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Walter Rudin meets Elias M. Stein
Get PDFWalter Rudin and Elias M. Stein were giants in the world of mathemat-
ics. They were loved and admired from students and researchers to teachers
and academics, both young and old. They touched many of us through their
inspiring books at the undergraduate and postgraduate level. Although they
were leading researchers in both harmonic analysis and several complex vari-
ables, we are not aware whether they interacted and discussed mathematics. In
this article, Rudin and Stein meet mathematically through a reformulation of
the beautiful theory of Fourier series with gaps that Rudin developed in the
1950s as an equivalent Fourier restriction problem from the 1970s, a problem
Stein proposed and which remains a fundamental, central problem in Euclidean
harmonic analysis today.
Walter Rudin was born in Vienna on 2 May, 1921 and emigrated to the US
in 1945, completing his PhD at Duke University in 1949. While a C. L. E.
Moore Instructor at MIT in the early 1950s, Walter was asked to teach a real
analysis course but he could not find a textbook that he liked so he decided to
write Principles of Mathematical Analysis which despite its age, has remained
the paragon of high quality. After a stint of teaching at the University of
Rochester, he took up a position at the University of Wisconsin, Madison in
1959 where he remained until his retirement as Vilas Professor in 1991. He died
at his home in Madison on 20 May, 2010.
Elias M. Stein (known to friends and colleagues as Eli) was born in Antwerp
on 13 January, 1931 and emigrated with his family to the US in 1941, settling in
New York where Eli attended high school. He went to the University of Chicago,
received his PhD in 1955, and then went to MIT as a C.L.E. Moore Instructor
before Antoni Zygmund told Eli “it’s time to return to Chicago.” In 1963, Stein
moved to Princeton University as a full professor where he remained until he
died on 23 December, 2018.
Between 2003 and 2011, Eli expanded the presentation of Walter’s Principles
and published a series of four books aimed at advanced undergraduates. This
series is quickly becoming an important part of any young analyst’s education.However the majority of books written by Rudin and Stein are postgraduate
textbooks and research monographs (too many to list here), mainly in the areas
of harmonic analysis and several complex variables where both men were central
figures.
In this article, these two luminaries meet in the world of mathematical anal-
ysis. We look back at some important work Rudin did in the 1950s and recast
it in terms of a far-reaching problem from the 1970s that Stein gave us
Layer-Wise Learning Framework for Efficient DNN Deployment in Biomedical Wearable Systems
Get PDFThe development of low-power wearable systems requires specialized techniques to accommodate their unique requirements and constraints. While significant advancements have been made in the inference phase of artificial intelligence, the training phase remains a challenge, particularly for biomedical wearable systems. Traditional training algorithms might not be suitable for these applications due to the substantial memory requirements and high computational costs associated with processing the large number of bits involved in neural network operations. In this paper, we introduce a novel learning procedure specifically designed for low-power wearable systems, dubbed Bio-BPfree (deep neural network training without backpropagation for low-power wearable systems). Using a two-class classification task, Bio-BPfree replaces conventional forward and backward backpropagation passes with four forward passes, two for data of the positive class and two for data of the negative class. Each layer is equipped with a unique objective function aimed at minimizing the distance between data points within the same class while maximizing the distance between data points from different classes. Our experimental results, which were obtained by conducting rigorous evaluations on the MIT-BIH dataset that features electrocardiogram (ECG) signals, effectively demonstrate the superior performance and suitability of Bio-BPfree for two-class classification tasks, particularly within the challenging environment of low-power wearable systems designed for continuous health monitoring and assessment.RYC2021-032853-
Robust Estimation of Distribution Algorithms via Fitness Landscape Analysis for Optimal Low-Thrust Orbital Maneuvers
Get PDFOne particular kind of evolutionary algorithms known as Estimation of Distribution Algorithms (EDAs) has gained the attention of the aerospace industry for its ability to solve nonlinear and complicated problems, particularly in the optimization of space trajectories during on-orbit operations of satellites. This article describes an effective method for optimizing the trajectory of a spacecraft using an evolutionary approach based on EDAs, incorporated with fitness landscape analysis (FLA). The approach utilizes flexible operators that are paired with seeding and selection mechanisms of EDAs. Initially, the orbit transfer problem is mathematically modeled and the objectives and constraints are identified. The landscape feature of the search space is analyzed via the dispersion metric to measure the modality and ruggedness of the search domain. The obtained information are used as feedback in developing adaptive operators for truncation factor and constraints separation threshold of the employed EDA. A framework for spacecraft trajectory optimization has been presented where the dispersion value for a space mission is estimated using a k-nearest neighbors (k-NN) algorithm. The suggested method is used to solve several problems related to low-thrust orbit transfer of satellites in Earth’s orbit. Results demonstrate that the suggested framework for trajectory design and optimization of space transfers is effective enough to offer fuel-efficient and energy-efficient maneuvers for different thrust levels of the propulsion system. Moreover, the performance of the proposed approach is evaluated against non-adaptive EDA and other advanced evolutionary algorithms. The obtained results certify that the proposed adaptive evolutionary approach is superior in identifying feasible minimum-fuel and minimum-energy transfer trajectories.BEAZ Bizkaia, 3/12/DP/2021/00150;
SPRI Group, Ekintzaile Program EK-00112-202
Minimum-Fuel Low-Thrust Trajectory Optimization Via a Direct Adaptive Evolutionary Approach
Get PDFSpace missions with low-thrust propulsion systems are of appreciable interest to space agencies because of their practicality due to higher specific impulses. This research proposes a technique to the solution of minimum-fuel non-coplanar orbit transfer problem. A direct adaptive method via Fitness Landscape Analysis (FLA) is coupled with a constrained evolutionary technique to explore the solution space for designing low-thrust orbit transfer trajectories. Taking advantage of the solution for multi-impulse orbit transfer problem, and parameterization of thrust vector, the orbital maneuver is transformed into a constrained continuous optimization problem. A constrained Estimation of Distribution Algorithms (EDA) is utilized to discover optimal transfer trajectories, while maintaining feasibility of the solutions. The low-thrust trajectory optimization problem is characterized via three parameters, referred to as problem identifiers, and the dispersion metric is utilized for analyzing the complexity of the solution domain. Two adaptive operators including the kernel density and outlier detection distance threshold within the framework of the employed EDA are developed, which work based on the landscape feature of the orbit transfer problem. Simulations are proposed to validate the efficacy of the proposed methodology in comparison to the non-adaptive approach. Results indicate that the adaptive approach possesses more feasibility ratio and higher optimality of the obtained solutions.BEAZ Bizkaia, 3/12/DP/2021/00150;
SPRI Group, Ekintzaile Program EK-00112-202
Pointwise localization and sharp weighted bounds for Rubio de Francia square functions
Get PDFLet be the Fourier restriction of ∈ 2 (R) to an interval ⊂ R. If Ω is an arbitrary collection of pairwise disjoint intervals, the square function of { : ∈ Ω} is termed the Rubio de Francia square function Ω . This article proves a pointwise bound for Ω by a sparse operator involving local 2-averages. A pointwise bound for the smooth version of Ω by a sparse square function is also proved. These pointwise localization principles lead toquantified(), >2andweak(), ≥2norminequalitiesforΩ.Inparticular,the obtained weak ( ) norm bounds are new for ≥ 2 and sharp for > 2. The proofs rely on sparse bounds for abstract balayages of Carleson sequences, local orthogonality and very elementary time-frequency analysis techniques.
The paper also contains two results related to the outstanding conjecture that Ω is bounded on 2() if and only if ∈ 1. The conjecture is verified for radially decreasing even 1 weights, and in full generality for the Walsh group analogue of Ω
Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of phi-functions of matrices via quadrature
Get PDFIn this article, we propose an algorithm for approximating the action of -functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the -functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art
Automated location of orofacial landmarks to characterize airway morphology in anaesthesia via deep convolutional neural networks
Get PDFBackground:A reliable anticipation of a difficult airway may notably enhance safety during anaesthesia. In current practice, clinicians use bedside screenings by manual measurements of patients’ morphology.
Objective:To develop and evaluate algorithms for the automated extraction of orofacial landmarks, which characterize airway morphology.
Methods:We defined 27 frontal + 13 lateral landmarks. We collected n=317 pairs of pre-surgery photos from patients undergoing general anaesthesia (140 females, 177 males). As ground truth reference for supervised learning, landmarks were independently annotated by two anaesthesiologists.
We trained two ad-hoc deep convolutional neural network architectures based on InceptionResNetV2 (IRNet) and MobileNetV2 (MNet), to predict simultaneously: (a) whether each landmark is visible or not (occluded, out of frame), (b) its 2D-coordinates (x, y). We implemented successive stages of transfer learning, combined with data augmentation. We added custom top layers on top of these networks, whose weights were fully tuned for our application. Performance in landmark extraction was evaluated by 10-fold cross-validation (CV) and compared against 5 state-of-the-art deformable models.
Results:With annotators’ consensus as the ‘gold standard’, our IRNet-based network performed comparably to humans in the frontal view: median CV loss L=1.277·10-3, inter-quartile range (IQR) [1.001, 1.660]; versus median 1.360, IQR [1.172, 1.651], and median 1.352, IQR [1.172, 1.619], for each annotator against consensus, respectively. MNet yielded slightly worse results: median 1.471, IQR [1.139, 1.982].
In the lateral view, both networks attained performances statistically poorer than humans: median CV loss L=2.141·10-3, IQR [1.676, 2.915], and median 2.611, IQR [1.898, 3.535], respectively; versus median 1.507, IQR [1.188, 1.988], and median 1.442, IQR [1.147, 2.010] for both annotators. However, standardized effect sizes in CV loss were small: 0.0322 and 0.0235 (non-significant) for IRNet, 0.1431 and 0.1518 (p<0.05) for MNet; therefore quantitatively similar to humans.
The best performing state-of-the-art model (a deformable regularized Supervised Descent Method, SDM) behaved comparably to our DCNNs in the frontal scenario, but notoriously worse in the lateral view.
Conclusions:We successfully trained two DCNN models for the recognition of 27 + 13 orofacial landmarks pertaining to the airway. Using transfer learning and data augmentation, they were able to generalize without overfitting, reaching expert-like performances in CV. Our IRNet-based methodology achieved a satisfactory identification and location of landmarks: particularly in the frontal view, at the level of anaesthesiologists. In the lateral view, its performance decayed, although with a non-significant effect size. Independent authors had also reported lower lateral performances; as certain landmarks may not be clear salient points, even for a trained human eye.BERC.2022-2025
BCAM Severo Ochoa accreditation CEX2021-001142-S / MICIN / AEI / 10.13039/50110001103
Wiener-Hopf Integral Equations in Mean First-passage Time Problems for Continuous-time Random Walks
Get PDFWe study the mean first-passage time (MFPT) for asymmetric continuous time random walks in continuous space characterised by finite mean waiting times and jump amplitudes with both finite average and finite variance. We derive an inhomogeneous Wiener-Hopf integral equation that allows the exact estimation of the MFPT, which depends on the whole distribution of the jump amplitudes, but on the average of the waiting times only. Thus, our findings hold for general non-Markovian processes, since Markovianity emerges solely with an exponential distribution of the waiting times. Through the paradigmatic case study of a
general class of asymmetric distributions of the jump-amplitudes that is exponential towards the boundary and arbitrary in the opposite direction, we show, that only the average of the jump amplitudes in the opposite direction of the boundary contributes to the MFPT. Moreover, we determine a length-scale, which depends only of the distribution of jumps in the direction of the boundary, such that for initial positions close to the boundary the MFPT depends on the specific whole distribution of jump amplitudes, in opposition to the appearing universality for initial positions far away from the boundary.PRE2018-08442
Estimation of the ROC curve and the area under it with complex survey data
Get PDFLogistic regression models are widely applied in daily practice. Hence, it is necessary to ensure they have an adequate predictive performance, which is usually estimated by means of the receiver operating characteristic (ROC) curve and the area under it (area under the curve [AUC]). Traditional estimators of these parameters are thought to be applied to simple random samples but are not appropriate for complex survey data. The goal of this work is to propose new weighted estimators for the ROC curve and AUC based on sampling weights which, in the context of complex survey data, indicate the number of units that each sampled observation represents in the population. The behaviour of the proposed estimators is evaluated and compared with the traditional unweighted ones by means of a simulation study. Finally, weighted and unweighted ROC curve and AUC estimators are applied to real survey data in order to compare the estimates in a real scenario. The results suggest the use of the weighted estimators proposed in this work in order to obtain unbiassed estimates for the ROC curve and AUC of logistic regression models fitted to complex survey data
Transformation of the Gibbs measure of the cubic NLS and fractional NLS under an approximated Birkhoff map
No full textWe study the Gibbs measure associated to the periodic cubic nonlinear Schrödinger equation. We establish a change of variable formula for this measure under the first step of the Birkhoff normal form reduction. We also consider the case of fractional dispersion