arXiv subjects

Gabriele Santin

Publications and source records attributed to Gabriele Santin.

At least 37 records · Page 2Linked to original sources

arXiv2023

Temporal clustering of social interactions trades-off disease spreading and knowledge diffusion

Non-pharmaceutical measures such as preventive quarantines, remote working, school and workplace closures, lockdowns, etc. have shown effectivenness from an epidemic control perspective; however they have also significant negative consequences on social life and relationships, work routines, and community engagement. In particular, complex ideas, work and school collaborations, innovative discoveries, and resilient norms formation and maintenance, which often require face-to-face interactions of two or more parties to be developed and synergically coordinated, are particularly affected. In this study, we propose an alternative hybrid solution that balances the slowdown of epidemic diffusion with the preservation of face-to-face interactions. Our approach involves a two-step partitioning of the population. First, we tune the level of node clustering, creating "social bubbles" with increased contacts within each bubble and fewer outside, while maintaining the average number of contacts in each network. Second, we tune the level of temporal clustering by pairing, for a certain time interval, nodes from specific social bubbles. Our results demonstrate that a hybrid approach can achieve better trade-offs between epidemic control and complex knowledge diffusion. The versatility of our model enables tuning and refining clustering levels to optimally achieve the desired trade-off, based on the potentially changing characteristics of a disease or knowledge diffusion process.

physics.soc-ph↗

arXiv2023

On the optimality of target-data-dependent kernel greedy interpolation in Sobolev Reproducing Kernel Hilbert Spaces

Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective, and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the $P$-greedy target-data-independent selection rule, and can additionally be proven to be optimal when they fully exploit adaptivity ($f$-greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in Reproducing Kernel Hilbert Spaces, as they allow us to compare adaptive interpolation with non-adaptive best nonlinear approximation.

math.NA↗

arXiv2023

Stability of convergence rates: Kernel interpolation on non-Lipschitz domains

Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces (RKHS) usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we leverage an analysis of greedy kernel algorithms to prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $Ω\subset \mathbb{R}^d$, thus allowing for non-Lipschitz domains including e.g. cusps and irregular boundaries. Especially we show that, when going to a smaller domain $\tildeΩ \subset Ω\subset \mathbb{R}^d$, the convergence rate does not deteriorate - i.e. the convergence rates are stable with respect to going to a subset. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness like the Gaussian kernel. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $Ω$ has an impact on the approximation properties. Numerical experiments illustrate and confirm the experiments.

math.NA↗

arXiv2023

Graph Neural Networks for temporal graphs: State of the art, open challenges, and opportunities

Graph Neural Networks (GNNs) have become the leading paradigm for learning on (static) graph-structured data. However, many real-world systems are dynamic in nature, since the graph and node/edge attributes change over time. In recent years, GNN-based models for temporal graphs have emerged as a promising area of research to extend the capabilities of GNNs. In this work, we provide the first comprehensive overview of the current state-of-the-art of temporal GNN, introducing a rigorous formalization of learning settings and tasks and a novel taxonomy categorizing existing approaches in terms of how the temporal aspect is represented and processed. We conclude the survey with a discussion of the most relevant open challenges for the field, from both research and application perspectives.

cs.LG↗

arXiv2023

Analysis of COVID-19 first wave in the US based on demographic, mobility, and environmental variables

COVID-19 had a strong and disruptive impact on our society, and yet further analyses on most relevant factors explaining the spread of the pandemic are needed. Interdisciplinary studies linking epidemiological, mobility, environmental, and socio-demographic data analysis can help understanding how historical conditions, concurrent social policies and environmental factors impacted on the evolution of the pandemic crisis. This work deals with a regression analysis linking COVID-19 mortality to socio-demographic, mobility, and environmental data in the US during the first half of 2020, i.e., during the COVID-19 pandemic first wave. This study can provide very useful insights about risk factors enhancing mortality rates before non-pharmaceutical interventions or vaccination campaigns took place. Our cross-sectional ecological regression analysis demonstrates that, when considering the entire US area, the socio-demographic variables globally play the most important role with respect to environmental and mobility variables in describing COVID-19 mortality. Compared to the complete generalized linear model considering all socio-demographic, mobility, and environmental data, the regression based only on socio-demographic data provides a better approximation and proves to be a better explanatory model when compared to the mobility-based and environmental-based models. However, when looking at single entries within each of the three groups, we see that the mobility data can become relevant descriptive predictors at local scale, as in New Jersey where the time spent at work is one of the most relevant explanatory variables, while environmental data play contradictory roles.

physics.soc-ph↗

arXiv2022

Interpolation with the polynomial kernels

The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their lack of strict positive definiteness. In particular they do not enjoy the usual property of unisolvency for arbitrary point sets, which is one of the key properties used to build kernel-based interpolation methods. This paper is devoted to establish some initial results for the study of these kernels, and their related interpolation algorithms, in the context of approximation theory. We will first prove necessary and sufficient conditions on point sets which guarantee the existence and uniqueness of an interpolant. We will then study the Reproducing Kernel Hilbert Spaces (or native spaces) of these kernels and their norms, and provide inclusion relations between spaces corresponding to different kernel parameters. With these spaces at hand, it will be further possible to derive generic error estimates which apply to sufficiently smooth functions, thus escaping the native space. Finally, we will show how to employ an efficient stable algorithm to these kernels to obtain accurate interpolants, and we will test them in some numerical experiment. After this analysis several computational and theoretical aspects remain open, and we will outline possible further research directions in a concluding section. This work builds some bridges between kernel and polynomial interpolation, two topics to which the authors, to different extents, have been introduced under the supervision or through the work of Stefano De Marchi. For this reason, they wish to dedicate this work to him in the occasion of his 60th birthday.

math.NA↗

arXiv2022

Equally spaced points are optimal for Brownian Bridge kernel interpolation

In this paper we show how ideas from spline theory can be used to construct a local basis for the space of translates of a general iterated Brownian Bridge kernel $k_{β,\varepsilon}$ for $β\in\mathbb{N}$, $\varepsilon\geq 0$. In the simple case $β=1$, we derive an explicit formula for the corresponding Lagrange basis, which allows us to solve interpolation problems without inverting any linear system. We use this basis to prove that interpolation with $k_{1,\varepsilon}$ is uniformly stable, i.e., the Lebesgue constant is bounded independently of the number an location of the interpolation points, and that equally spaced points are the unique minimizers of the associated power function, and are thus error optimal. In this derivation, we investigate the role of the shape parameter $\varepsilon>0$, and discuss its effect on these error and stability bounds. Some of the ideas discussed in this paper could be extended to more general Green kernels.

math.NA↗

arXiv2022

Convergence analysis for image interpolation in terms of the cSSIM

Assessing the similarity of two images is a complex task that attracts significant efforts in the image processing community. The widely used Structural Similarity Index Measure (SSIM) addresses this problem by quantifying a perceptual structural similarity. In this paper we consider a recently introduced continuous SSIM (cSSIM), which allows one to analyze sequences of images of increasingly fine resolutions, and further extend the definition of the index to encompass the locally weighted version that is used in practice. For both the local and the global versions, we prove that the continuous index includes the classical SSIM as a special case, and we provide a precise connection between image similarity measured by the cSSIM and by the $L_2$ norm. Using this connection, we derive bounds on the cSSIM by means of bounds on the $L_2$ error, and we even prove that the two error measures are equivalent in certain circumstances. We exploit these results to obtain precise rates of convergence with respect to the cSSIM for several concrete image interpolation methods, and we further validate these findings by different numerical experiments. This newly established connection paves the way to obtain novel insights into the features and limitations of the SSIM, including on the effect of the local weighted window on the index performances.

math.NA↗

arXiv2022

A Framework for Verifiable and Auditable Federated Anomaly Detection

Federated Leaning is an emerging approach to manage cooperation between a group of agents for the solution of Machine Learning tasks, with the goal of improving each agent's performance without disclosing any data. In this paper we present a novel algorithmic architecture that tackle this problem in the particular case of Anomaly Detection (or classification or rare events), a setting where typical applications often comprise data with sensible information, but where the scarcity of anomalous examples encourages collaboration. We show how Random Forests can be used as a tool for the development of accurate classifiers with an effective insight-sharing mechanism that does not break the data integrity. Moreover, we explain how the new architecture can be readily integrated in a blockchain infrastructure to ensure the verifiable and auditable execution of the algorithm. Furthermore, we discuss how this work may set the basis for a more general approach for the design of federated ensemble-learning methods beyond the specific task and architecture discussed in this paper.

cs.LG↗

arXiv2022

Reprogramming FairGANs with Variational Auto-Encoders: A New Transfer Learning Model

Fairness-aware GANs (FairGANs) exploit the mechanisms of Generative Adversarial Networks (GANs) to impose fairness on the generated data, freeing them from both disparate impact and disparate treatment. Given the model's advantages and performance, we introduce a novel learning framework to transfer a pre-trained FairGAN to other tasks. This reprogramming process has the goal of maintaining the FairGAN's main targets of data utility, classification utility, and data fairness, while widening its applicability and ease of use. In this paper we present the technical extensions required to adapt the original architecture to this new framework (and in particular the use of Variational Auto-Encoders), and discuss the benefits, trade-offs, and limitations of the new model.

cs.LG↗

arXiv2021

Greedy algorithms for learning via exponential-polynomial splines

Kernel-based schemes are state-of-the-art techniques for learning by data. In this work we extend some ideas about kernel-based greedy algorithms to exponential-polynomial splines, whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals ($f$-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function ($λ$-greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e. the spline nodes. However, while the {$f$-greedy} selection is tailored to one specific target function, the $λ$-greedy algorithm is independent of the function values and enables us to define a priori optimal interpolation nodes.

math.NA↗

arXiv2021

HJB-RBF based approach for the control of PDEs

Semi-lagrangian schemes for discretization of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for high-dimensional problems due to the curse of dimensionality. Here, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard's moving least squares approximation method on scattered grids. We propose a new method to generate a scattered mesh driven by the dynamics and the selection of the shape parameter in the RBF using an optimization routine. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method.

math.NA↗

arXiv2021

Analysis of target data-dependent greedy kernel algorithms: Convergence rates for $f$-, $f \cdot P$- and $f/P$-greedy

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory especially when compared to the case of the data-independent $P$-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. The rates are confirmed by a number of examples. These results are made possible by a new analysis of greedy algorithms in general Hilbert spaces.

math.NA↗

arXiv2021

Structured Deep Kernel Networks for Data-Driven Closure Terms of Turbulent Flows

Standard kernel methods for machine learning usually struggle when dealing with large datasets. We review a recently introduced Structured Deep Kernel Network (SDKN) approach that is capable of dealing with high-dimensional and huge datasets - and enjoys typical standard machine learning approximation properties. We extend the SDKN to combine it with standard machine learning modules and compare it with Neural Networks on the scientific challenge of data-driven prediction of closure terms of turbulent flows. We show experimentally that the SDKNs are capable of dealing with large datasets and achieve near-perfect accuracy on the given application.

cs.LG↗

arXiv2021

Kernel-Based Models for Influence Maximization on Graphs based on Gaussian Process Variance Minimization

The inference of novel knowledge, the discovery of hidden patterns, and the uncovering of insights from large amounts of data from a multitude of sources make Data Science (DS) to an art rather than just a mere scientific discipline. The study and design of mathematical models able to analyze information represents a central research topic in DS. In this work, we introduce and investigate a novel model for influence maximization (IM) on graphs using ideas from kernel-based approximation, Gaussian process regression, and the minimization of a corresponding variance term. Data-driven approaches can be applied to determine proper kernels for this IM model and machine learning methodologies are adopted to tune the model parameters. Compared to stochastic models in this field that rely on costly Monte-Carlo simulations, our model allows for a simple and cost-efficient update strategy to compute optimal influencing nodes on a graph. In several numerical experiments, we show the properties and benefits of this new model.

cs.LG↗

arXiv2020

Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem

For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold. Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system. In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics are quantified. We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.

math.NA↗

arXiv2020

Biomechanical surrogate modelling using stabilized vectorial greedy kernel methods

Greedy kernel approximation algorithms are successful techniques for sparse and accurate data-based modelling and function approximation. Based on a recent idea of stabilization of such algorithms in the scalar output case, we here consider the vectorial extension built on VKOGA. We introduce the so called $γ$-restricted VKOGA, comment on analytical properties and present numerical evaluation on data from a clinically relevant application, the modelling of the human spine. The experiments show that the new stabilized algorithms result in improved accuracy and stability over the non-stabilized algorithms.

math.NA↗

arXiv2020

Sampling based approximation of linear functionals in Reproducing Kernel Hilbert Spaces

In this paper we analyze a greedy procedure to approximate a linear functional defined in a Reproducing Kernel Hilbert Space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals, including integration functionals. For a large class of functionals, we prove convergence results for the approximation by means of uniform and greedy points which generalize in various ways several known results. A perturbation analysis of the weights and node computation is also discussed. Beyond the theoretical investigations, we demonstrate numerically that our algorithm is effective in treating various integration densities, and that it is even very competitive when compared to existing methods for Uncertainty Quantification.

math.NA↗