Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of the fields of graph signal processing (GSP) and graph machine learning (GML). Key graph-aware methods include Fourier transform, filtering, sampling, as well as topology identification and spatiotemporal processing. Although versatile, graphs can model only pairwise dependencies in the data. To this end, topological structures such as simplicial and cell complexes have emerged as algebraic representations for more intricate structure modeling in data-driven systems, fueling the rapid development of novel topological-based processing and learning methods. This paper first presents the core principles of topological signal processing through the Hodge theory, a framework instrumental in propelling the field forward thanks to principled connections with GSP-GML. It then outlines advances in topological signal representation, filtering, and sampling, as well as inferring topological structures from data, processing spatiotemporal topological signals, and connections with topological machine learning. The impact of topological signal processing and learning is finally highlighted in applications dealing with flow data over networks, geometric processing, statistical ranking, biology, and semantic communication.

In this paper, we propose a new method for joint ranging and Phase Offset (PO) estimation of multiple transponder-equipped aviation vehicles (TEAVs), including Manned Aerial Vehicles (MAVs) and Unmanned Aerial Vehicles (UAVs). The proposed method employs the overlapping uncoordinated Automatic Dependent Surveillance-Broadcast (ADS-B) packets broadcasted by the TEAVs for joint range and PO estimation prior to ADS-B packet decoding; thus, it can improve air safety when packet decoding is infeasible due to packet collision. Moreover, it enables coherent detection of ADS-B packets, which can result in more reliable multiple target tracking in aviation systems using cooperative sensors for sense and avoid. By minimizing the Kullback-Leibler Divergence (KLD), we show that the received complex baseband signal, coming from K uncoordinated TEAVs, which is corrupted by Additive White Gaussian Noise (AWGN) at a single antenna receiver can be approximated by an independent and identically distributed (i.i.d.) Gaussian Mixture (GM) with 2^K mixture components in the two-dimensional plane. The proposed estimator employs the Expectation-Maximization (EM) algorithm to estimate the modes of the 2D Gaussian mixture followed by a reordering estimation technique to jointly estimate range and PO. Simulation results show that the proposed joint estimator outperforms excising methods, such as the time segmentation method and the blind adaptive beamforming.

We consider the problem of recovering complex-valued block sparse signals with unknown borders. Such signals arise naturally in numerous applications. Several algorithms have been developed to solve the problem of unknown block partitions. In pattern-coupled sparse Bayesian learning (PCSBL), each coefficient involves its own hyperparameter and those of its immediate neighbors to exploit the block sparsity. Extended block sparse Bayesian learning (EBSBL) assumes the block sparse signal consists of correlated and overlapping blocks to enforce block correlations. We propose a simpler alternative to EBSBL and reveal the underlying relationship between the proposed method and a particular case of EBSBL. The proposed algorithm uses the fact that immediate neighboring sparse coefficients are correlated. The proposed model is similar to classical sparse Bayesian learning (SBL). However, unlike the diagonal correlation matrix in conventional SBL, the unknown correlation matrix has a tridiagonal structure to capture the correlation with neighbors. Due to the entanglement of the elements in the inverse tridiagonal matrix, instead of a direct closed-form solution, an approximate solution is proposed. The alternative algorithm avoids the high dictionary coherence in EBSBL, reduces the unknowns of EBSBL, and is computationally more efficient. The sparse reconstruction performance of the algorithm is evaluated with both correlated and uncorrelated block sparse coefficients. Simulation results demonstrate that the proposed algorithm outperforms PCSBL and correlation-based methods such as EBSBL in terms of reconstruction quality. The numerical results also show that the proposed correlated SBL algorithm can deal with isolated zeros and nonzeros as well as block sparse patterns.

A new method for joint ranging and Phase Offset (PO) estimation of multiple drones/aircrafts is proposed in this paper. The proposed method employs the superimposed uncoordinated Automatic Dependent Surveillance-Broadcast (ADS-B) packets broadcasted by drones/aircrafts for joint range and PO estimation. It jointly estimates range and PO prior to ADS-B packet decoding; thus, it can improve air safety when packet decoding is infeasible due to packet collision. Moreover, it enables coherent detection of ADS-B packets, which can result in more reliable multiple target tracking in aviation systems using cooperative sensors for detect and avoid (DAA). By minimizing the Kullback-Leibler Divergence (KLD) statistical distance measure, we show that the received complex baseband signal coming from K uncoordinated drones/aircrafts corrupted by Additive White Gaussian Noise (AWGN) at a single antenna receiver can be approximated by an independent and identically distributed (i.i.d.) Gaussian Mixture (GM) with 2^{K} mixture components in the two-dimensional (2D) plane. While direct joint Maximum Likelihood Estimation (MLE) of range and PO from the derived GM Probability Density Function (PDF) leads to an intractable maximization, our proposed method employs the Expectation-Maximization (EM) algorithm to estimate the modes of the 2D Gaussian mixture followed by a reordering estimation technique through combinatorial optimization to estimate range and PO. An extension to a multiple antenna receiver is also investigated in this article. While the proposed estimator can estimate the range of multiple drones/aircrafts with a single receive antenna, a larger number of drones/aircrafts can be supported with higher accuracy by the use of multiple antennas at the receiver. The effectiveness of the proposed estimator is supported by simulation results. We show that the proposed estimator can jointly estimate the range of multiple drones/aircrafts accurately.

Identifying overlapping communities from data is crucial for grasping the complex structure and dynamics of networks, amongst others in fields such as computational neuroscience. Research using fMRI has demonstrated that brain regions can change their functional network membership over time using temporal independent component analysis (tICA). However, reproducibility of such overlapping communities remains a challenge. Recently, several alternative approaches have been proposed to identify such overlapping communities. While results are promising, less is known about the model and assumptions that underlie these approaches. This paper shows that the bilinear model, combined with the assumption of quasi-stationary and uncorrelated sources, underlies novel methods for identifying overlapping brain networks. Furthermore, we propose a new algorithm, and through simulations, we investigate the robustness of our algorithm and several existing methods to solve the problem in noisy conditions with few available data samples. We conclude that quasi-stationary blind source separation-based techniques can have a promising advantage over tICA in terms of identifiability of overlapping communities and thus have the potential to contribute towards greater reproducibility of results.

In this paper, we present a novel convolution theorem which encompasses the well known convolution theorem in (graph) signal processing as well as the one related to time-varying filters. Specifically, we show how a node-wise convolution for signals supported on a graph can be expressed as another node-wise convolution in a frequency domain graph, different from the original graph. This is achieved through a parameterization of the filter coefficients following a basis expansion model. After showing how the presented theorem is consistent with the already existing body of literature, we discuss its implications in terms of non-stationarity. Finally, we propose a data-driven algorithm based on subspace fitting to learn the frequency domain graph, which is then corroborated by experimental results on synthetic and real data.

The main focus of this paper is an active sensing application that involves selecting transmit and receive sensors to optimize the Cramér-Rao bound (CRB) on target parameters. Although the CRB is non-convex in the transmit and receive selection, we demonstrate that it is convex in the virtual array weight vector, which describes the multiplicity of the virtual array elements. Based on this finding, we propose a novel algorithm that optimizes the virtual array weight vector first and then finds a matching transceiver array. This greatly enhances the efficiency of the transmit and receive sensor selection problem.

In this work, we deal with the problem of reconstructing a complete bandlimited graph signal from partially sampled noisy measurements. For a known graph structure, an efficient greedy algorithm is presented to partition the graph nodes into disjoint subsets such that sampling the graph signal from any subset leads to a sufficiently accurate reconstruction. Furthermore, we consider a scenario where the graph is massive and data processing centrally is no longer practical. To overcome this issue, a distributed framework is proposed that allows us to implement partitioning algorithms in a parallelized fashion. Finally, we provide numerical simulation results on synthetic and real-world data to show that our proposals outperform the state-of-the-art.

Four-dimensional ultrasound imaging of complex biological systems such as the brain is technically challenging because of the spatiotemporal sampling requirements. We present computational ultrasound imaging (cUSi), an imaging method that uses complex ultrasound fields that can be generated with simple hardware and a physical wave prediction model to alleviate the sampling constraints. cUSi allows for high-resolution four-dimensional imaging of brain hemodynamics in awake and anesthetized mice.

In this work, we consider the self-calibration problem of joint calibration and direction-of-Arrival (DOA) estimation using acoustic sensor arrays. Unlike many previous iterative approaches, we propose solvers that can be readily used for both linear and non-linear arrays for jointly estimating the sensor gain, phase errors, and the source DOAs. We derive these algorithms for both the conventional element-space and covariance data models. We focus on sparse and regular arrays formed using scalar sensors as well as vector sensors. The developed algorithms are obtained by transforming the underlying non-linear calibration model into a linear model, and subsequently by using convex relaxation techniques to estimate the unknown parameters. We also derive identifiability conditions for the existence of a unique solution to the self-calibration problem. To demonstrate the effectiveness of the developed techniques, numerical experiments, and comparisons to the state-of-The-Art methods are provided. Finally, the results from an experiment that was performed in an anechoic chamber using an acoustic vector sensor array are presented to demonstrate the usefulness of the proposed self-calibration techniques.

The edge flow reconstruction task consists of retreiving edge flow signals from corrupted or incomplete measurements. This is typically solved by a regularized optimization problem on higher-order networks such as simplicial complexes and the corresponding regularizers are chosen based on prior knowledge. Tailoring this prior to the setting of interest can be challenging or it may not even be possible. Thus, we consider to learn this prior knowledge via a model-based deep learning approach. We propose a new regularized optimization problem for the simplicial edge flow reconstruction task, the simplicial ElasticNet, which combines the advantages of the 1 and 2 norms. We solve the simplicial ElasticNet problem via the multi-block alternating direction method of multipliers (ADMM) algorithm and provide conditions on its convergence. By unrolling the ADMM iterative steps, we develop a model-based neural network with a low requirement on the number of training data. This unrolling network replaces the fixed parameters in the iterative algorithm by learnable weights, thus exploiting the neural network s learning capability while preserving the iterative algorithm s interpretability. We enhance this unrolling network via simplicial convolutional filters to aggregate information from the edge flow neighbors, ultimately, improving the network learning expressivity. Extensive experiments on real-world and synthetic datasets validate the proposed approaches and show considerable improvements over both baselines and traditional non-model-based neural networks.