Anna C. Gilbert

Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit

This paper demonstrates theoretically and empirically that a greedy algorithm called orthogonal matching pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results, which require O(m2) measurements. The new results for OMP are comparable with recent results for another approach called basis pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems.

Algorithms for simultaneous sparse approximation: part I: Greedy pursuit

A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the same time, the problem balances the error in approximation against the total number of elementary signals that participate. These elementary signals typically model coherent structures in the input signals, and they are chosen from a large, linearly dependent collection.The first part of this paper proposes a greedy pursuit algorithm, called simultaneous orthogonal matching pursuit (S-OMP), for simultaneous sparse approximation. Then it presents some numerical experiments that demonstrate how a sparse model for the input signals can be identified more reliably given several input signals. Afterward, the paper proves that the S-OMP algorithm can compute provably good solutions to several simultaneous sparse approximation problems.The second part of the paper develops another algorithmic approach called convex relaxation, and it provides theoretical results on the performance of convex relaxation for simultaneous sparse approximation.

Dynamics of IP traffic: a study of the role of variability and the impact of control

Using the ns-2-simulator to experiment with different aspects of user- or session-behaviors and network configurations and focusing on the qualitative aspects of a wavelet-based scaling analysis, we present a systematic investigation into how and why variability and feedback-control contribute to the intriguing scaling properties observed in actual Internet traces (as our benchmark data, we use measured Internet traffic from an ISP). We illustrate how variability of both user aspects and network environments (i) causes self-similar scaling behavior over large time scales, (ii) determines a more or less pronounced change in scaling behavior around a specific time scale, and (iii) sets the stage for the emergence of surprisingly rich scaling dynamics over small time scales; i.e., multifractal scaling. Moreover, our scaling analyses indicate whether or not open-loop controls such as UDP or closed-loop controls such as TCP impact the local or small-scale behavior of the traffic and how they contribute to the observed multifractal nature of measured Internet traffic. In fact, our findings suggest an initial physical explanation for why measured Internet traffic over small time scales is highly complex and suggest novel ways for detecting and identifying, for example, performance bottlenecks.This paper focuses on the qualitative aspects of a wavelet-based scaling analysis rather than on the quantitative use for which it was originally designed. We demonstrate how the presented techniques can be used for analyzing a wide range of different kinds of network-related measurements in ways that were not previously feasible. We show that scaling analysis has the ability to extract relevant information about the time-scale dynamics of Internet traffic, thereby, we hope, making these techniques available to a larger segment of the networking research community.

Data networks as cascades: investigating the multifractal nature of Internet WAN traffic

In apparent contrast to the well-documented self-similar (i.e., monofractal) scaling behavior of measured LAN traffic, recent studies have suggested that measured TCP/IP and ATM WAN traffic exhibits more complex scaling behavior, consistent with multifractals. To bring multifractals into the realm of networking, this paper provides a simple construction based on cascades (also known as multiplicative processes) that is motivated by the protocol hierarchy of IP data networks. The cascade framework allows for a plausible physical explanation of the observed multifractal scaling behavior of data traffic and suggests that the underlying multiplicative structure is a traffic invariant for WAN traffic that co-exists with self-similarity. In particular, cascades allow us to refine the previously observed self-similar nature of data traffic to account for local irregularities in WAN traffic that are typically associated with networking mechanisms operating on small time scales, such as TCP flow control.To validate our approach, we show that recent measurements of Internet WAN traffic from both an ISP and a corporate environment are consistent with the proposed cascade paradigm and hence with multifractality. We rely on wavelet-based time-scale analysis techniques to visualize and to infer the scaling behavior of the traces, both globally and locally. We also discuss and illustrate with some examples how this cascade-based approach to describing data network traffic suggests novel ways for dealing with networking problems and helps in building intuition and physical understanding about the possible implications of multifractality on issues related to network performance analysis.

Combining geometry and combinatorics: A unified approach to sparse signal recovery

There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach utilizes geometric properties of the measurement matrix Phi. A notable example is the Restricted Isometry Property, which states that the mapping Phi preserves the Euclidean norm of sparse signals; it is known that random dense matrices satisfy this constraint with high probability. On the other hand, the combinatorial approach utilizes sparse matrices, interpreted as adjacency matrices of sparse (possibly random) graphs, and uses combinatorial techniques to recover an approximation to the signal. In this paper we present a unification of these two approaches. To this end, we extend the notion of Restricted Isometry Property from the Euclidean lscr2 norm to the Manhattan lscr1 norm. Then we show that this new lscr1 -based property is essentially equivalent to the combinatorial notion of expansion of the sparse graph underlying the measurement matrix. At the same time we show that the new property suffices to guarantee correctness of both geometric and combinatorial recovery algorithms. As a result, we obtain new measurement matrix constructions and algorithms for signal recovery which, compared to previous algorithms, are superior in either the number of measurements or computational efficiency of decoders.

The changing nature of network traffic: scaling phenomena

In this paper, we report on some preliminary results from an in-depth, wavelet-based analysis of a set of high-quality, packet-level traffic measurements, collected over the last 6-7 years from a number of different wide-area networks (WANs). We first validate and confirm an earlier finding, originally due to Paxson and Floyd [14], that actual WAN traffic is consistent with statistical self-similarity for sufficiently large time scales. We then relate this large-time scaling phenomenon to the empirically observed characteristics of WAN traffic at the level of individual connections or applications. In particular, we present here original results about a detailed statistical analysis of Web-session characteristics, and report on an intriguing scaling property of measured WAN traffic at the transport layer (i.e., number of TCP connection arrivals per time unit). This scaling property of WAN traffic at the TCP layer was absent in the pre-Web period but has become ubiquitous in todays WWW-dominated WANs and is a direct consequence of the ever-increasing popularity of the Web (WWW) and its emergence as the major contributor to WAN traffic. Moreover, we show that this changing nature of WAN traffic can be naturally accounted for by self-similar traffic models, primarily because of their ability to provide physical explanations for empirically observed traffic phenomena in a networking context. Finally, we provide empirical evidence that actual WAN traffic traces also exhibit scaling properties over small time scales, but that the small-time scaling phenomenon is distinctly different from the observed large-time scaling property. We relate this newly observed characteristic of WAN traffic to the effects that the dominant network protocols (e.g., TCP) and controls have on the flow of packets across the network and discuss the potential that multifractals have in this context for providing a structural modeling approach for WAN traffic and for capturing in a compact and parsimonious manner the observed scaling phenomena at large as well as small time scales.

Sparse Recovery Using Sparse Matrices

In this paper, we survey algorithms for sparse recovery problems that are based on sparse random matrices. Such matrices has several attractive properties: they support algorithms with low computational complexity, and make it easy to perform incremental updates to signals. We discuss applications to several areas, including compressive sensing, data stream computing, and group testing.

One sketch for all: fast algorithms for compressed sensing

Compressed Sensing is a new paradigm for acquiring the compressible signals that arise in many applications. These signals can be approximated using an amount of information much smaller than the nominal dimension of the signal. Traditional approaches acquire the entire signal and process it to extract the information. The new approach acquires a small number of nonadaptive linear measurements of the signal and uses sophisticated algorithms to determine its information content. Emerging technologies can compute these general linear measurements of a signal at unit cost per measurement. This paper exhibits a randomized measurement ensemble and a signal reconstruction algorithm that satisfy four requirements: 1. The measurement ensemble succeeds for all signals, with high probability over the random choices in its construction. 2. The number of measurements of the signal is optimal, except for a factor polylogarithmic in the signal length. 3. The running time of the algorithm is polynomial in the amount of information in the signal and polylogarithmic in the signal length. 4. The recovery algorithm offers the strongest possible type of error guarantee. Moreover, it is a fully polynomial approximation scheme with respect to this type of error bound. Emerging applications demand this level of performance. Yet no otheralgorithm in the literature simultaneously achieves all four of these desiderata.

Near-optimal sparse fourier representations via sampling

(MATH) We give an algorithm for finding a Fourier representation <b>R</b> of <i>B</i> terms for a given discrete signal signal <b>A</b> of length <i>N</i>, such that

Random Sampling for Analog-to-Information Conversion of Wideband Signals

\|\signal-\repn\|_2^2