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The inherent "black box" nature of deep neural networks (DNNs) compromises their transparency and reliability. Recently, explainable AI (XAI) has garnered increasing attention from researchers. Several perturbation-based interpretations have emerged. However, these methods often fail to adequately consider feature dependencies. To solve this problem, we introduce a perturbation-based interpretation guided by feature coalitions, which leverages deep information of network to extract correlated features. Then, we proposed a carefully-designed consistency loss to guide network interpretation. Both quantitative and qualitative experiments are conducted to validate the effectiveness of our proposed method. Code is available at github.com/Teriri1999/Perturebation-on-Feature-Coalition., Comment: 4 pages, 4 figures, 2 tables

Many real-world networks are characterized by directionality; however, the absence of an appropriate Fourier basis hinders the effective implementation of graph signal processing techniques. Inspired by discrete signal processing, where embedding a line digraph into a cycle digraph facilitates the powerful Discrete Fourier Transform for signal analysis, addressing the structural complexities of general digraphs can help overcome the limitations of the Graph Fourier Transform (GFT) and unlock its potential. The Discrete Fourier Transform (DFT) serves as a Graph Fourier Transform for both cycle graphs and Cayley digraphs on the finite cyclic groups $\mathbb{Z}_N$. We propose a systematic method to identify a class of such Cayley digraphs that can encompass a given directed graph. By embedding the directed graph into these Cayley digraphs and opting for envelope extensions that naturally support the Graph Fourier Transform, the GFT functionalities of these extensions can be harnessed for signal analysis. Among the potential envelopes, optimal performance is achieved by selecting one that meets key properties. This envelope's structure closely aligns with the characteristics of the original digraph. The Graph Fourier Transform of this envelope is reliable in terms of numerical stability, and its columns approximately form an eigenbasis for the adjacency matrix associated with the original digraph. It is shown that the envelope extensions possess a convolution product, with their GFT fulfilling the convolution theorem. Additionally, shift-invariant graph filters (systems) are described as the convolution operator, analogous to the classical case. This allows the utilization of systems for signal analysis., Comment: 14 pages, 16 figures, 3 tables

Directed acyclic graphs (DAGs) are used for modeling causal relationships, dependencies, and flows in various systems. However, spectral analysis becomes impractical in this setting because the eigendecomposition of the adjacency matrix yields all eigenvalues equal to zero. This inherent property of DAGs results in an inability to differentiate between frequency components of signals on such graphs. This problem can be addressed by {alternating the Fourier basis or adding edges in a DAG}. However, these approaches change the physics of the considered problem. To address this limitation, we propose a \textit{graph zero-padding} approach. This approach involves augmenting the original DAG with additional vertices that are connected to the existing structure. The added vertices are characterized by signal values set to zero. The proposed technique enables the spectral evaluation of system outputs on DAGs (in almost all cases), that is the computation of vertex-domain convolution without the adverse effects of aliasing due to changes in a graph structure, { with the ultimate goal of preserving the output of the system on a graph as if the changes in the graph structure were not performed}., Comment: 28 pages, 13 figures

A comprehensive approach to the spectrum characterization (derivation of eigenvalues and the corresponding multiplicities) for non-normalized, symmetric discrete trigonometric transforms (DTT) is presented in the paper. Eight types of the DTT are analyzed. New explicit analytic expressions for the eigenvalues, together with their multiplicities, for the cases of three DTT (DCT$_{(1)}$, DCT$_{(5)}$, and DST$_{(8)}$), are the main contribution of this paper. Moreover, the presented theory is supplemented by new, original derivations for the closed-form expressions of the square and the trace of analyzed DTT matrices., Comment: 9 pages

Despite the tremendous success of convolutional neural networks (CNNs) in computer vision, the mechanism of CNNs still lacks clear interpretation. Currently, class activation mapping (CAM), a famous visualization technique to interpret CNN's decision, has drawn increasing attention. Gradient-based CAMs are efficient while the performance is heavily affected by gradient vanishing and exploding. In contrast, gradient-free CAMs can avoid computing gradients to produce more understandable results. However, existing gradient-free CAMs are quite time-consuming because hundreds of forward interference per image are required. In this paper, we proposed Cluster-CAM, an effective and efficient gradient-free CNN interpretation algorithm. Cluster-CAM can significantly reduce the times of forward propagation by splitting the feature maps into clusters in an unsupervised manner. Furthermore, we propose an artful strategy to forge a cognition-base map and cognition-scissors from clustered feature maps. The final salience heatmap will be computed by merging the above cognition maps. Qualitative results conspicuously show that Cluster-CAM can produce heatmaps where the highlighted regions match the human's cognition more precisely than existing CAMs. The quantitative evaluation further demonstrates the superiority of Cluster-CAM in both effectiveness and efficiency., Comment: 10 pages

Generative Adversarial Networks (GANs) can synthesize abundant photo-realistic synthetic aperture radar (SAR) images. Some recent GANs (e.g., InfoGAN), are even able to edit specific properties of the synthesized images by introducing latent codes. It is crucial for SAR image synthesis since the targets in real SAR images are with different properties due to the imaging mechanism. Despite the success of InfoGAN in manipulating properties, there still lacks a clear explanation of how these latent codes affect synthesized properties, thus editing specific properties usually relies on empirical trials, unreliable and time-consuming. In this paper, we show that latent codes are disentangled to affect the properties of SAR images in a non-linear manner. By introducing some property estimators for latent codes, we are able to provide a completely analytical nonlinear model to decompose the entangled causality between latent codes and different properties. The qualitative and quantitative experimental results further reveal that the properties can be calculated by latent codes, inversely, the satisfying latent codes can be estimated given desired properties. In this case, properties can be manipulated by latent codes as we expect., Comment: 13 pages, 14 figures

In this account, we describe the crazy-clock phenomenon involving the state I (low iodide and iodine concentration) to state II (high iodide and iodine concentration with new iodine phase) transition after a Briggs-Rauscher (BR) oscillatory process. While the BR crazy-clock phenomenon is known, it is the first time that crazy-clock behavior is linked and explained with the symmetry-breaking phenomenon, highlighting the entire process in a novel way. The presented phenomenon has been thoroughly investigated by running more than 60 experiments, and evaluated by using statistical cluster K-means analysis. The mixing rate, as well as the magnetic bar shape and dimensions, have a strong influence on the transition appearance. Although the transition for both mixing and no-mixing conditions are taking place completely randomly, by using statistical cluster analysis we obtain different numbers of clusters (showing the time-domains where the transition is more likely to occur). In the case of stirring, clusters are more compact and separated, revealed new hidden details regarding the chemical dynamics of nonlinear processes. The significance of the presented results is beyond oscillatory reaction kinetics since the described example belongs to the small class of chemical systems that shows intrinsic randomness in their response and it might be considered as a real example of a classical liquid random number generator.

Within the Compressive Sensing (CS) paradigm, sparse signals can be reconstructed based on a reduced set of measurements. Reliability of the solution is determined by the uniqueness condition. With its mathematically tractable and feasible calculation, coherence index is one of very few CS metrics with a considerable practical importance. In this paper, we propose an improvement of the coherence based uniqueness relation for the matching pursuit algorithms. Starting from a simple and intuitive derivation of the standard uniqueness condition based on the coherence index, we derive a less conservative coherence index-based lower bound for signal sparsity. The results are generalized to the uniqueness condition of the $l_0$-norm minimization for a signal represented in two orthonormal bases., Comment: 5 pages, 1 figure

In this paper, we present an approach to the reconstruction of signals exhibiting sparsity in a transformation domain, having some heavily disturbed samples. This sparsity-driven signal recovery exploits a carefully suited random sampling consensus (RANSAC) methodology for the selection of an inlier subset of samples. To this aim, two fundamental properties are used: a signal sample represents a linear combination of the sparse coefficients, whereas the disturbance degrade original signal sparsity. The properly selected samples are further used as measurements in the sparse signal reconstruction, performed using algorithms from the compressive sensing framework. Besides the fact that the disturbance degrades signal sparsity in the transformation domain, no other disturbance-related assumptions are made -- there are no special requirements regarding its statistical behavior or the range of its values. As a case study, the discrete Fourier transform (DFT) is considered as a domain of signal sparsity, owing to its significance in signal processing theory and applications. Numerical results strongly support the presented theory. In addition, exact relation for the signal-to-noise ratio (SNR) of the reconstructed signal is also presented. This simple result, which conveniently characterizes the RANSAC-based reconstruction performance, is numerically confirmed by a set of statistical examples., Comment: 4 figures, 15 pages

Many modern data analytics applications on graphs operate on domains where graph topology is not known a priori, and hence its determination becomes part of the problem definition, rather than serving as prior knowledge which aids the problem solution. Part III of this monograph starts by addressing ways to learn graph topology, from the case where the physics of the problem already suggest a possible topology, through to most general cases where the graph topology is learned from the data. A particular emphasis is on graph topology definition based on the correlation and precision matrices of the observed data, combined with additional prior knowledge and structural conditions, such as the smoothness or sparsity of graph connections. For learning sparse graphs (with small number of edges), the least absolute shrinkage and selection operator, known as LASSO is employed, along with its graph specific variant, graphical LASSO. For completeness, both variants of LASSO are derived in an intuitive way, and explained. An in-depth elaboration of the graph topology learning paradigm is provided through several examples on physically well defined graphs, such as electric circuits, linear heat transfer, social and computer networks, and spring-mass systems. As many graph neural networks (GNN) and convolutional graph networks (GCN) are emerging, we have also reviewed the main trends in GNNs and GCNs, from the perspective of graph signal filtering. Tensor representation of lattice-structured graphs is next considered, and it is shown that tensors (multidimensional data arrays) are a special class of graph signals, whereby the graph vertices reside on a high-dimensional regular lattice structure. This part of monograph concludes with two emerging applications in financial data processing and underground transportation networks modeling., Comment: 61 pages, 55 figures, 40 examples

The focus of Part I of this monograph has been on both the fundamental properties, graph topologies, and spectral representations of graphs. Part II embarks on these concepts to address the algorithmic and practical issues centered round data/signal processing on graphs, that is, the focus is on the analysis and estimation of both deterministic and random data on graphs. The fundamental ideas related to graph signals are introduced through a simple and intuitive, yet illustrative and general enough case study of multisensor temperature field estimation. The concept of systems on graph is defined using graph signal shift operators, which generalize the corresponding principles from traditional learning systems. At the core of the spectral domain representation of graph signals and systems is the Graph Discrete Fourier Transform (GDFT). The spectral domain representations are then used as the basis to introduce graph signal filtering concepts and address their design, including Chebyshev polynomial approximation series. Ideas related to the sampling of graph signals are presented and further linked with compressive sensing. Localized graph signal analysis in the joint vertex-spectral domain is referred to as the vertex-frequency analysis, since it can be considered as an extension of classical time-frequency analysis to the graph domain of a signal. Important topics related to the local graph Fourier transform (LGFT) are covered, together with its various forms including the graph spectral and vertex domain windows and the inversion conditions and relations. A link between the LGFT with spectral varying window and the spectral graph wavelet transform (SGWT) is also established. Realizations of the LGFT and SGWT using polynomial (Chebyshev) approximations of the spectral functions are further considered. Finally, energy versions of the vertex-frequency representations are introduced., Comment: 60 pages, 50 figures

A unitary shift operator (GSO) for signals on a graph is introduced, which exhibits the desired property of energy preservation over both backward and forward graph shifts. For rigour, the graph differential operator is also derived in an analytical form. The commutativity relation of the shift operator with the Fourier transform is next explored in conjunction with the proposed GSO to introduce a graph discrete Fourier transform (GDFT) which, unlike existing approaches, ensures the orthogonality of GDFT bases and admits a natural frequency-domain interpretation. The proposed GDFT is shown to allow for a coherent definition of the graph discrete Hilbert transform (GDHT) and the graph analytic signal. The advantages of the proposed GSO are demonstrated through illustrative examples., Comment: 5 pages, 3 figures

A class of doubly stochastic graph shift operators (GSO) is proposed, which is shown to exhibit: (i) lower and upper $L_{2}$-boundedness for locally stationary random graph signals; (ii) $L_{2}$-isometry for \textit{i.i.d.} random graph signals with the asymptotic increase in the incoming neighbourhood size of vertices; and (iii) preservation of the mean of any graph signal. These properties are obtained through a statistical consistency analysis of the graph shift, and by exploiting the dual role of the doubly stochastic GSO as a Markov (diffusion) matrix and as an unbiased expectation operator. Practical utility of the class of doubly stochastic GSOs is demonstrated in a real-world multi-sensor signal filtering setting., Comment: 5 pages, 1 figure

The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data., Comment: 49 pages, 40 figures

Graph signal processing deals with signals which are observed on an irregular graph domain. While many approaches have been developed in classical graph theory to cluster vertices and segment large graphs in a signal independent way, signal localization based approaches to the analysis of data on graph represent a new research direction which is also a key to big data analytics on graphs. To this end, after an overview of the basic definitions in graphs and graph signals, we present and discuss a localized form of the graph Fourier transform. To establish an analogy with classical signal processing, spectral- and vertex-domain definitions of the localization window are given next. The spectral and vertex localization kernels are then related to the wavelet transform, followed by a study of filtering and inversion of the localized graph Fourier transform. For rigour, the analysis of energy representation and frames in the localized graph Fourier transform is extended to the energy forms of vertex-frequency distributions, which operate even without the need to apply localization windows. Another link with classical signal processing is established through the concept of local smoothness, which is subsequently related to the particular paradigm of signal smoothness on graphs. This all represents a comprehensive account of the relation of general vertex-frequency analysis with classical time-frequency analysis, and important but missing link for more advanced applications of graphs signal processing. The theory is supported by illustrative and practically relevant examples., Comment: 16 pages, 12 figures

Influence of the finite-length registers and quantization effects on the reconstruction of sparse and approximately sparse signals is analyzed in this paper. For the nonquantized measurements, the compressive sensing (CS) framework provides highly accurate reconstruction algorithms that produce negligible errors when the reconstruction conditions are met. However, hardware implementations of signal processing algorithms involve the finite-length registers and quantization of the measurements. An analysis of the effects related to the measurements quantization with an arbitrary number of bits is the topic of this paper. A unified mathematical model for the analysis of the quantization noise and the signal nonsparsity on the CS reconstruction is presented. An exact formula for the expected energy of error in the CS-based reconstructed signal is derived. The theory is validated through various numerical examples with quantized measurements, including the cases of approximately sparse signals, noise folding, and floating-point arithmetics., Comment: 13 pages, 7 figures

With their ability to handle an increased amount of information, multivariate and multichannel signals can be used to solve problems normally not solvable with signals obtained from a single source. One such problem is the decomposition signals with several components whose domains of support significantly overlap in both the time and the frequency domain, including the joint time-frequency domain. Initially, we proposed a solution to this problem based on the Wigner distribution of multivariate signals, which requires the attenuation of the cross-terms. In this paper, an advanced solution based on an eigenvalue analysis of the multivariate signal autocorrelation matrix, followed by their time-frequency concentration measure minimization, is presented. This analysis provides less restrictive conditions for the signal decomposition than in the case of Wigner distribution. The algorithm for the components separation is based on the concentration measures of the eigenvector time-frequency representation, that are linear combinations of the overlapping signal components. With an increased number of sensors/channels, the robustness of the decomposition process to additive noise is also achieved. The theory is supported by numerical examples. The required channel dissimilarity is statistically investigated as well., Comment: 13 pages, 10 figures

Graphs are irregular structures which naturally account for data integrity, however, traditional approaches have been established outside Signal Processing, and largely focus on analyzing the underlying graphs rather than signals on graphs. Given the rapidly increasing availability of multisensor and multinode measurements, likely recorded on irregular or ad-hoc grids, it would be extremely advantageous to analyze such structured data as graph signals and thus benefit from the ability of graphs to incorporate spatial awareness of the sensing locations, sensor importance, and local versus global sensor association. The aim of this lecture note is therefore to establish a common language between graph signals, defined on irregular signal domains, and some of the most fundamental paradigms in DSP, such as spectral analysis of multichannel signals, system transfer function, digital filter design, parameter estimation, and optimal filter design. This is achieved through a physically meaningful and intuitive real-world example of geographically distributed multisensor temperature estimation. A similar spatial multisensor arrangement is already widely used in Signal Processing curricula to introduce minimum variance estimators and Kalman filters \cite{HM}, and by adopting this framework we facilitate a seamless integration of graph theory into the curriculum of existing DSP courses. By bridging the gap between standard approaches and graph signal processing, we also show that standard methods can be thought of as special cases of their graph counterparts, evaluated on linear graphs. It is hoped that our approach would not only help to demystify graph theoretic approaches in education and research but it would also empower practitioners to explore a whole host of otherwise prohibitive modern applications., Comment: 10 pages, 3 figures, 5 boxes

The existence and uniqueness conditions are a prerequisite for reliable reconstruction of sparse signals from reduced sets of measurements within the Compressive Sensing (CS) paradigm. However, despite their underpinning role for practical applications, existing uniqueness relations are either computationally prohibitive to implement (Restricted Isometry Property), or involve mathematical tools that are beyond the standard background of engineering graduates (Coherence Index). This can introduce conceptual and computational obstacles in the development of engineering intuition, the design of suboptimal practical solutions, or understanding of limitations. To this end, we introduce a simple but rigorous derivation of the coherence index condition, based on standard linear algebra, with the aim to empower signal processing practitioners with intuition in the design and ease in implementation of CS systems. Given that the coherence index is one of very few CS metrics that admits mathematically tractable and computationally feasible calculation, it is our hope that this work will help bridge the gap between the theory and applications of compressive sensing., Comment: 8 pages, 3 figures, 3 boxes

A signal is sparse in one of its representation domain if the number of nonzero coefficients in that domain is much smaller than the total number of coefficients. Sparse signals can be reconstructed from a very reduced set of measurements/observations. The topic of this paper are conditions for the unique reconstruction of sparse signals from a reduced set of observations. After the basic definitions are introduced, the unique reconstruction conditions are reviewed using the spark, restricted isometry, and coherence of the measurement matrix. Uniqueness of the reconstruction of signals sparse in the discrete Fourier domain (DFT), as the most important signal transformation domain, is considered as well., Comment: 32 pages, 3 figures, 4 MATLAB programs