probability measures within a Wasserstein error margin ɛ>0. Our findings reveal a balancing trade-off between p and L, with L scaling as 1+O(N/p) for data interpolation, and as 1+Op−1+(1+p)−1ɛ−d for measures.In the high-dimensional and wide setting where d,p>N, our result can be refined to achieve L=0. This naturally raises the problem of data interpolation in the autonomous regime, characterized by L=0. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when p=N we develop an inductive control strategy based on a separability assumption whose probability increases with d. In the second one, we establish an explicit error decay rate with respect to p which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating D.

Quantum advantage and stability to errors in analogue quantum simulators

Several quantum hardware platforms, while being unable to perform fully fault-tolerant quantum computation, can still be operated as analogue quantum simulators for addressing many-body problems. However, due to the presence of errors, it is not clear to what extent those devices can provide us with an advantage with respect to classical computers. In this work, we make progress on this problem for noisy analogue quantum simulators computing physically relevant properties of many-body systems both in equilibrium and undergoing dynamics. We first formulate a system-size independent notion of stability against extensive errors, which we prove for Gaussian fermion models, as well as for a restricted class of spin systems. Remarkably, for the Gaussian fermion models, our analysis shows the stability of critical models which have long-range correlations. Furthermore, we analyze how this stability may lead to a quantum advantage, for the problem of computing the thermodynamic limit of many-body models, in the presence of a constant error rate and without any explicit error correction.

Effective estimates of ergodic quantities illustrated on the Bolyai-Rényi map

We present a practical and effective method for rigorously estimating quantities associated to top eigenvalues of transfer operators to very high precision. The method combines explicit error bounds of the Lagrange-Chebyshev approximation with an established min-max method. We illustrate its applicability by significantly improving rigorous estimates on various ergodic quantities associated to the Bolyai–Rényi map.

Differentially Private Synthetic Control

Synthetic control is a causal inference tool used to estimate the treatment effects of an intervention by creating synthetic counterfactual data. This approach combines measurements from other similar observations (i.e., donor pool) to predict a counterfactual time series of interest (i.e., target unit) by analyzing the relationship between the target and the donor pool before the intervention. As synthetic control tools are increasingly applied to sensitive or proprietary data, formal privacy protections are often required. In this work, we provide the first algorithms for differentially private synthetic control with explicit error bounds. Our approach builds upon tools from non-private synthetic control and differentially private empirical risk minimization. We provide upper and lower bounds on the sensitivity of the synthetic control query and provide explicit error bounds on the accuracy of our private synthetic control algorithms. We show that our algorithms produce accurate predictions for the target unit and that the cost of privacy is small. Finally, we empirically evaluate the performance of our algorithm, and show favorable performance in a variety of parameter regimes, as well as provide guidance to practitioners for hyperparameter tuning.

Markov processes on quasi-random graphs

We study Markov population processes on large graphs, with the local state transition rates of a single vertex being a linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. An explicit error bound is given and it is 1N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{\\sqrt{N}}$$\\end{document} plus the largest discrepancy of the graph. For Erdős–Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process.

Forward layer-wise learning of convolutional neural networks through separation index maximizing

This paper proposes a forward layer-wise learning algorithm for CNNs in classification problems. The algorithm utilizes the Separation Index (SI) as a supervised complexity measure to evaluate and train each layer in a forward manner. The proposed method explains that gradually increasing the SI through layers reduces the input data’s uncertainties and disturbances, achieving a better feature space representation. Hence, by approximating the SI with a variant of local triplet loss at each layer, a gradient-based learning algorithm is suggested to maximize it. Inspired by the NGRAD (Neural Gradient Representation by Activity Differences) hypothesis, the proposed algorithm operates in a forward manner without explicit error information from the last layer. The algorithm’s performance is evaluated on image classification tasks using VGG16, VGG19, AlexNet, and LeNet architectures with CIFAR-10, CIFAR-100, Raabin-WBC, and Fashion-MNIST datasets. Additionally, the experiments are applied to text classification tasks using the DBPedia and AG’s News datasets. The results demonstrate that the proposed layer-wise learning algorithm outperforms state-of-the-art methods in accuracy and time complexity.

A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theorem

We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations. Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space. The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation. We establish the convergence of the scheme and determine the convergence rate with an explicit error bound, using the comparison principles for both the scheme and the equation together with a mollification procedure. The first application is obtaining the convergence rate of Peng's robust central limit theorem with an explicit bound of Berry-Esseen type. The second application is an approximation scheme with its convergence rate for the Black-Scholes-Barenblatt equation.

Improving Factual Error Correction by Learning to Inject Factual Errors

Factual error correction (FEC) aims to revise factual errors in false claims with minimal editing, making them faithful to the provided evidence. This task is crucial for alleviating the hallucination problem encountered by large language models. Given the lack of paired data (i.e., false claims and their corresponding correct claims), existing methods typically adopt the ‘mask-then-correct’ paradigm. This paradigm relies solely on unpaired false claims and correct claims, thus being referred to as distantly supervised methods. These methods require a masker to explicitly identify factual errors within false claims before revising with a corrector. However, the absence of paired data to train the masker makes accurately pinpointing factual errors within claims challenging. To mitigate this, we propose to improve FEC by Learning to Inject Factual Errors (LIFE), a three-step distantly supervised method: ‘mask-corrupt-correct’. Specifically, we first train a corruptor using the ‘mask-then-corrupt’ procedure, allowing it to deliberately introduce factual errors into correct text. The corruptor is then applied to correct claims, generating a substantial amount of paired data. After that, we filter out low-quality data, and use the remaining data to train a corrector. Notably, our corrector does not require a masker, thus circumventing the bottleneck associated with explicit factual error identification. Our experiments on a public dataset verify the effectiveness of LIFE in two key aspects: Firstly, it outperforms the previous best-performing distantly supervised method by a notable margin of 10.59 points in SARI Final (19.3% improvement). Secondly, even compared to ChatGPT prompted with in-context examples, LIFE achieves a superiority of 7.16 points in SARI Final.

On the Hill Discriminant of Lamé's Differential Equation

Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function ${\rm sn}$ depending on the modulus $k$, and two additional parameters $h$ and $\nu$. This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lamé's equations is determined by the value of its Hill discriminant $D(h,\nu,k)$. The Hill discriminant is compared to an explicitly known quantity including explicit error bounds. This result is derived from the observation that Lamé's equation with $k=1$ can be solved by hypergeometric functions because then the elliptic function ${\rm sn}$ reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of $D(h,\nu,k)$ when the modulus $k$ tends to $1$.

Discrepancy of rational points in simple algebraic groups

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

Higher order Turán inequalities for the distinct partition function

We prove that the number q(n) of partitions of n with distinct parts is log-concave for n≥33 and satisfies the third-order Turán inequalities for n≥121 conjectured by Craig and Pun. In doing so, we establish explicit error terms for q(n) and for q(n−1)q(n+1)/q(n)2 based on Chern's asymptotic formulas for η-quotients.

Role of ultrasensitivity in biomolecular circuitry for achieving homeostasis

Living systems have developed control mechanisms for achieving homeostasis. Here, we propose a plausible biological feedback architecture that exploits ultrasensitivity and shows adaptive responses without requiring error detection mechanism (i.e., by measuring an external reference signal and deviation from this). While standard engineering control systems are usually based on error measurements, this is not the case for biological systems. We find that a two-state negative feedback control system, without explicit error measurements, is able to track a reference signal that is implicitly determined by the tunable threshold and slope characterizing the sigmoidal ultrasensitive relationship implemented by the control system. We design different ultrasensitive control functions (ultrasensitive up- or down-regulation, or both) and, by performing sensitivity analysis, show that increasing the sensitivity level of the control allows achieving robust adaptive responses to the effects of parameter variations and step disturbances. Finally, we show that the devised control system architecture without error detection is implemented within the yeast osmoregulatory response network and allows achieving adaptive responses to osmotic stress, by exploiting the ubiquitous ultrasensitive features of the involved biomolecular circuitry.

Feedback Mechanisms of DepEd English Teachers in Modular Distance Learning: A Case Study

With the shift in the educational setting from traditional classrooms to online learning for most countries and modular distance learning in the Philippines, giving feedback to the students proves to be a challenge. Research conducted during the COVID-19 pandemic, whether online or through modular learning, only included feedback as a part of the problems encountered, not as the main aspect of the research. Thus, this study explored the nature of the feedback practices delivered by English DepEd teachers to their students through modular learning. Moreover, this is primarily concerned with the teachers' experiences and methods of developing feedback for their students in this modular distance learning scheme. The study employed a qualitative research design, utilizing a single explanatory case study. The study involved eight public English high school teachers and was conducted in two public secondary high schools in Leyte. The data was collected through an interview protocol, analysed using thematic analysis, and validated through triangulation. The findings of this study revealed that explicit error correction through annotation and positive feedback through commendation are the only feedback mechanisms/corrections used by the English DepEd teachers on their modular instruction. While these practices address some of the problems in English learning through modular distance learning, it is observed that they are severely limited by internal and external factors from both teachers and students. 
 
 Keywords: Feedback Practices; modular learning; teachers’ experiences and method; error correction; and positive feedback.

Optimal pointwise error estimates for piecewise functions expanded with Laguerre polynomials

This paper focuses on pointwise error estimates for piecewise functions expanded in the form of Laguerre polynomial series. By utilizing the reproducing kernel of Laguerre polynomials, we derive an explicit pointwise error formula, which enables the estimation of convergence orders of pointwise values for piecewise functions by using the Hilb-type formula and generalized van der Corput-type Lemmas. The presented convergence analyses illustrate that the piecewise functions exhibit local superconvergence away from the singular points. Plenty of numerical experiments are carried out to validate the accuracy of these theoretical results.