arXiv subjects

Ao Tang

Publications and source records attributed to Ao Tang.

27 records · Page 2Linked to original sources

arXiv2013

Sparse Recovery with Graph Constraints

Sparse recovery can recover sparse signals from a set of underdetermined linear measurements. Motivated by the need to monitor large-scale networks from a limited number of measurements, this paper addresses the problem of recovering sparse signals in the presence of network topological constraints. Unlike conventional sparse recovery where a measurement can contain any subset of the unknown variables, we use a graph to characterize the topological constraints and allow an additive measurement over nodes (unknown variables) only if they induce a connected subgraph. We provide explicit measurement constructions for several special graphs, and the number of measurements by our construction is less than that needed by existing random constructions. Moreover, our construction for a line network is provably optimal in the sense that it requires the minimum number of measurements. A measurement construction algorithm for general graphs is also proposed and evaluated. For any given graph $G$ with $n$ nodes, we derive bounds of the minimum number of measurements needed to recover any $k$-sparse vector over $G$ ($M^G_{k,n}$). Using the Erdős-Rényi random graph as an example, we characterize the dependence of $M^G_{k,n}$ on the graph structure.

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arXiv2013

Sparse Recovery from Nonlinear Measurements with Applications in Bad Data Detection for Power Networks

In this paper, we consider the problem of sparse recovery from nonlinear measurements, which has applications in state estimation and bad data detection for power networks. An iterative mixed $\ell_1$ and $\ell_2$ convex program is used to estimate the true state by locally linearizing the nonlinear measurements. When the measurements are linear, through using the almost Euclidean property for a linear subspace, we derive a new performance bound for the state estimation error under sparse bad data and additive observation noise. As a byproduct, in this paper we provide sharp bounds on the almost Euclidean property of a linear subspace, using the "escape-through-the-mesh" theorem from geometric functional analysis. When the measurements are nonlinear, we give conditions under which the solution of the iterative algorithm converges to the true state even though the locally linearized measurements may not be the actual nonlinear measurements. We numerically evaluate our iterative convex programming approach to perform bad data detections in nonlinear electrical power networks problems. We are able to use semidefinite programming to verify the conditions for convergence of the proposed iterative sparse recovery algorithms from nonlinear measurements.

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arXiv2011

Sparse Recovery with Graph Constraints: Fundamental Limits and Measurement Construction

This paper addresses the problem of sparse recovery with graph constraints in the sense that we can take additive measurements over nodes only if they induce a connected subgraph. We provide explicit measurement constructions for several special graphs. A general measurement construction algorithm is also proposed and evaluated. For any given graph $G$ with $n$ nodes, we derive order optimal upper bounds of the minimum number of measurements needed to recover any $k$-sparse vector over $G$ ($M^G_{k,n}$). Our study suggests that $M^G_{k,n}$ may serve as a graph connectivity metric.

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arXiv2011

On State Estimation with Bad Data Detection

In this paper, we consider the problem of state estimation through observations possibly corrupted with both bad data and additive observation noises. A mixed $\ell_1$ and $\ell_2$ convex programming is used to separate both sparse bad data and additive noises from the observations. Through using the almost Euclidean property for a linear subspace, we derive a new performance bound for the state estimation error under sparse bad data and additive observation noises. Our main contribution is to provide sharp bounds on the almost Euclidean property of a linear subspace, using the "escape-through-a-mesh" theorem from geometric functional analysis. We also propose and numerically evaluate an iterative convex programming approach to performing bad data detections in nonlinear electrical power networks problems.

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arXiv2011

On the Scaling Law for Compressive Sensing and its Applications

$\ell_1$ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity (the size of the support set), under which with high probability a sparse signal can be recovered from i.i.d. Gaussian measurements, have been computed and are referred to as "weak thresholds" \cite{D}. It was also known that there is a tradeoff between the sparsity and the $\ell_1$ minimization recovery stability. In this paper, we give a \emph{closed-form} characterization for this tradeoff which we call the scaling law for compressive sensing recovery stability. In a nutshell, we are able to show that as the sparsity backs off $\varpi$ ($0<\varpi<1$) from the weak threshold of $\ell_1$ recovery, the parameter for the recovery stability will scale as $\frac{1}{\sqrt{1-\varpi}}$. Our result is based on a careful analysis through the Grassmann angle framework for the Gaussian measurement matrix. We will further discuss how this scaling law helps in analyzing the iterative reweighted $\ell_1$ minimization algorithms. If the nonzero elements over the signal support follow an amplitude probability density function (pdf) $f(\cdot)$ whose $t$-th derivative $f^{t}(0) \neq 0$ for some integer $t \geq 0$, then a certain iterative reweighted $\ell_1$ minimization algorithm can be analytically shown to lift the phase transition thresholds (weak thresholds) of the plain $\ell_1$ minimization algorithm.

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arXiv2010

On the Performance of Sparse Recovery via L_p-minimization (0<=p <=1)

It is known that a high-dimensional sparse vector x* in R^n can be recovered from low-dimensional measurements y= A^{m*n} x* (m<n) . In this paper, we investigate the recovering ability of l_p-minimization (0<=p<=1) as p varies, where l_p-minimization returns a vector with the least l_p ``norm'' among all the vectors x satisfying Ax=y. Besides analyzing the performance of strong recovery where l_p-minimization needs to recover all the sparse vectors up to certain sparsity, we also for the first time analyze the performance of ``weak'' recovery of l_p-minimization (0<=p<1) where the aim is to recover all the sparse vectors on one support with fixed sign pattern. When m/n goes to 1, we provide sharp thresholds of the sparsity ratio that differentiates the success and failure via l_p-minimization. For strong recovery, the threshold strictly decreases from 0.5 to 0.239 as p increases from 0 to 1. Surprisingly, for weak recovery, the threshold is 2/3 for all p in [0,1), while the threshold is 1 for l_1-minimization. We also explicitly demonstrate that l_p-minimization (p<1) can return a denser solution than l_1-minimization. For any m/n<1, we provide bounds of sparsity ratio for strong recovery and weak recovery respectively below which l_p-minimization succeeds with overwhelming probability. Our bound of strong recovery improves on the existing bounds when m/n is large. Regarding the recovery threshold, l_p-minimization has a higher threshold with smaller p for strong recovery; the threshold is the same for all p for sectional recovery; and l_1-minimization can outperform l_p-minimization for weak recovery. These are in contrast to traditional wisdom that l_p-minimization has better sparse recovery ability than l_1-minimization since it is closer to l_0-minimization. We provide an intuitive explanation to our findings and use numerical examples to illustrate the theoretical predictions.

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arXiv2010

Compressive Sensing over Graphs

In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the properties of the edges from a graph. Unlike existing compressive sensing results, the collective additive measurements we are allowed to take must follow connected paths over the underlying graph. For a sufficiently connected graph with $n$ nodes, it is shown that, using $O(k \log(n))$ path measurements, we are able to recover any $k$-sparse link vector (with no more than $k$ nonzero elements), even though the measurements have to follow the graph path constraints. We further show that the computationally efficient $\ell_1$ minimization can provide theoretical guarantees for inferring such $k$-sparse vectors with $O(k \log(n))$ path measurements from the graph.

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arXiv2010

The Limits of Error Correction with lp Decoding

An unknown vector f in R^n can be recovered from corrupted measurements y = Af + e where A^(m*n)(m>n) is the coding matrix if the unknown error vector e is sparse. We investigate the relationship of the fraction of errors and the recovering ability of lp-minimization (0 < p <= 1) which returns a vector x minimizing the "lp-norm" of y - Ax. We give sharp thresholds of the fraction of errors that determine the successful recovery of f. If e is an arbitrary unknown vector, the threshold strictly decreases from 0.5 to 0.239 as p increases from 0 to 1. If e has fixed support and fixed signs on the support, the threshold is 2/3 for all p in (0, 1), while the threshold is 1 for l1-minimization.

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arXiv2010

A Unique "Nonnegative" Solution to an Underdetermined System: from Vectors to Matrices

This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. A vector solution is the unique solution to an underdetermined linear system only if the measurement matrix has a row-span intersecting the positive orthant. Focusing on two types of binary measurement matrices, Bernoulli 0-1 matrices and adjacency matrices of general expander graphs, we show that, in both cases, the support size of a unique nonnegative solution can grow linearly, namely O(n), with the problem dimension n. We also provide closed-form characterizations of the ratio of this support size to the signal dimension. For the matrix case, we show that under a necessary and sufficient condition for the linear compressed observations operator, there will be a unique positive semidefinite matrix solution to the compressed linear observations. We further show that a randomly generated Gaussian linear compressed observations operator will satisfy this condition with overwhelmingly high probability.

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