David I Shuman

The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains

In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogs to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.

Learning Parametric Dictionaries for Signals on Graphs

In sparse signal representation, the choice of a dictionary often involves a tradeoff between two desirable properties - the ability to adapt to specific signal data and a fast implementation of the dictionary. To sparsely represent signals residing on weighted graphs, an additional design challenge is to incorporate the intrinsic geometric structure of the irregular data domain into the atoms of the dictionary. In this work, we propose a parametric dictionary learning algorithm to design data-adapted, structured dictionaries that sparsely represent graph signals. In particular, we model graph signals as combinations of overlapping local patterns. We impose the constraint that each dictionary is a concatenation of subdictionaries, with each subdictionary being a polynomial of the graph Laplacian matrix, representing a single pattern translated to different areas of the graph. The learning algorithm adapts the patterns to a training set of graph signals. Experimental results on both synthetic and real datasets demonstrate that the dictionaries learned by the proposed algorithm are competitive with and often better than unstructured dictionaries learned by state-of-the-art numerical learning algorithms in terms of sparse approximation of graph signals. In contrast to the unstructured dictionaries, however, the dictionaries learned by the proposed algorithm feature localized atoms and can be implemented in a computationally efficient manner in signal processing tasks such as compression, denoising, and classification.

Chebyshev polynomial approximation for distributed signal processing

Unions of graph Fourier multipliers are an important class of linear operators for processing signals defined on graphs. We present a novel method to efficiently distribute the application of these operators to the high-dimensional signals collected by sensor networks. The proposed method features approximations of the graph Fourier multipliers by shifted Chebyshev polynomials, whose recurrence relations make them readily amenable to distributed computation. We demonstrate how the proposed method can be used in a distributed denoising task, and show that the communication requirements of the method scale gracefully with the size of the network.

A Wireless Soil Moisture Smart Sensor Web Using Physics-Based Optimal Control: Concept and Initial Demonstrations

This paper introduces a new concept for a smart wireless sensor web technology for optimal measurements of surface-to-depth profiles of soil moisture using in-situ sensors. The objective of the technology, supported by the NASA Earth Science Technology Office Advanced Information Systems Technology program, is to enable a guided and adaptive sampling strategy for the in-situ sensor network to meet the measurement validation objectives of spaceborne soil moisture sensors. A potential application for this technology is the validation of products from the Soil Moisture Active/Passive (SMAP) mission. Spatially, the total variability in soil-moisture fields comes from variability in processes on various scales. Temporally, variability is caused by external forcings, landscape heterogeneity, and antecedent conditions. Installing a dense in-situ network to sample the field continuously in time for all ranges of variability is impractical. However, a sparser but smarter network with an optimized measurement schedule can provide the validation estimates by operating in a guided fashion with guidance from its own sparse measurements. The feedback and control take place in the context of a dynamic physics-based hydrologic and sensor modeling system. The overall design of the smart sensor web-including the control architecture, physics-based hydrologic and sensor models, and actuation and communication hardware-is presented in this paper. We also present results illustrating sensor scheduling and estimation strategies as well as initial numerical and field demonstrations of the sensor web concept. It is shown that the coordinated operation of sensors through the control policy results in substantial savings in resource usage.

A Multiscale Pyramid Transform for Graph Signals

Multiscale transforms designed to process analog and discrete-time signals and images cannot be directly applied to analyze high-dimensional data residing on the vertices of a weighted graph, as they do not capture the intrinsic topology of the graph data domain. In this paper, we adapt the Laplacian pyramid transform for signals on Euclidean domains so that it can be used to analyze high-dimensional data residing on the vertices of a weighted graph. Our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling, graph reduction, and filtering and interpolation of signals on graphs. Equipped with appropriate notions of these operations, we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs.

A windowed graph Fourier transform

The prevalence of signals on weighted graphs is increasing; however, because of the irregular structure of weighted graphs, classical signal processing techniques cannot be directly applied to signals on graphs. In this paper, we define generalized translation and modulation operators for signals on graphs, and use these operators to adapt the classical windowed Fourier transform to the graph setting, enabling vertex-frequency analysis. When we apply this transform to a signal with frequency components that vary along a path graph, the resulting spectrogram matches our intuition from classical discrete-time signal processing. Yet, our construction is fully generalized and can be applied to analyze signals on any undirected, connected, weighted graph.

Optimal Sleep Scheduling for a Wireless Sensor Network Node

We consider the problem of conserving energy in a single node in a wireless sensor network by turning off the nodes radio for periods of a fixed time length. While packets may continue to arrive at the nodes buffer during the sleep periods, the node cannot transmit them until it wakes up. The objective is to design sleep control laws that minimize the expected value of a cost function representing both energy consumption costs and holding costs for backlogged packets. We consider a discrete time system with a Bernoulli arrival process. In this setting, we characterize optimal control laws under the finite horizon expected cost and infinite horizon expected average cost criteria.

Parametric dictionary learning for graph signals

We propose a parametric dictionary learning algorithm to design structured dictionaries that sparsely represent graph signals. We incorporate the graph structure by forcing the learned dictionaries to be concatenations of subdictionaries that are polynomials of the graph Laplacian matrix. The resulting atoms capture the main spatial and spectral components of the graph signals of interest, leading to adaptive representations with efficient implementations. Experimental results demonstrate the effectiveness of the proposed algorithm for the sparse approximation of graph signals.

Measurement Scheduling for Soil Moisture Sensing: From Physical Models to Optimal Control

In this paper, we consider the problem of monitoring soil moisture evolution using a wireless network of in situ sensors. Continuously sampling moisture levels with these sensors incurs high-maintenance and energy consumption costs, which are particularly undesirable for wireless networks. Our main hypothesis is that a sparser set of measurements can meet the monitoring objectives in an energy-efficient manner. The underlying idea is that we can trade off some inaccuracy in estimating soil moisture evolution for a significant reduction in energy consumption. We investigate how to dynamically schedule the sensor measurements so as to balance this tradeoff. Unlike many prior studies on sensor scheduling that make generic assumptions on the statistics of the observed phenomenon, we obtain statistics of soil moisture evolution from a physical model. We formulate the optimal measurement scheduling and estimation problem as a partially observable Markov decision problem (POMDP). We then utilize special features of the problem to approximate the POMDP by a computationally simpler finite-state Markov decision problem (MDP). The result is a scalable, implementable technology that we have tested and validated numerically and in the field.

Energy-Efficient Transmission Scheduling With Strict Underflow Constraints

This paper considers a single source transmitting data to one or more receivers/users over a shared wireless channel. Due to random fading, the wireless channel conditions vary with time and from user to user. Each user has a buffer to store received packets before they are drained. At each time step, the source determines how much power to use for transmission to each user. The sources objective is to dynamically allocate power in a manner that minimizes total power consumption and packet holding costs, while satisfying strict buffer underflow constraints and a joint power constraint in each slot. The primary application motivating this problem is wireless media streaming. For this application, the buffer underflow constraints prevent the user buffers from emptying, so as to maintain playout quality. In the case of a single user, a state-dependent modified base-stock policy is shown to be optimal with linear power-rate curves, and a state-dependent finite generalized base-stock policy is shown to be optimal with piecewise-linear convex power-rate curves. When certain technical conditions are satisfied, efficient methods to compute the critical numbers that complete the characterizations of the optimal control laws in each of these cases are presented. The structure of the optimal policy for the case of two users is then analyzed.