2063 research outputs found
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Normalized gradient flow optimization in the training of ReLU artificial neural networks
No full textThe training of artificial neural networks (ANNs) is nowadays a highly relevant
algorithmic procedure with many applications in science and industry. Roughly
speaking, ANNs can be regarded as iterated compositions between affine linear func-
tions and certain fixed nonlinear functions, which are usually multidimensional ver-
sions of a one-dimensional so-called activation function. The most popular choice of
such a one-dimensional activation function is the rectified linear unit (ReLU) activa-
tion function which maps a real number to its positive part R ∋ x 7→ max{x, 0} ∈ R.
In this article we propose and analyze a modified variant of the standard training
procedure of such ReLU ANNs in the sense that we propose to restrict the negative
gradient flow dynamics to a large submanifold of the ANN parameter space, which
is a strict C∞-submanifold of the entire ANN parameter space that seems to enjoy
better regularity properties than the entire ANN parameter space but which is also
sufficiently large and sufficiently high dimensional so that it can represent all ANN
realization functions that can be represented through the entire ANN parameter
space. In the special situation of shallow ANNs with just one-dimensional ANN lay-
ers we also prove for every Lipschitz continuous target function that every gradient
flow trajectory on this large submanifold of the ANN parameter space is globally
bounded. For the standard gradient flow on the entire ANN parameter space with
Lipschitz continuous target functions it remains an open problem of research to
prove or disprove the global boundedness of gradient flow trajectories even in the
situation of shallow ANNs with just one-dimensional ANN layers
Statistical Characterization of Wildfire Dynamics: Studying the Relation Between Burned Area and Head of the Fire
Get PDFWe show that the probability density function (PDF) of an area enclosed by a random perimeter is driven by the PDF of the integration bounds and the mean value of the perimeter function. With reference to wildfires, if the integration interval is aligned with the main direction of propagation, the PDF of the burned area is driven by the PDF of the position of the head of the fire times the mean value of the fire front. We show that if the random fire front perimeter is modelled as an ellipse-like curve with stochastic noise, the relation between the probability distribution of the burned area and the one of the head of the fire is linear, with the constant mean value of the perimeter as a factor. Different random noise models have been developed and implemented for this purpose. Different behaviours of the stochastic position of the head of the fire have been studied, including Viegas’ rate of spread model.
We show that for the realistic propagation model given by the operational wildfire cellular automata simulator Propagator [46], the PDF of the burned area is still driven by the position of the head of the fire and the mean value of the fire front perimeter, and the shape of the perimeter is not relevant. This has been shown for 5 case studies, ranging from easy ones to realistic complicated cases
Singular integrals along variable codimension one subspaces
Get PDFThis article deals with maximal operators on R formed by taking arbitrary rota-
tions of tensor products of a -dimensional Hörmander–Mihlin multiplier with the identity in
− coordinates, in the particular codimension 1 case = − 1. These maximal operators are
naturally connected to differentiation problems and maximally modulated singular integrals
such as Sjölin’s generalization of Carleson’s maximal operator. Our main result, a weak-type
2(R)-estimate on band-limited functions, leads to several corollaries. The first is a sharp
2(R) estimate for the maximal operator restricted to a finite set of rotations in terms of the
cardinality of the finite set. The second is a version of the Carleson–Sjölin theorem. In addi-
tion, we obtain that functions in the Besov space 0 (R), 2 ≤ < ∞, may be recovered from ,1
their averages along a measurable choice of codimension 1 subspaces, a form of Zygmund’s conjecture in general dimension
Are the statistical tests the best way to deal with the biomarker selection problem?
Get PDFStatistical tests are a powerful set of tools when applied correctly, but unfortunately the extended misuse of them has caused great concern. Among many other applications, they are used in the detection of biomarkers so as to use the resulting p-values as a reference with which the candidate biomarkers are ranked. Although statistical tests can be used to rank, they have not been designed for that use. Moreover, there is no need to compute any p-value to build a ranking of candidate biomarkers. Those two facts raise the question of whether or not alternative methods which are not based on the computation of statistical tests that match or improve their performances can be proposed. In this paper, we propose two alternative methods to statistical tests. In addition, we propose an evaluation framework to assess both statistical tests and alternative methods in terms of both the performance and the reproducibility. The results indicate that there are alternative methods that can match or surpass methods based on statistical tests in terms of the reproducibility when processing real data, while maintaining a similar performance when dealing with synthetic data. The main conclusion is that there is room for the proposal of such alternative methods
Deep learning enhanced principal component analysis for structural health monitoring
Get PDFThis paper proposes a Deep Learning enhanced Principal Component Analysis (PCA) approach for outlier detection to assess the structural condition of bridges. We employ a partially explainable autoencoder architecture to replicate and enhance the data compression and reconstruction ability of PCA. The particularity of the method lies in the addition of residual connections to account for nonlinearities. We apply the proposed method to monitoring data obtained from two bridges under real operation conditions and compare the results before and after adding the residual connections. Results show that the addition of residual connections enhances the outlier detection ability of the network, allowing to detect lighter damages.This work has received funding from: the European Union's Horizon 2020 research and innovation program under the grant agreement No 769373 (FORESEE project) and the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS); the Base Funding - UIDB/04708/2020 of the CONSTRUCT - Instituto de I\&D em Estruturas e Construções - funded by national funds through the FCT/MCTES (PIDDAC); the European Regional Development Fund (ERDF) through the Interreg V-A Spain-France-Andorra program POCTEFA 2014-2020 Project PIXIL (EFA362/19); the Spanish Ministry of Science and Innovation with references PID2019-108111RB-I00 MCIN/AEI/10.13039/501100011033 (FEDER/AEI) and the “BCAM Severo Ochoa” accreditation of excellence (SEV-2017-0718); and the Basque Government through the BERC 2018-2021 program, the two Elkartek projects 3KIA (KK-2020/00049) and MATHEO (KK-2019-00085), the grant "Artificial Intelligence in BCAM number EXP. 2019/00432", and the Consolidated Research Group MATHMODE (IT1294-19) given by the Department of Education
A Multivariate Time Series Streaming Classifier for Predicting Hard Drive Failures [Application Notes]
No full textDigital data storage systems such as hard drives can suffer breakdowns that cause the loss of stored data. Due to the cost of data and the damage that its loss entails, hard drive failure prediction is vital. In this context, the objective of this paper is to develop a method for detecting the beginning of hard drive malfunction using streaming SMART data, allowing the user to take actions before the breakdown occurs. This is a challenging task for two main reasons. First, there are not usually many examples of failed hard drives. Second, in these few available examples, hard drives are only identified and labeled as failed after complete breakdown occurs, but the exact moment when they begin to malfunction is usually unknown. Both these aspects significantly complicate the supervised learning of hard drive failure prediction models. To cope with these issues, the problem is addressed as a multidimensional time series streaming classification problem based on sliding windows. Moreover, as a solution to the highly imbalanced situation, the learned classifier is optimized to maximize the minimum recall of classes. Experimental results using the Backblaze benchmark dataset show that the proposed method reliably anticipates hard drive failures and obtains a higher balance between the recall values of both classes, failed and correct disks, compared to other state-of-the-art solutions.Digital KK-2019/00095
3KIA KK-2020/00049
IT1244-19
PID2019-104966GB-I0
Error representation of the time-marching DPG scheme
Get PDFIn this article, we introduce an error representation function to perform adaptivity in time of the recently developed time-marching Discontinuous Petrov–Galerkin (DPG) scheme. We first provide an analytical expression for the error that is the Riesz representation of the residual. Then, we approximate the error by enriching the test space in such a way that it contains the optimal test functions. The local error contributions can be efficiently computed by adding a few equations to the time-marching scheme. We analyze the quality of such approximation by constructing a Fortin operator and providing an a posteriori error estimate. The time-marching scheme proposed in this article provides an optimal solution along with a set of efficient and reliable local error contributions to perform adaptivity. We validate our method for both parabolic and hyperbolic problems
Deterministic and Stochastic Dynamics of COVID-19: The Case Study of Italy and Spain
Get PDFIn December 2019, a severe respiratory syndrome (COVID-19) caused by a new coronavirus (SARS-CoV-2) was identified in China and spread rapidly around the globe. COVID-19 was declared a pandemic by the World Health Organization (WHO) in March 2020. With eventually substantial global underestimation, more than 225 million cases were confirmed by the end of August 2021, counting more than 4.5 million deaths. COVID-19 symptoms range from mild (or no symptoms) to severe illness, with disease severity and death occurring according to a hierarchy of risks, with age and preexisting health conditions enhancing the risks of disease severity manifestation. In this paper, a mathematical model for COVID-19 transmission is proposed and analyzed. The model stratifies the studied population into two groups, older and younger. Applied to the COVID-19 outbreaks in Spain and in Italy, we find the disease-free equilibrium and the basic reproduction number for each case study. A sensitivity analysis to identify the key parameters which influence the basic reproduction number, and hence regulate the transmission dynamics of COVID-19, is also performed. Finally, the model is extended to its stochastic counterpart to encapsulate the variation or uncertainty found in the transmissibility of the disease. We observe the variability of the infectious population finding its distribution at a given time, demonstrating that for small populations, stochasticity will play an important role.Marie Skłodowska-Curie grant agreement No. 79249
Lower bounds on Hilbert-Kunz multiplicities and maximal F-signature
Get PDFABSTRACT. Hilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunz multiplicity.La Caixa Junior Leader Postdoctoral Fellowship: LCF/BQ/PI21/1183003