Generative Adversarial Networks (GANs) aim to produce realistic samples by mapping a low-dimensional latent space with known distribution to a high-dimensional data space, exploiting an adversarial training mechanism. However, without an effective characterisation of the generative process, such models face significant challenges in training and architecture selection. In this work, we propose a Topological Data Analysis based approach, using persistent homology, which can provide such a characterisation, where topological information of a data manifold is summarised by its Persistent Diagram and the evolution of its topological features is tracked throughout training. Our approach is applied across multiple GAN architectures using two benchmark datasets, where we demonstrate that conventional metrics such as Fréchet Inception Distance and intrinsic dimension estimates cannot adequately capture the quality of generated samples. Instead, our results confirm that a topological description of the generative process within GANs successfully captures training convergence and mode collapse. Finally, the layer-wise topological analysis determines the role each layer plays in the generative process, and may provide future guidance for refinement of architectures. Code available at https://github.com/bcorrad/genfold25.git .

Circle detection plays a pivotal role in computer vision, underpinning applications from industrial inspection and bioinformatics to autonomous driving. Traditional methods, however, often struggle with real–world complexities, as they demand extensive parameter tuning and adaptation across different domains. In this paper, we present the Synthetic Circle Dataset (SynCircle), a large synthetic image dataset designed to train a YOLO v10 network for circle detection. The YOLO v10 network, pre–trained solely on synthetic data, demonstrates remarkable off–the–shelf performance that surpasses conventional methods in various practical scenarios. Furthermore, we show that incorporating just a few labeled real images for fine–tuning can significantly boost performance, reducing the need for large annotated datasets. To promote reproducibility and streamline adoption, we publicly release both the trained YOLO v10 weights and the full SynCircle dataset.

Molecule generation has experienced multiple breakthroughs in recent years. While traditional techniques are very reliable they have low space exploration potential; therefore, machine learning techniques are needed to delve deep into the vast space of potential compounds, helping to find new ways of designing candidate molecules. This paper presents a sequential Markovian model based on graph neural networks for molecular generation. The model employs a Breadth–First Search (BFS) ordering strategy and a modular architecture to enhance independence between functions, with a focus on maintaining strict independence at each step of the Markovian process. Experimental results on ZINC (MOSES), ZINC (250 K) and Polymers datasets demonstrate the model’s ability to perform both unconditional and conditional molecule generation while preserving dataset properties. Notably, the step–wise approach achieves state–of–the–art results in terms of uniqueness and validity without using valency masks. • A novel sequential approach to molecular graph generation is presented. • The method exploits Graph Neural Networks to maximize information exploitation. • The employed Markov process and the architecture ensure a flexible learning procedure. • Conditional generation allows tailoring the generated molecules to the objectives of the specific studies. • The experimentation shows that the approach outperforms thestate-of-the-art.

Graph Neural Networks (GNNs) have emerged in recent years as a powerful tool to learn tasks across a wide range of graph domains in a data-driven fashion. Based on a message passing mechanism, GNNs have gained increasing popularity due to their intuitive formulation, closely linked to the Weisfeiler-Lehman (WL) test for graph isomorphism, to which they were demonstrated to be equivalent (Morris et al., 2019 and Xu et al., 2019). From a theoretical point of view, GNNs have been shown to be universal approximators, and their generalization capability - related to the Vapnik Chervonekis (VC) dimension (Scarselli et al., 2018) - has recently been investigated for GNNs with piecewise polynomial activation functions (Morris et al., 2023). The aim of our work is to extend this analysis on the VC dimension of GNNs to other commonly used activation functions, such as the sigmoid and hyperbolic tangent, using the framework of Pfaffian function theory. Bounds are provided with respect to the architecture parameters (depth, number of neurons, input size) as well as with respect to the number of colors resulting from the 1-WL test applied on the graph domain. The theoretical analysis is supported by a preliminary experimental study.

We systematically investigate the expressive power of path-based graph neural networks. While it has been shown that path-based graph neural networks can achieve strong empirical results, an investigation into their expressive power is lacking. Therefore, we propose PATH-WL, a general class of color refinement algorithms based on paths and shortest path distance information. We show that PATH-WL is incomparable to a wide range of expressive graph neural networks, can count cycles, and achieves strong empirical results on the notoriously difficult family of strongly regular graphs. Our theoretical results indicate that PATH-WL forms a new hierarchy of highly expressive graph neural networks.

Graph generative models often lack a proper reconstruction loss to evaluate the distance between the generated graph and the target graph. This is particularly important for molecular graph generators based on autoencoders, which should reconstruct the input graphs as precisely as possible. Though, the distance estimation can be useful for any graph generator based on reconstruction, including sequential methods relying on Graph Neural Networks. Since graphs are discrete entities by nature, general graph spaces lack a reliable, general, and computationally affordable distance function. Graph Edit Distance is of course an exact, general, and permutation–invariant method for evaluating the difference between two graphs defined in the same graph space. Since it needs all the possible combinations of pairs of nodes from the two graphs, its exact computation is a NP-complete problem and cannot be carried out for graphs larger than ten nodes. As a consequence, a comprehensive soft–estimation method for Graph Edit Distance based on siamese Graph Neural Networks is proposed. A theoretical discussion is carried out, showing that the proposed method can provide a reliable and precise soft–estimation of the Graph Edit Distance on molecular graphs. Molecular graph generators can therefore use this distance estimation as a powerful non–permutation–invariant reconstruction loss. Moreover, the experimental results show that the distance estimation is accurate, with a very low Mean Squared Error loss value.

Artificial Intelligence (AI)–generated images represent a significant threat in various fields, such as security, privacy, media forensics and content moderation. In this paper, a novel approach for the detection of StyleGAN2–generated human faces is presented, leveraging a Transfer Learning strategy to improve the Classification performance of the models. A modified version of the state– of–the–art semantic segmentation model DeepLabV3+, using either a ResNet50 or a MobileNetV3 Large as feature extraction backbones, is used to create both a face segmentation model and the synthetic image detector. To achieve this goal, the models are at first trained for face segmentation in a multi–class Classification task on a widely used semantic segmentation dataset, achieving remarkable results for both configurations. Then, the pre–trained models are retrained on a collection of real and generated images, gathered from different sources to solve a binary Classification task, namely to detect synthetic (i.e. generated) images, thus carrying out two different transfer learning strategies. The results indicate that this targeted methodology significantly improves the detection rates compared to analyzing the face as a whole, and underlines the importance of advanced image recognition technologies when tackling the challenge of detecting generated faces.

In the context of deep learning models, attention has recently been paid to studying the surface of the loss function in order to better understand training with methods based on gradient descent. This search for an appropriate description, both analytical and topological, has led to numerous efforts in identifying spurious minima and characterize gradient dynamics. Our work aims to contribute to this field by providing a topological measure for evaluating loss complexity in the case of multilayer neural networks. We compare deep and shallow architectures with common sigmoidal activation functions by deriving upper and lower bounds for the complexity of their respective loss functions and revealing how that complexity is influenced by the number of hidden units, training models, and the activation function used. Additionally, we found that certain variations in the loss function or model architecture, such as adding an ℓ2 regularization term or implementing skip connections in a feedforward network, do not affect loss topology in specific cases.

Abstract Graph neural networks (GNNs) are a broad class of connectionist models for graph processing. Recent studies have shown that GNNs can approximate any function on graphs, modulo the equivalence relation on graphs defined by the Weisfeiler–Lehman (WL) test. However, these results suffer from some limitations, both because they were derived using the Stone–Weierstrass theorem—which is existential in nature—and because they assume that the target function to be approximated must be continuous. Furthermore, all current results are dedicated to graph classification/regression tasks, where the GNN must produce a single output for the whole graph, while also node classification/regression problems, in which an output is returned for each node, are very common. In this paper, we propose an alternative way to demonstrate the approximation capability of GNNs that overcomes these limitations. Indeed, we show that GNNs are universal approximators in probability for node classification/regression tasks, as they can approximate any measurable function that satisfies the 1-WL-equivalence on nodes. The proposed theoretical framework allows the approximation of generic discontinuous target functions and also suggests the GNN architecture that can reach a desired approximation. In addition, we provide a bound on the number of the GNN layers required to achieve the desired degree of approximation, namely $$2r-1$$ 2 r - 1 , where r is the maximum number of nodes for the graphs in the domain.

Predicting drug side effects before they occur is a critical task for keeping the number of drug-related hospitalizations low and for improving drug discovery processes. Automatic predictors of side-effects generally are not able to process the structure of the drug, resulting in a loss of information. Graph neural networks have seen great success in recent years, thanks to their ability of exploiting the information conveyed by the graph structure and labels. These models have been used in a wide variety of biological applications, among which the prediction of drug side-effects on a large knowledge graph. Exploiting the molecular graph encoding the structure of the drug represents a novel approach, in which the problem is formulated as a multi-class multi-label graph-focused classification. We developed a methodology to carry out this task, using recurrent Graph Neural Networks, and building a dataset from freely accessible and well established data sources. The results show that our method has an improved classification capability, under many parameters and metrics, with respect to previously available predictors. The method is not ready for clinical tests yet, as the specificity is still below the preliminary 25% threshold. Future efforts will aim at improving this aspect.

Drug Side-Effects (DSEs) have a high impact on public health, care system costs, and drug discovery processes. Predicting the probability of side-effects, before their occurrence, is fundamental to reduce this impact, in particular on drug discovery. Candidate molecules could be screened before undergoing clinical trials, reducing the costs in time, money, and health of the participants. Drug side-effects are triggered by complex biological processes involving many different entities, from drug structures to protein-protein interactions. To predict their occurrence, it is necessary to integrate data from heterogeneous sources. In this work, such heterogeneous data is integrated into a graph dataset, expressively representing the relational information between different entities, such as drug molecules and genes. The relational nature of the dataset represents an important novelty for drug side-effect predictors. Graph Neural Networks (GNNs) are exploited to predict DSEs on our dataset with very promising results. GNNs are deep learning models that can process graph-structured data, with minimal information loss, and have been applied on a wide variety of biological tasks. Our experimental results confirm the advantage of using relationships between data entities, suggesting interesting future developments in this scope. The experimentation also shows the importance of specific subsets of data in determining associations between drugs and side-effects.

Many real-world domains involve information naturally represented by graphs, where nodes denote basic patterns while edges stand for relationships among them. The graph neural network (GNN) is a machine learning model capable of directly managing graph-structured data. In the original framework, GNNs are inductively trained, adapting their parameters based on a supervised learning environment. However, GNNs can also take advantage of transductive learning, thanks to the natural way they make information flow and spread across the graph, using relationships among patterns. In this paper, we propose a mixed inductive-transductive GNN model, study its properties and introduce an experimental strategy that allows us to understand and distinguish the role of inductive and transductive learning. The preliminary experimental results show interesting properties for the mixed model, highlighting how the peculiarities of the problems and the data can impact on the two learning strategies.

With a structural bioinformatic approach, we have explored amino acid compositions at PISA defined interfaces between small molecules and proteins that are contained in an optimized subset of 11,351 PDB files. The use of a series of restrictions, to prevent redundancy and biases from interactions between amino acids with charged side chains and ions, yielded a final data set of 45,230 protein-small molecule interfaces. We have compared occurrences of natural amino acids in surface exposed regions and binding sites for all the proteins of our data set. From our structural bioinformatic survey, the most relevant signal arose from the unexpected Gly abundance at enzyme catalytic sites. This finding suggested that Gly must have a fundamental role in stabilizing concave protein surface moieties. Subsequently, we have tried to predict the effect of in silico Gly mutations in hen egg white lysozyme to optimize those conditions that can reshape the protein surface with the appearance of new pockets. Replacing amino acids having bulky side chains with Gly in specific protein regions seems a feasible way for designing proteins with additional surface pockets, which can alter protein surface dynamics, therefore, representing controllable switches for protein activity.

Machine Learning (ML) achievements enabled automatic extraction of actionable information from data in a wide range of decisionmaking scenarios.This demands for improving both ML technical aspects (e.g., design and automation) and human-related metrics (e.g., fairness, robustness, privacy, and explainability), with performance guarantees at both levels.The aforementioned scenario posed three main challenges: (i) Learning from Complex Data (i.e., sequence, tree, and graph data), (ii) Learning Trustworthily, and (iii) Learning Automatically with Guarantees.The focus of this special session is on addressing one or more of these challenges with the final goal of Learning Trustworthily, Automatically, and with Guarantees from Complex Data.

Multi-parametric prostate MRI (mpMRI) is a powerful tool to diagnose prostate cancer, though difficult to interpret even for experienced radiologists. A common radiological procedure is to compare a magnetic resonance image with similarly diagnosed cases. To assist the radiological image interpretation process, computerized Content-Based Image Retrieval systems (CBIRs) can therefore be employed to improve the reporting workflow and increase its accuracy. In this article, we propose a new, supervised siamese deep learning architecture able to handle multi-modal and multi-view MR images with similar PIRADS score. An experimental comparison with well-established deep learning-based CBIRs (namely standard siamese networks and autoencoders) showed significantly improved performance with respect to both diagnostic (ROC-AUC), and information retrieval metrics (Precision-Recall, Discounted Cumulative Gain and Mean Average Precision). Finally, the new proposed multi-view siamese network is general in design, facilitating a broad use in diagnostic medical imaging retrieval.

Anisoptera are a suborder of insects belonging to the order of Odonata, commonly identified with the generic term dragonflies. They are characterized by a long and thin abdomen, two large eyes, and two pairs of transparent wings. Their ability to move the four wings independently allows dragonflies to fly forwards, backwards, to stop suddenly and to hover in mid–air, as well as to achieve high flight performance, with speed up to 50 km per hour. Thanks to these particular skills, many studies have been conducted on dragonflies, also using machine learning techniques. Some analyze the muscular movements of the flight to simulate dragonflies as accurately as possible, while others try to reproduce the neuronal mechanisms of hunting dragonflies. The lack of a consistent database and the difficulties in creating valid tools for such complex tasks have significantly limited the progress in the study of dragonflies. We provide two valuable results in this context: first, a dataset of carefully selected, pre–processed and labeled images, extracted from videos, has been released; then some deep neural network models, namely CNNs and LSTMs, have been trained to accurately distinguish the different phases of dragonfly flight, with very promising results.

In this paper, we study how to model taxi drivers' behavior and geographical information for an interesting and challenging task: the next destination prediction in a taxi journey. Predicting the next location is a well-studied problem in human mobility, which finds several applications in real-world scenarios, from optimizing the efficiency of electronic dispatching systems to predicting and reducing the traffic jam. This task is normally modeled as a multiclass classification problem, where the goal is to select, among a set of already known locations, the next taxi destination. We present a Recurrent Neural Network (RNN) approach that models the taxi drivers' behavior and encodes the semantics of visited locations by using geographical information from Location-Based Social Networks (LBSNs). In particular, the RNNs are trained to predict the exact coordinates of the next destination, overcoming the problem of producing, in output, a limited set of locations, seen during the training phase. The proposed approach was tested on the ECML/PKDD Discovery Challenge 2015 dataset-based on the city of Porto-, obtaining better results with respect to the competition winner, whilst using less information, and on Manhattan and San Francisco datasets.

During the last decade, deep learning and Convolutional Neural Networks (CNNs) have produced a devastating impact on computer vision, yielding exceptional results on a variety of problems, including analysis of medical images. Recently, these techniques have been extended to 3D images with the downside of a large increase in the computational load. In particular, state-of-the-art CNNs have been used for brain Nuclear Magnetic Resonance (NMR) imaging, with the aim of estimating the patients’ age. In fact, a large discrepancy between the real and the estimated age is a clear alarm for the onset of neurodegenerative diseases, such as some types of early dementia and Alzheimer’s disease. In this paper, we propose an effective alternative to 3D convolutions that guarantees a significant reduction of the computational requirements for this kind of analysis. The proposed architectures achieve comparable results with the competitor 3D methods, requiring only a fraction of the training time and GPU memory.

Urinary Tract Infections (UTIs) are very common in women, babies and the elderly. The most frequent cause is a bacterium, called Escherichia coli, which usually lives in the digestive system and in the bowel. Infections can target the urethra, bladder or kidneys. Traditional analysis methods, based on human experts' evaluation, are typically used to diagnose UTIs, an error prone and lengthy process, whereas an early treatment of common pathologies is fundamental to prevent the infection spreading to kidneys. This paper presents an image based Automated Bacterial Load Estimator (ABLE) system for the urinoculture screening, that provides quick and traceable results for UTIs. Infections are accurately detected and the bacterial load is evaluated through image processing techniques. First, digital color images of the Petri dishes are automatically captured, and cleaned from noisily elements due to laboratory procedures, then specific spatial clustering algorithms are applied to isolate the colonies from the culture ground and, finally, an accurate evaluation of the infection severity is performed. A dataset of 499 urine samples has been used during the experiments and the obtained results are fully discussed. The ABLE system speeds up the analysis, grants repeatable results, contributes to the process standardization, and guarantees a significant cost reduction.

Recently, researchers in the artificial neural network field have focused their attention on connectionist models composed by several hidden layers. In fact, experimental results and heuristic considerations suggest that deep architectures are more suitable than shallow ones for modern applications, facing very complex problems, e.g., vision and human language understanding. However, the actual theoretical results supporting such a claim are still few and incomplete. In this paper, we propose a new approach to study how the depth of feedforward neural networks impacts on their ability in implementing high complexity functions. First, a new measure based on topological concepts is introduced, aimed at evaluating the complexity of the function implemented by a neural network, used for classification purposes. Then, deep and shallow neural architectures with common sigmoidal activation functions are compared, by deriving upper and lower bounds on their complexity, and studying how the complexity depends on the number of hidden units and the used activation function. The obtained results seem to support the idea that deep networks actually implements functions of higher complexity, so that they are able, with the same number of resources, to address more difficult problems.

Graph Neural Networks (GNNs) are a powerful tool for processing graphs, that represent a natural way to collect information coming from several areas of science and engineering - e.g. data mining, computer vision, molecular chemistry, molecular biology, pattern recognition -, where data are intrinsically organized in entities and relationships among entities. Nevertheless, GNNs suffer, so as recurrent/recursive models, from the long-term dependency problem that makes the learning difficult in deep structures. In this paper, we present a new architecture, called Layered GNN (LGNN), realized by a cascade of GNNs: each layer is fed with the original data and with the state information calculated by the previous layer in the cascade. Intuitively, this allows each GNN to solve a subproblem, related only to those patterns that were misclassified by the previous GNNs. Some experimental results are reported, based on synthetic and real-world datasets, which assess a significant improvement in performances w.r.t. the standard GNN approach.

Recursive neural networks are a powerful tool for processing structured data. According to the recursive learning paradigm, the input information consists of directed positional acyclic graphs (DPAGs). In fact, recursive networks are fed following the partial order defined by the links of the graph. Unfortunately, the hypothesis of processing DPAGs is sometimes too restrictive, being the nature of some real-world problems intrinsically cyclic. In this paper, a methodology is proposed, which allows us to process any cyclic directed graph. Therefore, the computational power of recursive networks is definitely established, also clarifying the underlying limitations of the model.

Although the interest of a Web page is strictly related to its content and to the subjective readers' cultural background, a measure of the page authority can be provided that only depends on the topological structure of the Web. PageRank is a noticeable way to attach a score to Web pages on the basis of the Web connectivity. In this article, we look inside PageRank to disclose its fundamental properties concerning stability, complexity of computational scheme, and critical role of parameters involved in the computation. Moreover, we introduce a circuit analysis that allows us to understand the distribution of the page score, the way different Web communities interact each other, the role of dangling pages (pages with no outlinks), and the secrets for promotion of Web pages.

In this paper, a new recursive neural network model, able to process directed acyclic graphs with labeled edges, is introduced, in order to address the problem of object detection in images. In fact, the detection is a preliminary step in any object recognition system. The proposed method assumes a graph-based representation of images, that combines both spatial and visual features. In particular, after segmentation, an edge between two nodes stands for the adjacency relationship of two homogeneous regions, the edge label collects information on their relative positions, whereas node labels contain visual and geometric information on each region (area, color, texture, etc.). Such graphs are then processed by the recursive model in order to determine the eventual presence and the position of objects inside the image. Some experiments on face detection, carried out on scenes acquired by an indoor camera, are reported, showing very promising results. The proposed technique is general and can be applied in different object detection systems, since it does not include any a priori knowledge on the particular problem.

The definition of the ordering of the Web pages, returned on a given query, is a crucial topic, which gives rise to the notion of Web visibility. A fundamental contribution towards the conception of appropriate ordering criteria has been given by means of the introduction of PageRank, which takes into account only the hyper-linked structure of the Web, regardless of the content of the pages. We introduce a circuit analysis which allows us to understand the distribution of PageRank, and show some basic results for understanding the way it migrates amongst communities. In particular, we highlight some topological properties which suggest methods for the promotion of Web communities. These results confirm the importance and the effectiveness of PageRank for discovering relevant information but, at the same time, point out its vulnerability to spamming.

We propose solving nonlinear systems of equations by function optimization and we give an optimal algorithm which relies on a special canonical form of gradient descent. The algorithm can be applied under certain assumptions on the function to be optimized, that is, an upper bound must exist for the norm of the Hessian, whereas the norm of the gradient must be lower bounded. Due to its intrinsic structure, the algorithm looks particularly appealing for a parallel implementation. As a particular case, more specific results are given for linear systems. We prove that reaching a solution with a degree of precision /spl epsiv/ takes /spl Theta/(n/sup 2/k/sup 2/ log /sup k///sub /spl epsiv//), k being the condition number of A and n the problem dimension. Related results hold for systems of quadratic equations for which an estimation for the requested bounds can be devised. Finally, we report numerical results in order to establish the actual computational burden of the proposed method and to assess its performances with respect to classical algorithms for solving linear and quadratic equations.

Recursive neural networks are a powerful tool for processing structured data, thus filling the gap between connectionism, which is usually related to poorly organized data, and a great variety of real-world problems, where the information is naturally encoded in the relationships among the basic entities. In this paper, some theoretical results about linear recursive neural networks are presented that allow one to establish conditions on their dynamical properties and their capability to encode and classify structured information. A lot of the limitations of the linear model, intrinsically related to recursive processing, are inherited by the general model, thus establishing their computational capabilities and range of applicability. As a byproduct of our study some connections with the classical linear system theory are given where the processing is extended from sequences to graphs.

Recursive neural networks are conceived for processing graphs and extend the well-known recurrent model for processing sequences. In Frasconi et al. (1998), recursive neural networks can deal only with directed ordered acyclic graphs (DOAGs), in which the children of any given node are ordered. While this assumption is reasonable in some applications, it introduces unnecessary constraints in others. In this paper, it is shown that the constraint on the ordering can be relaxed by using an appropriate weight sharing, that guarantees the independence of the network output with respect to the permutations of the arcs leaving from each node. The method can be used with graphs having low connectivity and, in particular, few outcoming arcs. Some theoretical properties of the proposed architecture are given. They guarantee that the approximation capabilities are maintained, despite the weight sharing.

Recursive neural networks are a new connectionist model introduced for processing graphs. Linear recursive networks are a special subclass where the neurons have linear activation functions. The approximation properties of recursive networks are tightly connected to the possibility of distinguishing the patterns by generating a different internal encoding for each input of the domain. In this paper, it is shown that, even if linear recursive networks can distinguish the patterns of any finite set of trees, such a result requires a prohibitive memory consumption. However, it is also proved that the problem disappears when the domain is restricted to set of trees belonging to special sub-classes.

Gradient descent learning algorithms may get stuck in local minima, thus making the learning suboptimal. In this paper, we focus attention on multilayered networks used as autoassociators and show some relationships with classical linear autoassociators. In addition, by using the theoretical framework of our previous research, we derive a condition which is met at the end of the learning process and show that this condition has a very intriguing geometrical meaning in the pattern space.

Learning from examples plays a central role in artificial neural networks. The success of many learning schemes is not guaranteed, however, since algorithms like backpropagation may get stuck in local minima, thus providing suboptimal solutions. For feedforward networks, optimal learning can be achieved provided that certain conditions on the network and the learning environment are met. This principle is investigated for the case of networks using radial basis functions (RBF). It is assumed that the patterns of the learning environment are separable by hyperspheres. In that case, we prove that the attached cost function is local minima free with respect to all the weights. This provides us with some theoretical foundations for a massive application of RBF in pattern recognition.

Many researchers have recently focused their efforts on devising efficient algorithms, mainly based on optimization schemes, for learning the weights of recurrent neural networks. As in the case of feedforward networks, however, these learning algorithms may get stuck in local minima during gradient descent, thus discovering sub-optimal solutions. This paper analyses the problem of optimal learning in recurrent networks by proposing conditions that guarantee local minima free error surfaces. An example is given that also shows the constructive role of the proposed theory in designing networks suitable for solving a given task. Moreover, a formal relationship between recurrent and static feedforward networks is established such that the examples of local minima for feedforward networks already known in the literature can be associated with analogous ones in recurrent networks.