Search

In machine learning and data mining, outliers are data points that significantly differ from the dataset and often introduce irrelevant information that can induce bias in its statistics and models. Therefore, unsupervised methods are crucial to detect outliers if there is limited or no information about them. Global-Local Outlier Scores based on Hierarchies (GLOSH) is an unsupervised outlier detection method within HDBSCAN*, a state-of-the-art hierarchical clustering method. GLOSH estimates outlier scores for each data point by comparing its density to the highest density of the region they reside in the HDBSCAN* hierarchy. GLOSH may be sensitive to HDBSCAN*'s minpts parameter that influences density estimation. With limited knowledge about the data, choosing an appropriate minpts value beforehand is challenging as one or some minpts values may better represent the underlying cluster structure than others. Additionally, in the process of searching for ``potential outliers'', one has to define the number of outliers n a dataset has, which may be impractical and is often unknown. In this paper, we propose an unsupervised strategy to find the ``best'' minpts value, leveraging the range of GLOSH scores across minpts values to identify the value for which GLOSH scores can best identify outliers from the rest of the dataset. Moreover, we propose an unsupervised strategy to estimate a threshold for classifying points into inliers and (potential) outliers without the need to pre-define any value. Our experiments show that our strategies can automatically find the minpts value and threshold that yield the best or near best outlier detection results using GLOSH., Comment: Accepted at IEEE International Conference on Big Data, IEEE BigData 2024

We revisit the mean field parametrization of shallow neural networks, using signed measures on unbounded parameter spaces and duality pairings that take into account the regularity and growth of activation functions. This setting directly leads to the use of unbalanced Kantorovich-Rubinstein norms defined by duality with Lipschitz functions, and of spaces of measures dual to those of continuous functions with controlled growth. These allow to make transparent the need for total variation and moment bounds or penalization to obtain existence of minimizers of variational formulations, under which we prove a compactness result in strong Kantorovich-Rubinstein norm, and in the absence of which we show several examples demonstrating undesirable behavior. Further, the Kantorovich-Rubinstein setting enables us to combine the advantages of a completely linear parametrization and ensuing reproducing kernel Banach space framework with optimal transport insights. We showcase this synergy with representer theorems and uniform large data limits for empirical risk minimization, and in proposed formulations for distillation and fusion applications., Comment: 39 pages, 1 table

This paper deals with the target localisation problem in search and rescue scenarios in which the technology is based on electromagnetic transceivers. The noise floor and the shape of the electromagnetic radiation pattern make this problem challenging. Indeed, on the one hand, the signal-to-noise ratio reduces with the inverse of the distance from the electromagnetic source thus impacting estimation-based techniques applicability. On the other hand, non-isotropic radiation patterns lessen the efficacy of gradient-based policies. In this work, we manage a fleet of autonomous agents, equipped with electromagnetic sensors, by combining gradient-based and estimation-based techniques to speed up the transmitter localisation. Simulations specialized in the ARTVA technology used in search and rescue in avalanche scenarios confirm that our scheme outperforms current solutions., Comment: accepted for presentation at the 2024 IEEE 63rd IEEE Conference on Decision and Control (CDC2024)

This paper responds to a commentary by Neal (2024) regarding the Distinctiveness centrality metrics introduced by Fronzetti Colladon and Naldi (2020). Distinctiveness centrality offers a novel reinterpretation of degree centrality, particularly emphasizing the significance of direct connections to loosely connected peers within (social) networks. This response paper presents a more comprehensive analysis of the correlation between Distinctiveness and the Beta and Gamma measures. All five distinctiveness measures are considered, as well as a more meaningful range of the alpha parameter and different network topologies, distinguishing between weighted and unweighted networks. Findings indicate significant variability in correlations, supporting the viability of Distinctiveness as alternative or complementary metrics within social network analysis. Moreover, the paper presents computational complexity analysis and simplified R code for practical implementation. Encouraging initial findings suggest potential applications in diverse domains, inviting further exploration and comparative analyses.

This paper proposes a data-driven approach for constructing firmly nonexpansive operators. We demonstrate its applicability in Plug-and-Play methods, where classical algorithms such as forward-backward splitting, Chambolle--Pock primal-dual iteration, Douglas--Rachford iteration or alternating directions method of multipliers (ADMM), are modified by replacing one proximal map by a learned firmly nonexpansive operator. We provide sound mathematical background to the problem of learning such an operator via expected and empirical risk minimization. We prove that, as the number of training points increases, the empirical risk minimization problem converges (in the sense of Gamma-convergence) to the expected risk minimization problem. Further, we derive a solution strategy that ensures firmly nonexpansive and piecewise affine operators within the convex envelope of the training set. We show that this operator converges to the best empirical solution as the number of points in the envelope increases in an appropriate sense. Finally, the experimental section details practical implementations of the method and presents an application in image denoising.

In this work we apply the recently introduced framework of degenerate preconditioned proximal point algorithms to the hybrid proximal extragradient (HPE) method for maximal monotone inclusions. The latter is a method that allows inexact proximal (or resolvent) steps where the error is controlled by a relative-error criterion. Recently the HPE framework has been extended to the Douglas-Rachford method by Eckstein and Yao. In this paper we further extend the applicability of the HPE framework to splitting methods. To that end we use the framework of degenerate preconditioners that allows to write a large class of splitting methods as preconditioned proximal point algorithms. In this way we modify many splitting methods such that one or more of the resolvents can be computed inexactly with an error that is controlled by an adaptive criterion. Further, we illustrate the algorithmic framework in the case of Chambolle-Pock's primal dual hybrid gradient method and the Davis-Yin's forward Douglas-Rachford method. In both cases, the inexact computation of the resolvent shows clear advantages in computing time and accuracy., Comment: 33 pages, 6 figures

The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, of the set of vertices $V$ of a tetrahedron $T$ is the minimal closed set containing $V$ such that its complement is union of open balls of radius $R$. When $R$ is greater than the circumradius of $T$, the boundary of the $R$-hulloid consists of $V$ and possibly of four spherical subsets of well defined spheres of radius $R$ through the vertices of $T$. The existence of a value $R^*$ such that these subsets collapse into a point $O^*$, in the interior of $T$, is investigated; in such case $O^*$ belongs to four spheres of radius $R^*$, each one through three vertices of $T$ and not containing the fourth one. As a consequence, the range of $\rho$ such that $V$ is a $\rho$-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem., Comment: 16 pages, 2 figures

A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance coordinates to the vertices of a tetrahedron $T \subset \mathbb{R}^3$ is introduced to represent the configurations of four spheres of radius $R^*$, which intersect in one point, each sphere containing three vertices of $T$ but not the fourth one. This problem is related to that of computing the largest value $r$ for which the set of vertices of $T$ is an $r$-body. For triangular pyramids we completely describe the set of geometric configurations with the required four balls of radius $R^*$. The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one $R^*$ is unique, in the second one three values $R^*$ there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of $r$-bodies., Comment: 24 pages, 6 figures

This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical solvers that work with rational numbers can only find an approximate solution. We study the following question: is it possible to certify feasibility of a given SDP using an approximate solution that is sufficiently close to some exact solution? Existing approaches make the assumption that there exist rational feasible solutions (and use techniques such as rounding and lattice reduction algorithms). We propose an alternative approach that does not need this assumption. More specifically, we show how to construct a system of polynomial equations whose set of real solutions is guaranteed to have an isolated correct solution (assuming that the target exact solution is maximum-rank). This allows, in particular, to use algorithms from real algebraic geometry for solving systems of polynomial equations, yielding a hybrid (or symbolic-numerical) method for SDPs. We experimentally compare it with a pure symbolic method in [Henrion, Naldi, Safey El Din; SIAM J. Optim., 2016]; the hybrid method was able to certify feasibility of many SDP instances on which [Henrion, Naldi, Safey El Din; SIAM J. Optim., 2016] failed. We argue that our approach may have other uses, such as refining an approximate solution using methods of numerical algebraic geometry for systems of polynomial equations.

In Beyond5G and 6G networks, a common theme is that sensing will play a more significant role than ever before. Over this trend, Integrated Sensing and Communications (ISAC) is focused on unifying the sensing functionalities and the communications ones and to pursue direct tradeoffs between them as well as mutual performance gains. We frame the resource tradeoff between the SAC functionalities within an economic setting. We model a service provision by one operator to the users, the utility of which is derived from both SAC functionalities. The tradeoff between the resources that the operator assigns to the SAC functionalities is analyzed from the point of view of the service prices, quantities and profits. We demonstrate that equilibrium quantities and prices exist. And we provide relevant recommendations for enforcing regulatory limits of both power and bandwidth.

Fast radio bursts (FRBs) are energetic, millisecond-duration radio pulses observed at extragalactic distances and whose origins are still a subject of heated debate. A fraction of the FRB population have shown repeating bursts, however it's still unclear whether these represent a distinct class of sources. We investigated the bursting behaviour of FRB 20220912A, one of the most active repeating FRBs known thus far. In particular, we focused on its burst energy distribution, linked to the source energetics, and its emission spectrum, with the latter directly related to the underlying emission mechanism. We monitored FRB 20220912A at $408$ MHz with the Northern Cross radio telescope and at $1.4$ GHz using the $32$-m Medicina Grueff radio telescope. Additionally, we conducted $1.2$ GHz observations taken with the upgraded Giant Meter Wave Radio Telescope (uGMRT) searching for a persistent radio source coincident with FRB 20220912A, which included high energy observations in the $0.3-10$ keV, $0.4-100$ MeV and $0.03-30$ GeV energy range. We report $16$ new bursts from FRB 20220912A at $408$ MHz during the period between October 16$^{\rm th}$ 2022 and December 31$^{\rm st}$ 2023. Their cumulative spectral energy distribution follows a power law with slope $\alpha_E = -1.3 \pm 0.2$ and we measured a repetition rate of $0.19 \pm 0.03$ hr$^{-1}$ for bursts having a fluence of $\mathcal{F} \geq 17$ Jy ms. Furthermore, we report no detections at 1.4 GHz for $\mathcal{F} \geq 20$ Jy ms. These non-detections imply an upper limit of $\beta < -2.3$, with $\beta$ being the $408$ MHz $-$ $1.4$ GHz spectral index of FRB 20220912A. This is inconsistent with positive $\beta$ values found for the only two known cases in which an FRB has been detected in separate spectral bands. We find that FRB 20220912A shows a decline of four orders of magnitude in its bursting activity at $1.4$ GHz over a .. (abridged), Comment: accepted for publication, 13 pages, 4 figures

In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carath\'eodory., Comment: 6 pages

Fast radio bursts (FRBs) are millisecond radio transients observed at cosmological distances. The nature of their progenitors is still a matter of debate, although magnetars are invoked by most models. The proposed FRB-magnetar connection was strengthened by the discovery of an FRB-like event from the Galactic magnetar SGR J1935+2154. In this work, we aim to investigate how prevalent magnetars such as SGR J1935+2154 are within FRB progenitors. We carried out an FRB search in a sample of seven nearby (< 12 Mpc) galaxies with the Northern Cross radio telescope for a total of 692 h. We detected one 1.8 ms burst in the direction of M101 with a fluence of $58 \pm 5$ Jy ms. Its dispersion measure of 303 pc cm$^{-3}$ places it most-likely beyond M101. Considering that no significant detection comes indisputably from the selected galaxies, we place a 38 yr$^{-1}$ upper limit on the total burst rate (i.e. including the whole sample) at the 95\% confidence level. This upper limit constrains the event rate per magnetar $\lambda_{\rm mag} < 0.42$ magnetar$^{-1}$ yr$^{-1}$ or, if combined with literature observations of a similar sample of nearby galaxies, it yields a joint constraint of $\lambda_{\rm mag} < 0.25$ magnetar$^{-1}$ yr$^{-1}$. We also provide the first constraints on the expected rate of FRBs hypothetically originating from ultraluminous X-ray (ULX) sources, since some of the galaxies observed during our observational campaign host confirmed ULXs. We obtain $< 13$ yr$^{-1}$ per ULX for the total sample of galaxies observed. Our results indicate that bursts with energies $E>10^{34}$ erg from magnetars like SGR J1935+2154 appear more rarely compared to previous observations and further disfavour them as unique progenitors for the cosmological FRB population, leaving more space open to the contribution from a population of more exotic magnetars, not born via core-collapsed supernovae., Comment: 9 pages, 4 figures, published in A&A

A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Pad\'e approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fr\"oberg's Conjecture in commutative algebra., Comment: 20 pages

In this paper the degenerate preconditioned proximal point algorithm will be combined with the idea of varying preconditioners leading to the degenerate variable metric proximal point algorithm. The weak convergence of the resulting iteration will be proven. From the perspective of the degenerate variable metric proximal point algorithm, a version of the primal-dual Douglas-Rachford method with varying preconditioners will be derived and a proof of its weak convergence which is based on the previous results for the proximal point algorithm, is provided, too. After that, we derive a heuristic on how to choose those varying preconditioners in order to increase the convergence speed of the method.

The truncated moment problem consists of determining whether a given finitedimensional vector of real numbers y is obtained by integrating a basis of the vector space of polynomials of bounded degree with respect to a non-negative measure on a given set K of a finite-dimensional Euclidean space. This problem has plenty of applications e.g. in optimization, control theory and statistics. When K is a compact semialgebraic set, the duality between the cone of moments of non-negative measures on K and the cone of non-negative polynomials on K yields an alternative: either y is a moment vector, or y is not a moment vector, in which case there exists a polynomial strictly positive on K making a linear functional depending on y vanish. Such a polynomial is an algebraic certificate of moment unrepresentability. We study the complexity of computing such a certificate using computer algebra algorithms.

In this paper, we propose several graph-based extensions of the Douglas-Rachford splitting (DRS) method to solve monotone inclusion problems involving the sum of $N$ maximal monotone operators. Our construction is based on a two-layer architecture that we refer to as bilevel graphs, to which we associate a generalization of the DRS algorithm that presents the prescribed structure. The resulting schemes can be understood as unconditionally stable frugal resolvent splitting methods with a minimal lifting in the sense of Ryu [Math Program 182(1):233-273, 2020], as well as instances of the (degenerate) Preconditioned Proximal Point method, which provides robust convergence guarantees. We further describe how the graph-based extensions of the DRS method can be leveraged to design new fully distributed protocols. Applications to a congested optimal transport problem and to distributed Support Vector Machines show interesting connections with the underlying graph topology and highly competitive performances with state-of-the-art distributed optimization approaches., Comment: 23 pages, 4 figures

This work is concerned with epidemiological models defined on networks, which highlight the prominent role of the social contact network of a given population in the spread of infectious diseases. In particular, we address the modelling and analysis of very large networks. As a basic epidemiological model, we focus on a SEIR (Susceptible-Exposed-Infective-Removed) model governing the behaviour of infectious disease among a population of individuals, which is partitioned into sub-populations. We study the long-time behaviour of the dynamic for this model, also taking into account the heterogeneity of the infections and the social network. By relying on the theory of graphons, we address the natural question of the large population limit and investigate the behaviour of the model as the size of the network tends to infinitely. After establishing the existence and uniqueness of solutions to the selected models, we discuss the use of the graphon-based limit model as a generative model for a network with particular statistical properties related to the distribution of connections. We also provide some preliminary numerical tests.

A classical result from topology called Uryshon's lemma asserts the existence of a continuous separator of two disjoint closed sets in a sufficiently regular topological space. In this work we make a search for this separator constructive and efficient in the context of real algebraic geometry. Namely, given two compact disjoint basic semialgebraic sets which are contained in an $n$-dimensional box, we provide an algorithm that computes a separating polynomial greater than or equal to 1 on the first set and less than or equal to 0 on the second one., Comment: 4 pages, 1 figure, submitted as en extended abstract for the last POEMA workshop