Uncertainty principles for signals defined on graphs: Bounds and characterizations

Agaskar, Ameya; Lu, Yue M.

doi:10.1109/icassp.2012.6288669

Cited by 18 publications

(31 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Varying the parameter allows one to "trace" and obtain the entire curve . A rigorous and complete proof of Theorem 1 is provided in this paper, extending the rough argument given in [25]. Based on the convexity of , we show in Section III-C that the sandwich algorithm [26] can be used to efficiently produce a piecewise linear approximation for the uncertainty curve that differs from the true curve by at most (under a suitable error metric) and requires solving typically sparse eigenvalue problems.…”

Section: B Contributions and Paper Organizationmentioning

confidence: 70%

“…After justifying the use of the Laplacian eigenvectors as a Fourier basis on graphs, we define in Section II-C the graph spread about a vertex , and the spectral spread, , of a signal defined on a graph. These two quantities, which we first introduced in some preliminary work [24], [25], are defined in analogy to the standard time and frequency spreads, respectively.…”

Section: B Contributions and Paper Organizationmentioning

confidence: 99%

“…However, the bound was not tight and applied only under restrictive conditions for the graph and the signal on it. In [25], we took a new approach to characterize a more general and precise relationship between the two kinds of uncertainty. In this paper, we continue this line of investigation, and provide a rigorous basis for the arguments presented in [25], in addition to some new results.…”

Section: B Contributions and Paper Organizationmentioning

confidence: 99%

See 2 more Smart Citations

A Spectral Graph Uncertainty Principle

Agaskar

2013

IEEE Trans. Inform. Theory

Self Cite

149

175

View full text Add to dashboard Cite

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral "spreads" are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $O(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure

show abstract

Section: B Contributions and Paper Organizationmentioning

confidence: 70%

Section: B Contributions and Paper Organizationmentioning

confidence: 99%

Section: B Contributions and Paper Organizationmentioning

confidence: 99%

See 1 more Smart Citation

A Spectral Graph Uncertainty Principle

Agaskar

2013

IEEE Trans. Inform. Theory

Self Cite

149

175

View full text Add to dashboard Cite

show abstract

“…Examples include uncertainty principle on graphs [1], on spheres [14], and on Riemannian manifolds [8]. Studies on the periodic frequency spread can be found in [4] and [35].…”

Section: Related Workmentioning

confidence: 99%

Sequences with minimal time–frequency uncertainty

Parhizkar

Barbotin

Vetterli

2015

Applied and Computational Harmonic Analysis

View full text Add to dashboard Cite

A central problem in signal processing and communications is to design signals that are compact both in time and frequency. Heisenberg's uncertainty principle states that a given function cannot be arbitrarily compact both in time and frequency, defining an "uncertainty" lower bound. Taking the variance as a measure of localization in time and frequency, Gaussian functions reach this bound for continuous-time signals. For sequences, however, this is not true; it is known that Heisenberg's bound is generally unachievable. For a chosen frequency variance, we formulate the search for "maximally compact sequences" as an exactly and efficiently solved convex optimization problem, thus providing a sharp uncertainty principle for sequences. Interestingly, the optimization formulation also reveals that maximally compact sequences are derived from Mathieu's harmonic cosine function of order zero. We further provide rational asymptotic expansions of this sharp uncertainty bound. We use the derived bounds as a benchmark to compare the compactness of well-known window functions with that of the optimal Mathieu's functions.

show abstract

“…Examples include uncertainty principle on graphs [10] and on spheres [11]. Initial studies on the periodic frequency spread were undertaken by [3] and [12].…”

Section: Related Workmentioning

confidence: 99%

Sequences with minimal time-frequency spreads

Parhizkar

Barbotin

Vetterli

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

View full text Add to dashboard Cite

For a given time or frequency spread, one can always find continuoustime signals, which achieve the Heisenberg uncertainty principle bound. This is known, however, not to be the case for discrete-time sequences; only widely spread sequences asymptotically achieve this bound. We provide a constructive method for designing sequences that are maximally compact in time for a given frequency spread. By formulating the problem as a semidefinite program, we show that maximally compact sequences do not achieve the classic Heisenberg bound. We further provide analytic lower bounds on the time-frequency spread of such signals.

show abstract

Uncertainty principles for signals defined on graphs: Bounds and characterizations

Cited by 18 publications

References 7 publications

A Spectral Graph Uncertainty Principle

A Spectral Graph Uncertainty Principle

Sequences with minimal time–frequency uncertainty

Sequences with minimal time-frequency spreads

Contact Info

Product

Resources

About