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Hypercomplex algebras have recently been gaining prominence in the field of deep learning owing to the advantages of their division algebras over real vector spaces and their superior results when dealing with multidimensional signals in real-world 3D and 4D paradigms. This paper provides a foundational framework that serves as a roadmap for understanding why hypercomplex deep learning methods are so successful and how their potential can be exploited. Such a theoretical framework is described in terms of inductive bias, i.e., a collection of assumptions, properties, and constraints that are built into training algorithms to guide their learning process toward more efficient and accurate solutions. We show that it is possible to derive specific inductive biases in the hypercomplex domains, which extend complex numbers to encompass diverse numbers and data structures. These biases prove effective in managing the distinctive properties of these domains, as well as the complex structures of multidimensional and multimodal signals. This novel perspective for hypercomplex deep learning promises to both demystify this class of methods and clarify their potential, under a unifying framework, and in this way promotes hypercomplex models as viable alternatives to traditional real-valued deep learning for multidimensional signal processing., Comment: Accepted for Publication in IEEE Signal Processing Magazine

The primary goal of the L3DAS23 Signal Processing Grand Challenge at ICASSP 2023 is to promote and support collaborative research on machine learning for 3D audio signal processing, with a specific emphasis on 3D speech enhancement and 3D Sound Event Localization and Detection in Extended Reality applications. As part of our latest competition, we provide a brand-new dataset, which maintains the same general characteristics of the L3DAS21 and L3DAS22 datasets, but with first-order Ambisonics recordings from multiple reverberant simulated environments. Moreover, we start exploring an audio-visual scenario by providing images of these environments, as perceived by the different microphone positions and orientations. We also propose updated baseline models for both tasks that can now support audio-image couples as input and a supporting API to replicate our results. Finally, we present the results of the participants. Further details about the challenge are available at https://www.l3das.com/icassp2023., Comment: Accepted to 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2023)

Neural network generalizability is becoming a broad research field due to the increasing availability of datasets from different sources and for various tasks. This issue is even wider when processing medical data, where a lack of methodological standards causes large variations being provided by different imaging centers or acquired with various devices and cofactors. To overcome these limitations, we introduce a novel, generalizable, data- and task-agnostic framework able to extract salient features from medical images. The proposed quaternion wavelet network (QUAVE) can be easily integrated with any pre-existing medical image analysis or synthesis task, and it can be involved with real, quaternion, or hypercomplex-valued models, generalizing their adoption to single-channel data. QUAVE first extracts different sub-bands through the quaternion wavelet transform, resulting in both low-frequency/approximation bands and high-frequency/fine-grained features. Then, it weighs the most representative set of sub-bands to be involved as input to any other neural model for image processing, replacing standard data samples. We conduct an extensive experimental evaluation comprising different datasets, diverse image analysis, and synthesis tasks including reconstruction, segmentation, and modality translation. We also evaluate QUAVE in combination with both real and quaternion-valued models. Results demonstrate the effectiveness and the generalizability of the proposed framework that improves network performance while being flexible to be adopted in manifold scenarios and robust to domain shifts. The full code is available at: https://github.com/ispamm/QWT., Comment: This paper is currently under review

Neural models based on hypercomplex algebra systems are growing and prolificating for a plethora of applications, ranging from computer vision to natural language processing. Hand in hand with their adoption, parameterized hypercomplex neural networks (PHNNs) are growing in size and no techniques have been adopted so far to control their convergence at a large scale. In this paper, we study PHNNs convergence and propose parameterized hypercomplex identity initialization (PHYDI), a method to improve their convergence at different scales, leading to more robust performance when the number of layers scales up, while also reaching the same performance with fewer iterations. We show the effectiveness of this approach in different benchmarks and with common PHNNs with ResNets- and Transformer-based architecture. The code is available at https://github.com/ispamm/PHYDI., Comment: Accepted at IEEE MLSP 2023 (Honorable Mention TOP 5% Outstanding Papers)

Catastrophic forgetting (CF) occurs when a neural network loses the information previously learned while training on a set of samples from a different distribution, i.e., a new task. Existing approaches have achieved remarkable results in mitigating CF, especially in a scenario called task incremental learning. However, this scenario is not realistic, and limited work has been done to achieve good results on more realistic scenarios. In this paper, we propose a novel regularization method called Centroids Matching, that, inspired by meta-learning approaches, fights CF by operating in the feature space produced by the neural network, achieving good results while requiring a small memory footprint. Specifically, the approach classifies the samples directly using the feature vectors produced by the neural network, by matching those vectors with the centroids representing the classes from the current task, or all the tasks up to that point. Centroids Matching is faster than competing baselines, and it can be exploited to efficiently mitigate CF, by preserving the distances between the embedding space produced by the model when past tasks were over, and the one currently produced, leading to a method that achieves high accuracy on all the tasks, without using an external memory when operating on easy scenarios, or using a small one for more realistic ones. Extensive experiments demonstrate that Centroids Matching achieves accuracy gains on multiple datasets and scenarios., Comment: Submitted to Transactions on Machine Learning Research (TMLR)

Image-to-image translation (I2I) aims at transferring the content representation from an input domain to an output one, bouncing along different target domains. Recent I2I generative models, which gain outstanding results in this task, comprise a set of diverse deep networks each with tens of million parameters. Moreover, images are usually three-dimensional being composed of RGB channels and common neural models do not take dimensions correlation into account, losing beneficial information. In this paper, we propose to leverage hypercomplex algebra properties to define lightweight I2I generative models capable of preserving pre-existing relations among image dimensions, thus exploiting additional input information. On manifold I2I benchmarks, we show how the proposed Quaternion StarGANv2 and parameterized hypercomplex StarGANv2 (PHStarGANv2) reduce parameters and storage memory amount while ensuring high domain translation performance and good image quality as measured by FID and LPIPS scores. Full code is available at: https://github.com/ispamm/HI2I.

Spatial audio methods are gaining a growing interest due to the spread of immersive audio experiences and applications, such as virtual and augmented reality. For these purposes, 3D audio signals are often acquired through arrays of Ambisonics microphones, each comprising four capsules that decompose the sound field in spherical harmonics. In this paper, we propose a dual quaternion representation of the spatial sound field acquired through an array of two First Order Ambisonics (FOA) microphones. The audio signals are encapsulated in a dual quaternion that leverages quaternion algebra properties to exploit correlations among them. This augmented representation with 6 degrees of freedom (6DOF) involves a more accurate coverage of the sound field, resulting in a more precise sound localization and a more immersive audio experience. We evaluate our approach on a sound event localization and detection (SELD) benchmark. We show that our dual quaternion SELD model with temporal convolution blocks (DualQSELD-TCN) achieves better results with respect to real and quaternion-valued baselines thanks to our augmented representation of the sound field. Full code is available at: https://github.com/ispamm/DualQSELD-TCN., Comment: Paper accepted for publication in Elsevier Pattern Recognition Letters

The L3DAS22 Challenge is aimed at encouraging the development of machine learning strategies for 3D speech enhancement and 3D sound localization and detection in office-like environments. This challenge improves and extends the tasks of the L3DAS21 edition. We generated a new dataset, which maintains the same general characteristics of L3DAS21 datasets, but with an extended number of data points and adding constrains that improve the baseline model's efficiency and overcome the major difficulties encountered by the participants of the previous challenge. We updated the baseline model of Task 1, using the architecture that ranked first in the previous challenge edition. We wrote a new supporting API, improving its clarity and ease-of-use. In the end, we present and discuss the results submitted by all participants. L3DAS22 Challenge website: www.l3das.com/icassp2022., Comment: Accepted to 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2022). arXiv admin note: substantial text overlap with arXiv:2104.05499

Catastrophic forgetting (CF) happens whenever a neural network overwrites past knowledge while being trained on new tasks. Common techniques to handle CF include regularization of the weights (using, e.g., their importance on past tasks), and rehearsal strategies, where the network is constantly re-trained on past data. Generative models have also been applied for the latter, in order to have endless sources of data. In this paper, we propose a novel method that combines the strengths of regularization and generative-based rehearsal approaches. Our generative model consists of a normalizing flow (NF), a probabilistic and invertible neural network, trained on the internal embeddings of the network. By keeping a single NF throughout the training process, we show that our memory overhead remains constant. In addition, exploiting the invertibility of the NF, we propose a simple approach to regularize the network's embeddings with respect to past tasks. We show that our method performs favorably with respect to state-of-the-art approaches in the literature, with bounded computational power and memory overheads., Comment: arXiv admin note: substantial text overlap with arXiv:2007.02443

Recent research has found that neural networks are vulnerable to several types of adversarial attacks, where the input samples are modified in such a way that the model produces a wrong prediction that misclassifies the adversarial sample. In this paper we focus on black-box adversarial attacks, that can be performed without knowing the inner structure of the attacked model, nor the training procedure, and we propose a novel attack that is capable of correctly attacking a high percentage of samples by rearranging a small number of pixels within the attacked image. We demonstrate that our attack works on a large number of datasets and models, that it requires a small number of iterations, and that the distance between the original sample and the adversarial one is negligible to the human eye.

In this paper, we investigate the degree of explainability of graph neural networks (GNNs). Existing explainers work by finding global/local subgraphs to explain a prediction, but they are applied after a GNN has already been trained. Here, we propose a meta-learning framework for improving the level of explainability of a GNN directly at training time, by steering the optimization procedure towards what we call `interpretable minima'. Our framework (called MATE, MetA-Train to Explain) jointly trains a model to solve the original task, e.g., node classification, and to provide easily processable outputs for downstream algorithms that explain the model's decisions in a human-friendly way. In particular, we meta-train the model's parameters to quickly minimize the error of an instance-level GNNExplainer trained on-the-fly on randomly sampled nodes. The final internal representation relies upon a set of features that can be `better' understood by an explanation algorithm, e.g., another instance of GNNExplainer. Our model-agnostic approach can improve the explanations produced for different GNN architectures and use any instance-based explainer to drive this process. Experiments on synthetic and real-world datasets for node and graph classification show that we can produce models that are consistently easier to explain by different algorithms. Furthermore, this increase in explainability comes at no cost for the accuracy of the model.

In this paper, we propose a novel ensembling technique for deep neural networks, which is able to drastically reduce the required memory compared to alternative approaches. In particular, we propose to extract multiple sub-networks from a single, untrained neural network by solving an end-to-end optimization task combining differentiable scaling over the original architecture, with multiple regularization terms favouring the diversity of the ensemble. Since our proposal aims to detect and extract sub-structures, we call it Structured Ensemble. On a large experimental evaluation, we show that our method can achieve higher or comparable accuracy to competing methods while requiring significantly less storage. In addition, we evaluate our ensembles in terms of predictive calibration and uncertainty, showing they compare favourably with the state-of-the-art. Finally, we draw a link with the continual learning literature, and we propose a modification of our framework to handle continuous streams of tasks with a sub-linear memory cost. We compare with a number of alternative strategies to mitigate catastrophic forgetting, highlighting advantages in terms of average accuracy and memory., Comment: Article accepted at Neural Networks

Graph representation learning has become a ubiquitous component in many scenarios, ranging from social network analysis to energy forecasting in smart grids. In several applications, ensuring the fairness of the node (or graph) representations with respect to some protected attributes is crucial for their correct deployment. Yet, fairness in graph deep learning remains under-explored, with few solutions available. In particular, the tendency of similar nodes to cluster on several real-world graphs (i.e., homophily) can dramatically worsen the fairness of these procedures. In this paper, we propose a novel biased edge dropout algorithm (FairDrop) to counter-act homophily and improve fairness in graph representation learning. FairDrop can be plugged in easily on many existing algorithms, is efficient, adaptable, and can be combined with other fairness-inducing solutions. After describing the general algorithm, we demonstrate its application on two benchmark tasks, specifically, as a random walk model for producing node embeddings, and to a graph convolutional network for link prediction. We prove that the proposed algorithm can successfully improve the fairness of all models up to a small or negligible drop in accuracy, and compares favourably with existing state-of-the-art solutions. In an ablation study, we demonstrate that our algorithm can flexibly interpolate between biasing towards fairness and an unbiased edge dropout. Furthermore, to better evaluate the gains, we propose a new dyadic group definition to measure the bias of a link prediction task when paired with group-based fairness metrics. In particular, we extend the metric used to measure the bias in the node embeddings to take into account the graph structure., Comment: Submitted to a journal for the peer-review process

Nonlinear models are known to provide excellent performance in real-world applications that often operate in non-ideal conditions. However, such applications often require online processing to be performed with limited computational resources. To address this problem, we propose a new class of efficient nonlinear models for online applications. The proposed algorithms are based on linear-in-the-parameters (LIP) nonlinear filters using functional link expansions. In order to make this class of functional link adaptive filters (FLAFs) efficient, we propose low-complexity expansions and frequency-domain adaptation of the parameters. Among this family of algorithms, we also define the partitioned-block frequency-domain FLAF, whose implementation is particularly suitable for online nonlinear modeling problems. We assess and compare frequency-domain FLAFs with different expansions providing the best possible tradeoff between performance and computational complexity. Experimental results prove that the proposed algorithms can be considered as an efficient and effective solution for online applications, such as the acoustic echo cancellation, even in the presence of adverse nonlinear conditions and with limited availability of computational resources., Comment: This work has been accepted for publication in IEEE Transactions on Systems, Man, and Cybernetics: Systems. Copyright may be transferred without notice, after which this version may no longer be accessible

The L3DAS21 Challenge is aimed at encouraging and fostering collaborative research on machine learning for 3D audio signal processing, with particular focus on 3D speech enhancement (SE) and 3D sound localization and detection (SELD). Alongside with the challenge, we release the L3DAS21 dataset, a 65 hours 3D audio corpus, accompanied with a Python API that facilitates the data usage and results submission stage. Usually, machine learning approaches to 3D audio tasks are based on single-perspective Ambisonics recordings or on arrays of single-capsule microphones. We propose, instead, a novel multichannel audio configuration based multiple-source and multiple-perspective Ambisonics recordings, performed with an array of two first-order Ambisonics microphones. To the best of our knowledge, it is the first time that a dual-mic Ambisonics configuration is used for these tasks. We provide baseline models and results for both tasks, obtained with state-of-the-art architectures: FaSNet for SE and SELDNet for SELD. This report is aimed at providing all needed information to participate in the L3DAS21 Challenge, illustrating the details of the L3DAS21 dataset, the challenge tasks and the baseline models., Comment: Documentation paper for the L3DAS21 Challenge for IEEE MLSP 2021. Further information on www.l3das.com/mlsp2021

Deep probabilistic generative models have achieved incredible success in many fields of application. Among such models, variational autoencoders (VAEs) have proved their ability in modeling a generative process by learning a latent representation of the input. In this paper, we propose a novel VAE defined in the quaternion domain, which exploits the properties of quaternion algebra to improve performance while significantly reducing the number of parameters required by the network. The success of the proposed quaternion VAE with respect to traditional VAEs relies on the ability to leverage the internal relations between quaternion-valued input features and on the properties of second-order statistics which allow to define the latent variables in the augmented quaternion domain. In order to show the advantages due to such properties, we define a plain convolutional VAE in the quaternion domain and we evaluate its performance with respect to its real-valued counterpart on the CelebA face dataset., Comment: Accepted for publication at the 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

Nonlinear adaptive filters often show some sparse behavior due to the fact that not all the coefficients are equally useful for the modeling of any nonlinearity. Recently, a class of proportionate algorithms has been proposed for nonlinear filters to leverage sparsity of their coefficients. However, the choice of the norm penalty of the cost function may be not always appropriate depending on the problem. In this paper, we introduce an adaptive combined scheme based on a block-based approach involving two nonlinear filters with different regularization that allows to achieve always superior performance than individual rules. The proposed method is assessed in nonlinear system identification problems, showing its effectiveness in taking advantage of the online combined regularization., Comment: This is a corrected version of the paper presented at EUSIPCO 2018 and published on IEEE https://ieeexplore.ieee.org/document/8552955

Catastrophic forgetting (CF) happens whenever a neural network overwrites past knowledge while being trained on new tasks. Common techniques to handle CF include regularization of the weights (using, e.g., their importance on past tasks), and rehearsal strategies, where the network is constantly re-trained on past data. Generative models have also been applied for the latter, in order to have endless sources of data. In this paper, we propose a novel method that combines the strengths of regularization and generative-based rehearsal approaches. Our generative model consists of a normalizing flow (NF), a probabilistic and invertible neural network, trained on the internal embeddings of the network. By keeping a single NF conditioned on the task, we show that our memory overhead remains constant. In addition, exploiting the invertibility of the NF, we propose a simple approach to regularize the network's embeddings with respect to past tasks. We show that our method performs favorably with respect to state-of-the-art approaches in the literature, with bounded computational power and memory overheads., Comment: A preliminary unpublished version of this work was presented in the LifelongML workshop, at ICML 2020

Deep neural networks are generally designed as a stack of differentiable layers, in which a prediction is obtained only after running the full stack. Recently, some contributions have proposed techniques to endow the networks with early exits, allowing to obtain predictions at intermediate points of the stack. These multi-output networks have a number of advantages, including: (i) significant reductions of the inference time, (ii) reduced tendency to overfitting and vanishing gradients, and (iii) capability of being distributed over multi-tier computation platforms. In addition, they connect to the wider themes of biological plausibility and layered cognitive reasoning. In this paper, we provide a comprehensive introduction to this family of neural networks, by describing in a unified fashion the way these architectures can be designed, trained, and actually deployed in time-constrained scenarios. We also describe in-depth their application scenarios in 5G and Fog computing environments, as long as some of the open research questions connected to them., Comment: Published in Cognitive Computation

Bayesian Neural Networks (BNNs) are trained to optimize an entire distribution over their weights instead of a single set, having significant advantages in terms of, e.g., interpretability, multi-task learning, and calibration. Because of the intractability of the resulting optimization problem, most BNNs are either sampled through Monte Carlo methods, or trained by minimizing a suitable Evidence Lower BOund (ELBO) on a variational approximation. In this paper, we propose a variant of the latter, wherein we replace the Kullback-Leibler divergence in the ELBO term with a Maximum Mean Discrepancy (MMD) estimator, inspired by recent work in variational inference. After motivating our proposal based on the properties of the MMD term, we proceed to show a number of empirical advantages of the proposed formulation over the state-of-the-art. In particular, our BNNs achieve higher accuracy on multiple benchmarks, including several image classification tasks. In addition, they are more robust to the selection of a prior over the weights, and they are better calibrated. As a second contribution, we provide a new formulation for estimating the uncertainty on a given prediction, showing it performs in a more robust fashion against adversarial attacks and the injection of noise over their inputs, compared to more classical criteria such as the differential entropy.

Graph convolutional networks (GCNs) are a family of neural network models that perform inference on graph data by interleaving vertex-wise operations and message-passing exchanges across nodes. Concerning the latter, two key questions arise: (i) how to design a differentiable exchange protocol (e.g., a 1-hop Laplacian smoothing in the original GCN), and (ii) how to characterize the trade-off in complexity with respect to the local updates. In this paper, we show that state-of-the-art results can be achieved by adapting the number of communication steps independently at every node. In particular, we endow each node with a halting unit (inspired by Graves' adaptive computation time) that after every exchange decides whether to continue communicating or not. We show that the proposed adaptive propagation GCN (AP-GCN) achieves superior or similar results to the best proposed models so far on a number of benchmarks, while requiring a small overhead in terms of additional parameters. We also investigate a regularization term to enforce an explicit trade-off between communication and accuracy. The code for the AP-GCN experiments is released as an open-source library., Comment: Published in IEEE Transaction on Neural Networks and Learning Systems

Continual learning of deep neural networks is a key requirement for scaling them up to more complex applicative scenarios and for achieving real lifelong learning of these architectures. Previous approaches to the problem have considered either the progressive increase in the size of the networks, or have tried to regularize the network behavior to equalize it with respect to previously observed tasks. In the latter case, it is essential to understand what type of information best represents this past behavior. Common techniques include regularizing the past outputs, gradients, or individual weights. In this work, we propose a new, relatively simple and efficient method to perform continual learning by regularizing instead the network internal embeddings. To make the approach scalable, we also propose a dynamic sampling strategy to reduce the memory footprint of the required external storage. We show that our method performs favorably with respect to state-of-the-art approaches in the literature, while requiring significantly less space in memory and computational time. In addition, inspired inspired by to recent works, we evaluate the impact of selecting a more flexible model for the activation functions inside the network, evaluating the impact of catastrophic forgetting on the activation functions themselves.

Multiple sclerosis is one of the most common chronic neurological diseases affecting the central nervous system. Lesions produced by the MS can be observed through two modalities of magnetic resonance (MR), known as T2W and FLAIR sequences, both providing useful information for formulating a diagnosis. However, long acquisition time makes the acquired MR image vulnerable to motion artifacts. This leads to the need of accelerating the execution of the MR analysis. In this paper, we present a deep learning method that is able to reconstruct subsampled MR images obtained by reducing the k-space data, while maintaining a high image quality that can be used to observe brain lesions. The proposed method exploits the multimodal approach of neural networks and it also focuses on the data acquisition and processing stages to reduce execution time of the MR analysis. Results prove the effectiveness of the proposed method in reconstructing subsampled MR images while saving execution time., Comment: 29th Italian Neural Networks Workshop (WIRN 2019)

In recent years, hyper-complex deep networks (such as complex-valued and quaternion-valued neural networks) have received a renewed interest in the literature. They find applications in multiple fields, ranging from image reconstruction to 3D audio processing. Similar to their real-valued counterparts, quaternion neural networks (QVNNs) require custom regularization strategies to avoid overfitting. In addition, for many real-world applications and embedded implementations, there is the need of designing sufficiently compact networks, with few weights and neurons. However, the problem of regularizing and/or sparsifying QVNNs has not been properly addressed in the literature as of now. In this paper, we show how to address both problems by designing targeted regularization strategies, which are able to minimize the number of connections and neurons of the network during training. To this end, we investigate two extensions of l1 and structured regularization to the quaternion domain. In our experimental evaluation, we show that these tailored strategies significantly outperform classical (real-valued) regularization approaches, resulting in small networks especially suitable for low-power and real-time applications., Comment: Published on CAAI Transactions on Intelligence Technology, https://digital-library.theiet.org/content/journals/10.1049/trit.2020.0020

Recently, data augmentation in the semi-supervised regime, where unlabeled data vastly outnumbers labeled data, has received a considerable attention. In this paper, we describe an efficient technique for this task, exploiting a recent framework we proposed for missing data imputation called graph imputation neural network (GINN). The key idea is to leverage both supervised and unsupervised data to build a graph of similarities between points in the dataset. Then, we augment the dataset by severely damaging a few of the nodes (up to 80\% of their features), and reconstructing them using a variation of GINN. On several benchmark datasets, we show that our method can obtain significant improvements compared to a fully-supervised model, and we are able to augment the datasets up to a factor of 10x. This points to the power of graph-based neural networks to represent structural affinities in the samples for tasks of data reconstruction and augmentation., Comment: Presented at the 2019 Italian Workshop on Neural Networks (WIRN'19)

Missing data imputation (MDI) is a fundamental problem in many scientific disciplines. Popular methods for MDI use global statistics computed from the entire data set (e.g., the feature-wise medians), or build predictive models operating independently on every instance. In this paper we propose a more general framework for MDI, leveraging recent work in the field of graph neural networks (GNNs). We formulate the MDI task in terms of a graph denoising autoencoder, where each edge of the graph encodes the similarity between two patterns. A GNN encoder learns to build intermediate representations for each example by interleaving classical projection layers and locally combining information between neighbors, while another decoding GNN learns to reconstruct the full imputed data set from this intermediate embedding. In order to speed-up training and improve the performance, we use a combination of multiple losses, including an adversarial loss implemented with the Wasserstein metric and a gradient penalty. We also explore a few extensions to the basic architecture involving the use of residual connections between layers, and of global statistics computed from the data set to improve the accuracy. On a large experimental evaluation, we show that our method robustly outperforms state-of-the-art approaches for MDI, especially for large percentages of missing values., Comment: Published in Neural Networks (2020)

In this brief we investigate the generalization properties of a recently-proposed class of non-parametric activation functions, the kernel activation functions (KAFs). KAFs introduce additional parameters in the learning process in order to adapt nonlinearities individually on a per-neuron basis, exploiting a cheap kernel expansion of every activation value. While this increase in flexibility has been shown to provide significant improvements in practice, a theoretical proof for its generalization capability has not been addressed yet in the literature. Here, we leverage recent literature on the stability properties of non-convex models trained via stochastic gradient descent (SGD). By indirectly proving two key smoothness properties of the models under consideration, we prove that neural networks endowed with KAFs generalize well when trained with SGD for a finite number of steps. Interestingly, our analysis provides a guideline for selecting one of the hyper-parameters of the model, the bandwidth of the scalar Gaussian kernel. A short experimental evaluation validates the proof., Comment: Submitted as a brief paper to IEEE TNNLS

Complex-valued neural networks (CVNNs) have been shown to be powerful nonlinear approximators when the input data can be properly modeled in the complex domain. One of the major challenges in scaling up CVNNs in practice is the design of complex activation functions. Recently, we proposed a novel framework for learning these activation functions neuron-wise in a data-dependent fashion, based on a cheap one-dimensional kernel expansion and the idea of kernel activation functions (KAFs). In this paper we argue that, despite its flexibility, this framework is still limited in the class of functions that can be modeled in the complex domain. We leverage the idea of widely linear complex kernels to extend the formulation, allowing for a richer expressiveness without an increase in the number of adaptable parameters. We test the resulting model on a set of complex-valued image classification benchmarks. Experimental results show that the resulting CVNNs can achieve higher accuracy while at the same time converging faster., Comment: Accepted at ICASSP 2019

Learning from data in the quaternion domain enables us to exploit internal dependencies of 4D signals and treating them as a single entity. One of the models that perfectly suits with quaternion-valued data processing is represented by 3D acoustic signals in their spherical harmonics decomposition. In this paper, we address the problem of localizing and detecting sound events in the spatial sound field by using quaternion-valued data processing. In particular, we consider the spherical harmonic components of the signals captured by a first-order ambisonic microphone and process them by using a quaternion convolutional neural network. Experimental results show that the proposed approach exploits the correlated nature of the ambisonic signals, thus improving accuracy results in 3D sound event detection and localization., Comment: Submitted to ICASSP 2019

Gated recurrent neural networks have achieved remarkable results in the analysis of sequential data. Inside these networks, gates are used to control the flow of information, allowing to model even very long-term dependencies in the data. In this paper, we investigate whether the original gate equation (a linear projection followed by an element-wise sigmoid) can be improved. In particular, we design a more flexible architecture, with a small number of adaptable parameters, which is able to model a wider range of gating functions than the classical one. To this end, we replace the sigmoid function in the standard gate with a non-parametric formulation extending the recently proposed kernel activation function (KAF), with the addition of a residual skip-connection. A set of experiments on sequential variants of the MNIST dataset shows that the adoption of this novel gate allows to improve accuracy with a negligible cost in terms of computational power and with a large speed-up in the number of training iterations., Comment: Accepted for presentation at 2018 IEEE International Workshop on Machine Learning for Signal Processing (MLSP)

Graph neural networks (GNNs) are a class of neural networks that allow to efficiently perform inference on data that is associated to a graph structure, such as, e.g., citation networks or knowledge graphs. While several variants of GNNs have been proposed, they only consider simple nonlinear activation functions in their layers, such as rectifiers or squashing functions. In this paper, we investigate the use of graph convolutional networks (GCNs) when combined with more complex activation functions, able to adapt from the training data. More specifically, we extend the recently proposed kernel activation function, a non-parametric model which can be implemented easily, can be regularized with standard $\ell_p$-norms techniques, and is smooth over its entire domain. Our experimental evaluation shows that the proposed architecture can significantly improve over its baseline, while similar improvements cannot be obtained by simply increasing the depth or size of the original GCN., Comment: Submitted to EUSIPCO 2018

Complex-valued neural networks (CVNNs) are a powerful modeling tool for domains where data can be naturally interpreted in terms of complex numbers. However, several analytical properties of the complex domain (e.g., holomorphicity) make the design of CVNNs a more challenging task than their real counterpart. In this paper, we consider the problem of flexible activation functions (AFs) in the complex domain, i.e., AFs endowed with sufficient degrees of freedom to adapt their shape given the training data. While this problem has received considerable attention in the real case, a very limited literature exists for CVNNs, where most activation functions are generally developed in a split fashion (i.e., by considering the real and imaginary parts of the activation separately) or with simple phase-amplitude techniques. Leveraging over the recently proposed kernel activation functions (KAFs), and related advances in the design of complex-valued kernels, we propose the first fully complex, non-parametric activation function for CVNNs, which is based on a kernel expansion with a fixed dictionary that can be implemented efficiently on vectorized hardware. Several experiments on common use cases, including prediction and channel equalization, validate our proposal when compared to real-valued neural networks and CVNNs with fixed activation functions., Comment: Submitted to IEEE Transactions on Emerging Topics in Computational Intelligence