Summary of the Emerging Field of Signal Processing on Graphs - Paper Example

Social, transportation, energy, neutral networks, sensor and high-dimensional data naturally reside on the vertices of the weighted graphs. The resultant field of signal processing on graph merges spectral and algebraic graph theoretical concepts having the computational harmonic analysis that will process the signals on the graphs. The main purpose for the generation of the graphs is for data representation in such a way that they describe the geometric structures of the data domains in various applications. There are different graph fields depending on the engineering field. Common data processing tasks in the various applications are filtering, in painting, denoising and compressing graph signals. There are a lot of considerations that are taken into place before generation of the signal graphs. There are some challenges of signal processing on graphs. Both signals on a classical discrete-time signal having N samples and a graph with N vertices can be considered as vectors in RN. The challenge to the application of signal processing technique in the graph is that processing the graphed signal in a way similar to discrete time signal ignores the key dependencies that arise from the irregular data domain. Furthermore, there are many but simple fundamental concepts that underlie the classical signal processing methods that turn to be more challenging in the graph setting.

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Graphs in general

The graph spectral domains focuses on the construction, analysis and manipulation of graph compared to signal graphs. In signal processing of graphs, the spectral graph theory has been considered as a tool to determine the frequency spectra and expansion bases for graphs. The weighted graphs and graphs signals are used when analyzing signals defined on connected, weighted graph G ={V, E, W}, that entails finite set of vertices V with [V] = N weighted adjacency matrix W and a set of edges E. The graph Laplacian which is also known as the combinatorial graph Laplacian is defined as L := D -W, where the degree of the matrix L stands for the diagnol matrix that its I th diagnol element d I is the same to the sum of the weights of all the edges that is incident to the vertex i. This graph is a real symmetric matrix and therefore has an entire set of orthonormal eigen values. The eigen values that are used have real, non-negative eigen values. The zero will appear as an eigen value that has multiplicity same to the number of connected components to the graphs.

Graph representation in two domains is of great help in describing data on the graphs. The graph fourier change and its inverse makes it easy to represent a signal in two different domains, the graph spectral domain and the vertex domain. We do start with signal g in the vertex domain. However, it is important also to define signal g directly in the graph spectral domain. They are referring to as kernels. Discrete calculus and signal smoothness with respect to the intrinsic structure of the graph helps in analyzing and emphasizing on the smoothness with respect to the intrinsic structure of the data domain that is the weighted graph. Differential geometry gives tools to incorporate the geometric structure symmetry of the underlying manifolds into the analysis of continuous signals on differentiable manifold operators that makes it convenient to operate the machinery.

Generalized Operators for Signal on Graphs

The operators include translation, filtering, modulation, down sampling and modulation to the graph setting. They help in developing and localized multiscale transforms. Filtering entails the extending nortion of frequency filtering to the graph setting. Frequency filtering refers to the representation of an input as a linear combination of complex exponential and attenuating or amplifying the contributions to some of the component complex exponentials. Filtering in the vertex domain , the output is written in the vertex I as a linear combination of the components of the input signal a the vertices within a K-hop local neighborhood of vertex i. Translation as an operator is seen through the transformation of variable (Ty f ) (t) :=f (t -y), that cannot directly show the graph setting. This is unlike the convolution where one cannot directly generalize its definition. This is because of the term h(t -x) in the convolution product of graph setting. The only way generalized convolution product can be defined for signal is through replacing the complex exponential in with the graph laplacian eigen values. Modulation and dilation is also used to localize the signals frequency content using the classical modulation operator: (M~f ) (t) := e2 i t f (t) that can be represented as (M~f (E) =f (p -~), 6p ! R. Down sampling and reduction are the operators that are used when various multiscale transforms for signals on graphs that needs successfully coarser versions of the original graph that can retain the original properties of the initial graph such as the intrinsic geometric structure. This can be done by identifying the reduced set of vertices and then assigning the edges and the weights that have been reduced to link with the new set.

Summary of Big data analysis with signal processing on graphs

There are numerous situations that we are faced to analyze a he set of data. This is mostly common in government, scientific, commercial and industrial domains. There are a number of innovative approaches that have ben devised for extraction of important data from large data. Some of these approaches include singular value decomposition, principal component analysis, and spectral analysis among others. Some of the proposed methods that have been put forward for big data analysis have complex structures on them. Low dimensional representations of high-dimentional data have many approaches and in all these approaches, the data sets are regarded as graphs in high-dimensional spaces that is created by small subsets of the graph Laplacian eigenvalues. Signal processing on graphs goes beyond the classical signal processing theory to produce graphs. DSPs are one of the best methodologies that can be used in the big data analysis. It is important to understand fundamental signal processing techniques such as filtering and the right graph model.

Signal processing on graphs can be used for big data analysis. DSPG looks into the analysis and processing of data sets where the data elements are related by similarity, dependency as well as physical proximity. This is expressed as G = ( , V A), where V = { , v v 0 1 f, } N- is the set of N nodes and A is the weighted adjacency matrix of the graph. A nonzero weight may be taken to show the present of a directed age from Vm to Vn. Signal graphs are defined by the map s: V " C v s n n 7 , and C stands for the set of complex numbers. It is easier to write the graph signal as vectors. The finite periodic time series are indexed directly to the cyclic graphs so that all the edges can be directed to have the same weight and each node to correspond to the time sample. The data that has been collected by the sensor is also an example of graph signal as each node stand for a web site. Graph shift comes in when during the analysis with DSPG, a signal shift is applied as the time delays. sun n = s -1 mod N can be used to find the delayed finite period of length N. DSPG goes beyond the concept of shift to general graphs by showing the graph shift as a local operation that substitutes a signal value Sn at node Vn. Being that DSPG graph shift is determined by the axiomatically, other options for the operation of a graph shift are acceptable. The benefit that is associated with this kind is that it leads to a significant processing framework for commutative and linear graph filters. In signal processing, filter refers to a system that takes a signal as an output and input signal. A Fourier transform with respect to a set of operators is the extension of a signal into fundamentals of the operators of the eigen values. The DSPG helps in data analysis by defining the Fourier transform with respect to the graph filters. There are a number of alternative choices of graph Fourier basis. The challenges that are involved in data analysis are addressed by DSPG by representation of data set structure with graphs then amounting the data into graph signals.