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63
Compressed network monitoring
"... This paper describes a procedure for estimating a full set of network path metrics, such as loss or delay, from a limited number of measurements. The approach exploits the strong spatial and temporal correlation observed in pathlevel metric data, which arises due to shared links and stationary comp ..."
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This paper describes a procedure for estimating a full set of network path metrics, such as loss or delay, from a limited number of measurements. The approach exploits the strong spatial and temporal correlation observed in pathlevel metric data, which arises due to shared links and stationary components of the observed phenomena. We design diffusion wavelets based on the routing matrix to generate a basis in which the signals are compressible. This allows us to exploit powerful nonlinear estimation algorithms that strive for sparse solutions. We demonstrate our results using measurements of endtoend delay in the Abilene network. Our results show that we can recover network mean endtoend delay with 95 % accuracy while monitoring only 4 % of the routes. Index Terms — network monitoring, diffusion wavelets, compressed sensing 1.
Compressed network monitoring for IP and alloptical networks
 In Proceedings of ACM IMC 2007
, 2007
"... We address the problem of efficient endtoend network monitoring of path metrics in communication networks. Our goal is to minimize the number of measurements or monitors required to maintain an acceptable estimation accuracy. We present a framework based on diffusion wavelets and nonlinear estimat ..."
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We address the problem of efficient endtoend network monitoring of path metrics in communication networks. Our goal is to minimize the number of measurements or monitors required to maintain an acceptable estimation accuracy. We present a framework based on diffusion wavelets and nonlinear estimation. Our procedure involves the development of a diffusion wavelet basis that is adapted to the monitoring problem. This basis exploits spatial and temporal correlations in the measured phenomena to provide a compressible representation of the path metrics. The framework employs nonlinear estimation techniques using ℓ1 minimization to generate estimates for the unmeasured paths. We describe heuristic approaches for the selection of the paths that should be monitored, or equivalently, where hardware monitors should be located. We demonstrate how our estimation framework can improve the efficiency of endtoend delay estimation in IP networks and reduce the number of hardware monitors required to track biterror rates in alloptical networks (networks with no electrical regenerators).
Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs
 59141M. SPIE, 2005. URL HTTP://LINK.AIP.ORG/LINK/?PSI/5914/59141M/1
, 2005
"... Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {T t}t≥0, for which T t has low rank for large t. 1 This includes important classes of diffusionlike operators, in any dimension, ..."
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Cited by 8 (6 self)
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Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {T t}t≥0, for which T t has low rank for large t. 1 This includes important classes of diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. The dyadic powers of an operator are used to induce a multiresolution analysis, analogous to classical LittlewoodPaley 14 and wavelet theory, while associated wavelet packets can also be constructed. 2 This extends multiscale function and operator analysis and signal processing to a large class of spaces, such as manifolds and graphs, with efficient algorithms. Powers and functions of T (notably its Green’s function) are efficiently computed, represented and compressed. This construction is related and generalizes certain Fast Multipole Methods, 3 the wavelet representation of CalderónZygmund and pseudodifferential operators, 4 and also relates to algebraic multigrid techniques. 5 The original diffusion wavelet construction yields orthonormal bases for multiresolution spaces {Vj}. The orthogonality requirement has some advantages from the numerical perspective, but several drawbacks in terms of the space and frequency localization of the basis functions. Here we show how to relax this requirement in order to construct biorthogonal bases of diffusion scaling functions and wavelets. This yields more compact representations of the powers of the operator, better localized basis functions. This new construction also applies to non selfadjoint semigroups, arising in many applications.
Universal local parametrizations via heat kernels and eigenfunctions of the laplacian
, 2010
"... We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with C α metric). These coordinates are biLipschitz on embedded balls of the domain or manifold, with distortion constan ..."
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Cited by 7 (1 self)
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We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with C α metric). These coordinates are biLipschitz on embedded balls of the domain or manifold, with distortion constants that depend only on natural geometric properties of the domain or manifold. The proof of these results relies on estimates, from above and below, for the heat kernel and its gradient, as well as for the eigenfunctions of the Laplacian and their gradient. These estimates hold in the nonsmooth category, and are stable with respect to perturbations
Lowrank variance approximation in GMRF models: Single and multiscale approaches
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2008
"... We present a versatile framework for tractable computation of approximate variances in largescale Gaussian Markov random field estimation problems. In addition to its efficiency and simplicity, it also provides accuracy guarantees. Our approach relies on the construction of a certain lowrank alia ..."
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We present a versatile framework for tractable computation of approximate variances in largescale Gaussian Markov random field estimation problems. In addition to its efficiency and simplicity, it also provides accuracy guarantees. Our approach relies on the construction of a certain lowrank aliasing matrix with respect to the Markov graph of the model. We first construct this matrix for singlescale models with shortrange correlations and then introduce spliced wavelets and propose a construction for the longrange correlation case, and also for multiscale models. We describe the accuracy guarantees that the approach provides and apply the method to a large interpolation problem from oceanography with sparse, irregular, and noisy measurements, and to a gravity inversion problem.
Geometry of probability spaces
 Constr. Approx
"... Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However this avenue has been limited in many areas where calculus is obstructed as in singular spaces, and function spaces of functions on a space X ..."
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Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However this avenue has been limited in many areas where calculus is obstructed as in singular spaces, and function spaces of functions on a space X where X itself is a function space. Examples of the last
Wiener’s lemma for localized integral operators
 Appl. Comput. Harmonic Anal
"... Abstract. In this paper, we introduce two classes of localized integral operators on L 2 (R d) with the Wiener class W and the Kurbatov class K of integral operators as their models. We show that those two classes of localized integral operators are pseudoinverse closed nonunital subalgebra of B 2 ..."
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Cited by 6 (4 self)
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Abstract. In this paper, we introduce two classes of localized integral operators on L 2 (R d) with the Wiener class W and the Kurbatov class K of integral operators as their models. We show that those two classes of localized integral operators are pseudoinverse closed nonunital subalgebra of B 2, the Banach algebra of all bounded operators on L 2 (R d) with usual operator norm. 1.
Value function approximation on nonlinear manifolds for robot motor control
 In Proceedings of the IEEE Conference on Robots and Automation (ICRA
, 2007
"... Abstract — The least squares approach works efficiently in value function approximation, given appropriate basis functions. Because of its smoothness, the Gaussian kernel is a popular and useful choice as a basis function. However, it does not allow for discontinuity which typically arises in realwo ..."
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Cited by 5 (1 self)
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Abstract — The least squares approach works efficiently in value function approximation, given appropriate basis functions. Because of its smoothness, the Gaussian kernel is a popular and useful choice as a basis function. However, it does not allow for discontinuity which typically arises in realworld reinforcement learning tasks. In this paper, we propose a new basis function based on geodesic Gaussian kernels, which exploits the nonlinear manifold structure induced by the Markov decision processes. The usefulness of the proposed method is successfully demonstrated in a simulated robot arm control and Khepera robot navigation. I.
Approximating functions of few variables in high dimensions , Constructive Approximation
"... Let f be a continuous function defined on Ω: = [0,1] N which depends on only ℓ coordinate variables, f(x1,...,xN) = g(xi1,...,xiℓ). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i1,...,iℓ are known to us, then by askin ..."
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Cited by 5 (3 self)
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Let f be a continuous function defined on Ω: = [0,1] N which depends on only ℓ coordinate variables, f(x1,...,xN) = g(xi1,...,xiℓ). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i1,...,iℓ are known to us, then by asking for the values of f at m = L ℓ uniformly spaced points, we could recover f to the accuracy gLip1L −1 in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i1,...,iℓ are not known to us. A prototypical result of this paper is that by asking for C(ℓ)L ℓ (log 2 N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy gLip1L −1 when g ∈ Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance ǫ (which is not known) by functions of ℓ variables. 1