Learning parametric dictionaries for graph signals

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.Comment: 8 pages, 5 figures, to appear in the Proceedings of the IEEE International Conference on Distributed Computing in Sensor Systems (DCOSS), June, 2011, Barcelona, Spai

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 geometric structure of the underlying 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.Comment: 16 pages, 13 figure

Energy-Efficient Transmission Scheduling with Strict Underflow Constraints

We consider 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 source's objective is to allocate power in a manner that minimizes an expected cost measure, while satisfying strict buffer underflow constraints and a total power constraint in each slot. The expected cost measure is composed of costs associated with power consumption from transmission and packet holding costs. 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 with linear power-rate curves, we show that a modified base-stock policy is optimal under the finite horizon, infinite horizon discounted, and infinite horizon average expected cost criteria. For a single user with piecewise-linear convex power-rate curves, we show that a finite generalized base-stock policy is optimal under all three expected cost criteria. We also present the sequences of critical numbers that complete the characterization of the optimal control laws in each of these cases when some additional technical conditions are satisfied. We then analyze the structure of the optimal policy for the case of two users. We conclude with a discussion of methods to identify implementable near-optimal policies for the most general case of M users.Comment: 109 pages, 11 pdf figures, template.tex is main file. We have significantly revised the paper from version 1. Additions include the case of a single receiver with piecewise-linear convex power-rate curves, the case of two receivers, and the infinite horizon average expected cost proble

Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames

We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues. We use this approach to construct computationally efficient, spectrum-adapted, tight vertex-frequency and graph wavelet frames. We give numerous examples of the resulting spectrum-adapted graph filters, and also present an illustrative example of vertex-frequency analysis using the proposed construction

From Sleeping to Stockpiling: Energy Conservation via Stochastic Scheduling in Wireless Networks.

Motivated by the need to conserve energy in wireless networks, we study three stochastic dynamic scheduling problems. In the first problem, we consider a wireless sensor node that can turn its radio off for fixed durations of time in order to conserve energy. We formulate finite horizon expected cost and infinite horizon average expected cost problems to model the fundamental tradeoff between packet delay and energy consumption. Through analysis of the dynamic programming equations, we derive structural results on the optimal policies for both formulations. For the infinite horizon problem, we identify a threshold decision rule to determine the optimal control action when the queue is empty. In the second problem, we consider a sensor node with an inaccurate timer in the ultra-low power sleep mode. The loss in timing accuracy in the sleep mode can result in unnecessary energy consumption from two unsynchronized devices trying to communicate. We develop a novel method for the node to calibrate its timer: occasionally waking up to measure the ambient temperature, upon which the timer speed depends. The objective is to dynamically schedule a limited number of temperature measurements in a manner most useful to improving the accuracy of the timer. We formulate optimization problems with both continuous and discrete underlying time scales, and implement a numerical solution to an equivalent reduction of the second formulation. In the third problem, we consider a single source transmitting data to one or more receivers over a shared wireless channel. Each receiver has a buffer to store received packets before they are drained. The transmitter's goal is to minimize total power consumption by exploiting the temporal and spatial variation of the channel, while preventing the receivers' buffers from emptying. In the case of a single receiver, we show that modified base-stock and finite generalized base-stock policies are optimal when the power-rate curves are linear and piecewise-linear convex, respectively. We also present the sequences of critical numbers that complete the characterizations of the optimal policies when additional technical conditions are satisfied. We then analyze the structure of the optimal policy for the case of two receivers.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77839/1/dishuman_1.pd

Distributed Signal Processing via Chebyshev Polynomial Approximation

Unions of graph multiplier operators 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. The proposed method features approximations of the graph multipliers by shifted Chebyshev polynomials, whose recurrence relations make them readily amenable to distributed computation. We demonstrate how the proposed method can be applied to distributed processing tasks such as smoothing, denoising, inverse filtering, and semi-supervised classification, and show that the communication requirements of the method scale gracefully with the size of the network