Do users from Carnegie Mellon University form social communities on Facebook? Do signal processing researchers tightly collaborate with each other? Do Chinese restaurants in Manhattan cluster together? These seemingly different problems share a common structure: an attribute that may be localized on a graph. In other words, nodes activated by an attribute form a subgraph that can be easily separated from other nodes. In this paper, we thus focus on the task of detecting localized attributes on a graph. We are particularly interested in categorical attributes such as attributes in online social networks, ratings in recommender systems, and viruses in cyber-physical systems because they are widely used in numerous data mining applications. To solve the task, we formulate a statistical hypothesis testing problem to decide whether a given attribute is localized or not. We propose two statistics: Graph wavelet statistic and graph scan statistic, both of which are provably effective in detecting localized attributes. We validate the robustness of the proposed statistics on both simulated data and two real-world applications: High air-pollution detection and keyword ranking in a coauthorship network collected from IEEE Xplore. Experimental results show that the proposed graph wavelet statistic and graph scan statistic are effective and efficient.
@cite_67: The synthesis of a new category of spatial filters that produces sharp output correlation peaks with controlled peak values is considered. The sharp nature of the correlation peak is the major feature emphasized, since it facilitates target detection. Since these filters minimize the average correlation plane energy as the first step in filter synthesis, we refer to them as minimum average correlation energy filters. Experimental laboratory results from optical implementation of the filters are also presented and discussed.
@cite_55: A selective sampling procedure called distilled sensing (DS) is proposed, and shown to be an efiective method for recovering sparse signals in noise. Based on the notion that it is often easier to rule out locations that do not contain signal than it is to directly identify non-zero signal components, DS is a sequential method that systematically focuses sensing resources towards the signal subspace. This adaptivity in sensing results in rather surprising gains in sparse signal recovery| dramatically weaker sparse signals can be recovered using DS compared with conventional non-adaptive sensing procedures.
@cite_49: The smallest signal discernible through background noise is formulated in terms of the pulse energy, its repetition rate, the receiver design, and the choice of integrating and indicating means. The smallest signal visible on a Type-A scan can be improved upon by the use of electromechanical integrators. Integration before detection is, in theory, ultimately the most effective, but runs into serious practical difficulties. These are avoided when the integration is performed after detection. Optimum predetector selectivity is formulated. The optimum detector (square law) is found, and other detectors compared. A transmitter criterion provides a basis for comparing the effectiveness of transmitters, and shows how, at the expense of range resolution, longer pulses can increase visibility.
@cite_80: 
@cite_15: We introduce a new signal model, called (K,C)-sparse, to capture K-sparse signals in N dimensions whose nonzero coefficients are contained within at most C clusters, with C < K << N. In contrast to the existing work in the sparse approximation and compressive sensing literature on block sparsity, no prior knowledge of the locations and sizes of the clusters is assumed. We prove that O (K + C log(N C))) random projections are sufficient for (K,C)-model sparse signal recovery based on subspace enumeration. We also provide a robust polynomialtime recovery algorithm for (K,C)-model sparse signals with provable estimation guarantees.
@cite_17: 1. On rainbows and spectra 2. From Euclid to Hilbert: 2.1 Introduction 2.2 Vector spaces 2.3 Hilbert spaces 2.4 Approximations, projections, and decompositions 2.5 Bases and frames 2.6 Computational aspects 2.A Elements of analysis and topology 2.B Elements of linear algebra 2.C Elements of probability 2.D Basis concepts Exercises with solutions Exercises 3. Sequences and discrete-time systems: 3.1 Introduction 3.2 Sequences 3.3 Systems 3.4 Discrete-time Fourier Transform 3.5 z-Transform 3.6 Discrete Fourier Transform 3.7 Multirate sequences and systems 3.8 Stochastic processes and systems 3.9 Computational aspects 3.A Elements of analysis 3.B Elements of algebra Exercises with solutions Exercises 4. Functions and continuous-time systems: 4.1 Introduction 4.2 Functions 4.3 Systems 4.4 Fourier Transform 4.5 Fourier series 4.6 Stochastic processes and systems Exercises with solutions Exercises 5. Sampling and interpolation: 5.1 Introduction 5.2 Finite-dimensional vectors 5.3 Sequences 5.4 Functions 5.5 Periodic functions 5.6 Computational aspects Exercises with solutions Exercises 6. Approximation and compression: 6.1 Introduction 6.2 Approximation of functions on finite intervals by polynomials 6.3 Approximation of functions by splines 6.4 Approximation of functions and sequences by series truncation 6.5 Compression 6.6 Computational aspects Exercises with solutions Exercises 7. Localization and uncertainty: 7.1 Introduction 7.2 Localization for functions 7.3 Localization for sequences 7.4 Tiling the time-frequency plane 7.5 Examples of local Fourier and wavelet bases 7.6 Recap and a glimpse forward Exercises with solutions Exercises.
In classical signal processing, a localized signal is constant over local connected regions separated by lower-dimensional boundaries. It is often related to concepts such as impulse function, step function, square wave and Haar basis @cite_17 . Detecting localized signals has been considered through signal noise discrimination @cite_49 , edge detection @cite_80 , pattern matching @cite_67 and support recovery of sparse signals @cite_55 @cite_15 , among others. We here look at the counterpart problem on graphs. A localized attribute (graph signal) is constant over a subgraph that is easily separated from the rest of the nodes. Similarly to localized signals in classical signal processing, a localized attribute emphasizes fast transitions (corresponding to boundaries) and localization in the graph vertex domain (corresponding to attributes that are nonzero in a local neighborhood).