Search

The design of sparse neural networks, i.e., of networks with a reduced number of parameters, has been attracting increasing research attention in the last few years. The use of sparse models may significantly reduce the computational and storage footprint in the inference phase. In this context, the lottery ticket hypothesis (LTH) constitutes a breakthrough result, that addresses not only the performance of the inference phase, but also of the training phase. It states that it is possible to extract effective sparse subnetworks, called winning tickets, that can be trained in isolation. The development of effective methods to play the lottery, i.e., to find winning tickets, is still an open problem. In this article, we propose a novel class of methods to play the lottery. The key point is the use of concave regularization to promote the sparsity of a relaxed binary mask, which represents the network topology. We theoretically analyze the effectiveness of the proposed method in the convex framework. Then, we propose extended numerical tests on various datasets and architectures, that show that the proposed method can improve the performance of state-of-the-art algorithms.

Laser-based displays are highly sought after for their superior brightness and color performance, especially in advanced applications like augmented reality (AR). However, their broader adoption has been hindered by bulky projector designs and complex optical module assemblies. Here, we introduce a new laser display architecture enabled by large-scale visible photonic integrated circuits (PICs) to address these challenges. Unlike previous projector-style laser displays, this architecture features an ultra-thin, flat-panel form factor, replacing bulky free-space illumination modules with a single, high-performance photonic chip. Centimeter-scale PIC devices, which integrate thousands of distinct optical components on-chip, are carefully tailored to achieve high display uniformity, contrast, and efficiency. We demonstrate a 2 mm-thick flat-panel laser display combining the PIC with a liquid-crystal-on-silicon (LCoS) panel, achieving 211% of the color gamut and more than 80% volume reduction compared to traditional LCoS displays. We further showcase its application in a see-through AR system. Our work represents a major advancement in the integration of nanophotonics with display technology, enabling a range of new display concepts, from high-performance immersive displays to slim-panel 3D holography.

We propose an optimization problem to minimize the base stations transmission powers in OFDMA heterogeneous networks, while respecting users' individual throughput demands. The decision variables are the users' working bandwidths, their association, and the base stations transmission powers. To deal with wireless channel uncertainty, the channel gains are treated as random variables respecting a log-normal distribution, leading to a non-convex chance constrained mixed-integer optimization problem, which is then formulated as a mixed-integer Robust Geometric Program. The efficacy of the proposed method is shown in a real-world scenario of a large European city.

Federated learning has emerged in the last decade as a distributed optimization paradigm due to the rapidly increasing number of portable devices able to support the heavy computational needs related to the training of machine learning models. Federated learning utilizes gradient-based optimization to minimize a loss objective shared across participating agents. To the best of our knowledge, the literature mostly lacks elegant solutions that naturally harness the reciprocal statistical similarity between clients to redesign the optimization procedure. To address this gap, by conceiving the federated network as a similarity graph, we propose a novel modified framework wherein each client locally performs a perturbed gradient step leveraging prior information about other statistically affine clients. We theoretically prove that our procedure, due to a suitably introduced adaptation in the update rule, achieves a quantifiable speedup concerning the exponential contraction factor in the strongly convex case compared with popular algorithms FedAvg and FedProx, here analyzed as baselines. Lastly, we legitimize our conclusions through experimental results on the CIFAR10 and FEMNIST datasets, where we show that our algorithm speeds convergence up to a margin of 30 global rounds compared with FedAvg while modestly improving generalization on unseen data in heterogeneous settings.

As the global focus on combating environmental pollution intensifies, the transition to sustainable energy sources, particularly in the form of electric vehicles (EVs), has become paramount. This paper addresses the pressing need for Smart Charging for EVs by developing a comprehensive mathematical model aimed at optimizing charging station management. The model aims to efficiently allocate the power from charging sockets to EVs, prioritizing cost minimization and avoiding energy waste. Computational simulations demonstrate the efficacy of the mathematical optimization model, which can unleash its full potential when the number of EVs at the charging station is high., Comment: Paper submitted and accepted to ESCC 2024 - "11th International Conference on Energy, Sustainability and Climate Crisis August 26 - 30, 2024, Corfu, Greece"

In this paper, we deal with the equilibrium selection problem, which amounts to steering a population of individuals engaged in strategic game-theoretic interactions to a desired collective behavior. In the literature, this problem has been typically tackled by means of open-loop strategies, whose applicability is however limited by the need of accurate a priori information on the game and scarce robustness to uncertainty and noise. Here, we overcome these limitations by adopting a closed-loop approach using an adaptive-gain control scheme within a replicator equation -a nonlinear ordinary differential equation that models the evolution of the collective behavior of the population. For most classes of 2-action matrix games we establish sufficient conditions to design a controller that guarantees convergence of the replicator equation to the desired equilibrium, requiring limited a-priori information on the game. Numerical simulations corroborate and expand our theoretical findings., Comment: Under Review

Heterogeneous networks have emerged as a popular solution for accommodating the growing number of connected devices and increasing traffic demands in cellular networks. While offering broader coverage, higher capacity, and lower latency, the escalating energy consumption poses sustainability challenges. In this paper a novel optimization approach for OFDMA heterogeneous networks is proposed to minimize transmission power while respecting individual users throughput constraints. The problem is formulated as a mixed integer geometric program, and optimizes at once multiple system variables such as user association, working bandwidth, and base stations transmission powers. Crucially, the proposed approach becomes a convex optimization problem when user-base station associations are provided. Evaluations in multiple realistic scenarios from the production mobile network of a major European operator and based on precise channel gains and throughput requirements from measured data validate the effectiveness of the proposed approach. Overall, our original solution paves the road for greener connectivity by reducing the energy footprint of heterogeneous mobile networks, hence fostering more sustainable communication systems.

In this paper we analyze the resilience of a network of banks to joint price fluctuations of the external assets in which they have shared exposures, and evaluate the worst-case effects of the possible default contagion. Indeed, when the prices of certain external assets either decrease or increase, all banks exposed to them experience varying degrees of simultaneous shocks to their balance sheets. These coordinated and structured shocks have the potential to exacerbate the likelihood of defaults. In this context, we introduce first a concept of {default resilience margin}, $\epsilon^*$, i.e., the maximum amplitude of asset prices fluctuations that the network can tolerate without generating defaults. Such threshold value is computed by considering two different measures of price fluctuations, one based on the maximum individual variation of each asset, and the other based on the sum of all the asset's absolute variations. For any price perturbation having amplitude no larger than $\epsilon^*$, the network absorbs the shocks remaining default free. When the perturbation amplitude goes beyond $\epsilon^*$, however, defaults may occur. In this case we find the worst-case systemic loss, that is, the total unpaid debt under the most severe price variation of given magnitude. Computation of both the threshold level $\epsilon^*$ and of the worst-case loss and of a corresponding worst-case asset price scenario, amounts to solving suitable linear programming problems.}

Controlling evolutionary game-theoretic dynamics is a problem of paramount importance for the systems and control community, with several applications spanning from social science to engineering. Here, we study a population of individuals who play a generic 2-action matrix game, and whose actions evolve according to a replicator equation -- a nonlinear ordinary differential equation that captures salient features of the collective behavior of the population. Our objective is to steer such a population to a specified equilibrium that represents a desired collective behavior -- e.g., to promote cooperation in the prisoner's dilemma. To this aim, we devise an adaptive-gain controller, which regulates the system dynamics by adaptively changing the entries of the payoff matrix of the game. The adaptive-gain controller is tailored according to distinctive features of the game, and conditions to guarantee global convergence to the desired equilibrium are established., Comment: 6 pages, Accepted for presentation at the 2023 IEEE CDC

Existing empirical work has focused on assessing the effectiveness of non-pharmaceutical interventions on human mobility to contain the spread of COVID-19. Less is known about the ways in which the COVID-19 pandemic has reshaped the spatial patterns of population movement within countries. Anecdotal evidence of an urban exodus from large cities to rural areas emerged during early phases of the pandemic across western societies. Yet, these claims have not been empirically assessed. Traditional data sources, such as censuses offer coarse temporal frequency to analyse population movement over short-time intervals. Drawing on a data set of 21 million observations from Facebook users, we aim to analyse the extent and evolution of changes in the spatial patterns of population movement across the rural-urban continuum in Britain over an 18-month period from March, 2020 to August, 2021. Our findings show an overall and sustained decline in population movement during periods of high stringency measures, with the most densely populated areas reporting the largest reductions. During these periods, we also find evidence of higher-than-average mobility from highly dense population areas to low densely populated areas, lending some support to claims of large-scale population movements from large cities. Yet, we show that these trends were temporary. Overall mobility levels trended back to pre-coronavirus levels after the easing of non-pharmaceutical interventions. Following these interventions, we also found a reduction in movement to low density areas and a rise in mobility to high density agglomerations. Overall, these findings reveal that while COVID-19 generated shock waves leading to temporary changes in the patterns of population movement in Britain, the resulting vibrations have not significantly reshaped the prevalent structures in the national pattern of population movement., Comment: 4 figs

The current global financial system forms a highly interconnected network where a default in one of its nodes can propagate to many other nodes, causing a catastrophic avalanche effect. In this paper we consider the problem of reducing the financial contagion by introducing some targeted interventions that can mitigate the cascaded failure effects. We consider a multi-step dynamic model of clearing payments and introduce an external control term that represents corrective cash injections made by a ruling authority. The proposed control model can be cast and efficiently solved as a linear program. We show via numerical examples that the proposed approach can significantly reduce the default propagation by applying small targeted cash injections.

A Pontryagin-based approach to solve a class of constrained Nonlinear Model Predictive Control problems is proposed which employs the method of barrier functions for dealing with the state constraints. Unlike the existing works in literature the proposed method is able to cope with nonlinear input and state constraints without any significant modification of the optimization algorithm. A stability analysis of the closed-loop system is carried out by using the L-2 norm of the predicted state tracking error as a Lyapunov function. Theoretical results are tested and confirmed by numerical simulations on the Lotka-Volterra prey/predator system., Comment: 11 pages, 5 figures

This paper proposes a novel dynamical model for determining clearing payments in financial networks. We extend the classical Eisenberg-Noe model of financial contagion to multiple time periods, allowing financial operations to continue after possible initial pseudo defaults, thus permitting nodes to recover and eventually fulfil their liabilities. Optimal clearing payments in our model are computed by solving a suitable linear program, both in the full matrix payments case and in the pro-rata constrained case. We prove that the proposed model obeys the \emph{priority of debt claims} requirement, that is, each node at every step either pays its liabilities in full, or it pays out all its balance. In the pro-rata case, the optimal dynamic clearing payments are unique, and can be determined via a time-decoupled sequential optimization approach.

Consensus of autonomous agents is a benchmark problem in multi-agent control. In this paper, we consider continuous-time averaging consensus policies (or Laplacian flows) and their discrete-time counterparts over time-varying graphs in presence of unknown but bounded communication delays. It is known that consensus is established (no matter how large the delays are) if the graph is periodically, or uniformly quasi-strongly connected (UQSC). The UQSC condition is often believed to be the weakest sufficient condition under which consensus can be proved. We show that the UQSC condition can actually be substantially relaxed and replaced by a condition that we call aperiodic quasi-strong connectivity (AQSC), which, in some sense, proves to be very close to the necessary condition of integral connectivity. Furthermore, in some special situations such as undirected or type-symmetric graph, we find a necessary and sufficient condition for consensus in presence of bounded delay; the relevant results have been previously proved only in the undelayed case. The consensus criteria established in this paper generalize a number of results known in the literature., Comment: a shortened version is submitted to IEEE TAC

The dataset described in this paper contains daily data about COVID-19 cases that occurred in Italy over the period from Jan. 28, 2020 to March 20, 2021, divided into ten age classes of the population, the first class being 0-9 years, the tenth class being 90 years and over. The dataset contains eight columns, namely: date (day), age class, number of new cases, number of newly hospitalized patients, number of patients entering intensive care, number of deceased patients, number of recovered patients, number of active infected patients. This data has been officially released for research purposes by the Italian authority for COVID-19 epidemiologic surveillance (Istituto Superiore di Sanit\`a - ISS), upon formal request by the authors, in accordance with the Ordonnance of the Chief of the Civil Protection Department n. 691 dated Aug. 4 2020. A separate file contains the numerosity of the population in each age class, according to the National Institute of Statistics (ISTAT) data of the resident population of Italy as of Jan. 2020. This data has potential use, for instance, in epidemiologic studies of the effects of the COVID-19 contagion in Italy, in mortality analysis by age class, and in the development and testing of dynamical models of the contagion., Comment: six pages, one table, three figures

Financial networks are characterized by complex structures of mutual obligations. These obligations are fulfilled entirely or in part (when defaults occur) via a mechanism called clearing, which determines a set of payments that settle the claims by respecting rules such as limited liability, absolute priority, and proportionality (pro-rated payments). In the presence of shocks on the financial system, however, the clearing mechanism may lead to cascaded defaults and eventually to financial disaster. In this paper, we first study the clearing model under pro-rated payments of Eisenberg and Noe, and we derive novel necessary and sufficient conditions for the uniqueness of the clearing payments, valid for an arbitrary topology of the financial network. Then, we argue that the proportionality rule is one of the factors responsible for cascaded defaults, and that the overall system loss can be reduced if this rule is lifted. The proposed approach thus shifts the focus from the individual interest to the overall system's interest to control and contain adverse effects of cascaded failures, and we show that clearing payments in this setting can be computed by solving suitable convex optimization problems.

Consensus of autonomous agents is a benchmark problem in cooperative control. In this paper, we consider standard continuous-time averaging consensus policies (or Laplacian flows) over time-varying graphs and focus on robustness of consensus against communication delays. Such a robustness has been proved under the assumption of uniform quasi-strong connectivity of the graph. It is known, however, that the uniform connectivity is not necessary for consensus. For instance, in the case of undirected graph and undelayed communication consensus requires a much weaker condition of integral connectivity. In this paper, we show that the latter results remain valid in presence of unknown but bounded communication delays, furthermore, the condition of undirected graph can be substantially relaxed and replaced by the conditions of non-instantaneous type-symmetry. Furthermore, consensus can be proved for any feasible solution of the delay differential inequalities associated to the consensus algorithm. Such inequalities naturally arise in problems of containment control, distributed optimization and models of social dynamics., Comment: An extended version of a conference paper to be presented on IEEE Conference on Decision and Control 2020

The purpose of this work is to give a contribution to the understanding of the COVID-19 contagion in Italy. To this end, we developed a modified Susceptible-Infected-Recovered (SIR) model for the contagion, and we used official data of the pandemic up to March 30th, 2020 for identifying the parameters of this model. The non standard part of our approach resides in the fact that we considered as model parameters also the initial number of susceptible individuals, as well as the proportionality factor relating the detected number of positives with the actual (and unknown) number of infected individuals. Identifying the contagion, recovery and death rates as well as the mentioned parameters amounts to a non-convex identification problem that we solved by means of a two-dimensional grid search in the outer loop, with a standard weighted least-squares optimization problem as the inner step.

Within the Private Equity (PE) market, the event of a private company undertaking an Initial Public Offering (IPO) is usually a very high-return one for the investors in the company. For this reason, an effective predictive model for the IPO event is considered as a valuable tool in the PE market, an endeavor in which publicly available quantitative information is generally scarce. In this paper, we describe a data-analytic procedure for predicting the probability with which a company will go public in a given forward period of time. The proposed method is based on the interplay of a neural network (NN) model for estimating the overall event probability, and Survival Analysis (SA) for further modeling the probability of the IPO event in any given interval of time. The proposed neuro-survival model is tuned and tested across nine industrial sectors using real data from the Thomson Reuters Eikon PE database.

The nearest-centroid classifier is a simple linear-time classifier based on computing the centroids of the data classes in the training phase, and then assigning a new datum to the class corresponding to its nearest centroid. Thanks to its very low computational cost, the nearest-centroid classifier is still widely used in machine learning, despite the development of many other more sophisticated classification methods. In this paper, we propose two sparse variants of the nearest-centroid classifier, based respectively on $\ell_1$ and $\ell_2$ distance criteria. The proposed sparse classifiers perform simultaneous classification and feature selection, by detecting the features that are most relevant for the classification purpose. We show that training of the proposed sparse models, with both distance criteria, can be performed exactly (i.e., the globally optimal set of features is selected) and at a quasi-linear computational cost. The experimental results show that the proposed methods are competitive in accuracy with state-of-the-art feature selection techniques, while having a significantly lower computational cost.

We consider the problem of deriving from experimental data an approximation of an unknown function, whose derivatives also approximate the unknown function derivatives. Solving this problem is useful, for instance, in the context of nonlinear system identification for obtaining models that are more accurate and reliable than the traditional ones, based on plain function approximation. Indeed, models identified by accounting for the derivatives can provide a better performance in several tasks, such as multi-step prediction, simulation, Nonlinear Model Predictive Control, and control design in general. In this paper, we propose a novel approach based on convex optimization, allowing us to solve the aforementioned identification problem. We develop an optimality analysis, showing that models derived using this approach enjoy suitable optimality properties in Sobolev spaces. The optimality analysis also leads to the derivation of tight uncertainty bounds on the unknown function and its derivatives. We demonstrate the effectiveness of the approach with two numerical examples. The first one is concerned with approximation of an univariate function, the second one with multi-step prediction of the Chua chaotic circuit.

Many multi-agent control algorithms and dynamic agent-based models arising in natural and social sciences are based on the principle of iterative averaging. Each agent is associated to a value of interest, which may represent, for instance, the opinion of an individual in a social group, the velocity vector of a mobile robot in a flock, or the measurement of a sensor within a sensor network. This value is updated, at each iteration, to a weighted average of itself and of the values of the adjacent agents. It is well known that, under natural assumptions on the network's graph connectivity, this local averaging procedure eventually leads to global consensus, or synchronization of the values at all nodes. Applications of iterative averaging include, but are not limited to, algorithms for distributed optimization, for solution of linear and nonlinear equations, for multi-robot coordination and for opinion formation in social groups. Although these algorithms have similar structures, the mathematical techniques used for their analysis are diverse, and conditions for their convergence and differ from case to case. In this paper, we review many of these algorithms and we show that their properties can be analyzed in a unified way by using a novel tool based on recurrent averaging inequalities (RAIs). We develop a theory of RAIs and apply it to the analysis of several important multi-agent algorithms recently proposed in the literature.

Predicting the exit (e.g. bankrupt, acquisition, etc.) of privately held companies is a current and relevant problem for investment firms. The difficulty of the problem stems from the lack of reliable, quantitative and publicly available data. In this paper, we contribute to this endeavour by constructing an exit predictor model based on qualitative data, which blends the outcomes of three classifiers, namely, a Logistic Regression model, a Random Forest model, and a Support Vector Machine model. The output of the combined model is selected on the basis of the majority of the output classes of the component models. The models are trained using data extracted from the Thomson Reuters Eikon repository of 54697 US and European companies over the 1996-2011 time span. Experiments have been conducted for predicting whether the company eventually either gets acquired or goes public (IPO), against the complementary event that it remains private or goes bankrupt, in the considered time window. Our model achieves a 63\% predictive accuracy, which is quite a valuable figure for Private Equity investors, who typically expect very high returns from successful investments.

We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node (LSE networks) is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of networks maps to a family of subtraction-free ratios of generalized posynomials, which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of Difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for design that possess a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing effective optimization-based design. We illustrate the proposed approach by applying it to data-driven design of a diet for a patient with type-2 diabetes.

Motivated by the increasing interest of the control community towards social sciences and the study of opinion formation and belief systems, in this paper we address the problem of exploiting voting data for inferring the underlying affinity of individuals to competing ideology groups. In particular, we mine key voting records of the Italian Senate during the XVII legislature, in order to extract the hidden information about the closeness of senators to political parties, based on a parsimonious feature extraction method that selects the most relevant bills. Modeling the voting data as outcomes of a mixture of random variables and using sparse learning techniques, we cast the problem in a probabilistic framework and derive an information theoretic measure, which we refer to as Political Data-aNalytic Affinity (Political DNA). The advantages of this new affinity measure are discussed in the paper. The results of the numerical analysis on voting data unveil underlying relationships among political exponents of the Italian Senate., Comment: 13 Pages. Submitted as invited paper to ECC19

We show in this paper that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is an universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named LSET. Under a suitable exponential transformation, the class of LSET functions maps to a family of generalized posynomials GPOST, which we similarly show to be universal approximators for log-log-convex functions. A key feature of an LSET network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a GPOST model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models.

Pose Graph Optimization involves the estimation of a set of poses from pairwise measurements and provides a formalization for many problems arising in mobile robotics and geometric computer vision. In this paper, we consider the case in which a subset of the measurements fed to pose graph optimization is spurious. Our first contribution is to develop robust estimators that can cope with heavy-tailed measurement noise, hence increasing robustness to the presence of outliers. Since the resulting estimators require solving nonconvex optimization problems, we further develop convex relaxations that approximately solve those problems via semidefinite programming. We then provide conditions under which the proposed relaxations are exact. Contrarily to existing approaches, our convex relaxations do not rely on the availability of an initial guess for the unknown poses, hence they are more suitable for setups in which such guess is not available (e.g., multi-robot localization, recovery after localization failure). We tested the proposed techniques in extensive simulations, and we show that some of the proposed relaxations are indeed tight (i.e., they solve the original nonconvex problem 10 exactly) and ensure accurate estimation in the face of a large number of outliers., Comment: 10 pages, 5 figures, accepted for publication in the IEEE Robotics and Automation Letters, 2018

The discovery of the topological nature of free-space light and its quantum chiral behavior has recently raised large attention. This important scientific endeavor features spin-based integrated quantum technologies. Herein, we discuss a novel phenomenon based on a resonantly-enhanced quantum spin-Hall transport of light observed in a dielectric resonator operating near the bound-state-in-continuum (BIC) regime. The BIC mode is characterized by a transverse photonic spin angular momentum density extended on a macroscopic area. As such, the experimental excited mode in near-BIC regime generates resonant surface waves characterized by spin-momentum locking and that propagate along the symmetry axes of the structure. In addition, the generated side waves are interpreted as an abrupt nonparaxial redirection of the exciting far field light, which is responsible for the spin-to-orbital angular momentum conversion evidenced in the spin-orbit asymmetry measured in the intensity of the side waves. The experimental results are in excellent agreement with a model that combines geometric parallel transport of light polarization and spin-momentum locking. In addition, breaking the excitation symmetry leads to a total spin-directive coupling. Our results reveal the possibility of a BIC-enhanced macroscopic spin-directive coupling, a novel fundamental mechanism of light-spin manipulation that will have strong impact on emerging quantum technologies.

In this paper, we discuss a stochastic decision problem of optimally selecting the order in which to try $n$ opportunities that may yield an uncertain reward in the future. The motivation came out from pure curiosity, after an informal conversation about what could be the best way to date friends. The problem structure turned out to be suitable also for other situations, such as the problem of optimally selecting the order of submission of a paper to journals. Despite the seemingly combinatorial nature of the problem, we show that optimal-tradeoff solutions can be found by simply ordering a sequence of $n$ real numbers.

This paper deals with the problem of finding a low-complexity estimate of the impulse response of a linear time-invariant discrete-time dynamic system from noise-corrupted input-output data. To this purpose, we introduce an identification criterion formed by the average (over the input perturbations) of a standard prediction error cost, plus a weighted l1 regularization term which promotes sparse solutions. While it is well known that such criteria do provide solutions with many zeros, a critical issue in our identification context is where these zeros are located, since sensible low-order models should be zero in the tail of the impulse response. The flavor of the key results in this paper is that, under quite standard assumptions (such as i.i.d. input and noise sequences and system stability), the estimate of the impulse response resulting from the proposed criterion is indeed identically zero from a certain time index (named the leading order) onwards, with arbitrarily high probability, for a sufficiently large data cardinality. Numerical experiments are reported that support the theoretical results, and comparisons are made with some other state-of-the-art methodologies.

This document constitutes the final report of the contractual activity between Directa SIM and Dipartimento di Automatica e Informatica, Politecnico di Torino, on the research topic titled "quantificazione del rischio di un portafoglio di strumenti finanziari per trading online su device fissi e mobili."

In this Letter we present a Fresnel lens fabricated on the end of an optical fiber. The lens is fabricated using nanoimprint lithography of a functional high refractive index material, which is suitable for mass production. The main advantage of the presented Fresnel lens compared to a conventional fiber lens is its high refractive index (n=1.69), which enables efficient light focusing even inside other media such as water or adhesive. Measurement of the lens performance in an immersion liquid (n=1.51) shows a near diffraction limited focal spot of 810 nm in diameter at the 1/e2 intensity level for a wavelength of 660 nm. Applications of such fiber lenses include integrated optics, optical trapping and fiber probes., Comment: 4 pages , 5 figures

Integration of complex photonic structures onto optical fiber facets enables powerful platforms with unprecedented optical functionalities. Conventional nanofabrication technologies, however, do not permit viable integration of complex photonic devices onto optical fibers owing to their low throughput and high cost. In this paper we report the fabrication of a three dimensional structure achieved by direct Nanoimprint Lithography on the facet of an optical fiber. Nanoimprint processes and tools were specifically developed to enable a high lithographic accuracy and coaxial alignment of the optical device with respect to the fiber core. To demonstrate the capability of this new approach, a 3D beam splitter has been designed, imprinted and optically characterized. Scanning electron microscopy and optical measurements confirmed the excellent lithographic capabilities of the proposed approach as well as the desired optical performance of the imprinted structure. The inexpensive solution presented here should enable advancements in areas such as integrated optics and sensing, achieving enhanced portability and versatility of fiber optic components., Comment: 7 pages, 3 figures