We study Semidefinite Programming, relaxations for Sensor Network Localization, with anchors and with noisy distance information. The main point of the paper is to view as a (nearest) Euclidean Distance Matrix, , completion problem and to show the advantages for using this latter, well studied model. We first show that the current popular relaxation is equivalent to known relaxations in the literature for completions. The existence of anchors in the problem is not special. The set of anchors simply corresponds to a given fixed clique for the graph of the problem. We next propose a method of projection when a large clique or a dense subgraph is identified in the underlying graph. This projection reduces the size, and improves the stability, of the relaxation. In addition, viewing the problem as an completion problem yields better low rank approximations for the low dimensional realizations. And, the projection reduction procedure can be repeated for other given cliques of sensors or for sets of sensors, where many distances are known. Thus, further size reduction can be obtained. Optimality duality conditions and a primal-dual interior-exterior path following algorithm are derived for the relaxations We discuss the relative stability and strength of two formulations and the corresponding algorithms that are used. In particular, we show that the quadratic formulation arising from the relaxation is better conditioned than the linearized form, that is used in the literature and that arises from applying a Schur complement.
The relaxations solve a closest matrix problem and generally use the @math norm. The @math norm is used in @cite_29 , where the noise in the radio signal is assumed to come from a multivariate normal distribution with mean @math and variance-covariance matrix @math , i.e. from a spherical normal distribution so that the least squares estimates are the maximum likelihood estimates. (We use the @math norm as well in this paper. Our approach follows that in @cite_3 for completion without anchors.)
@cite_29 @cite_3
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We consider the problem of estimating an unknown signal @math from noisy linear observations @math . In many practical instances, @math has a certain structure that can be captured by a structure inducing convex function @math . For example, @math norm can be used to encourage a sparse solution. To estimate @math with the aid of @math , we consider the well-known LASSO method and provide sharp characterization of its performance. We assume the entries of the measurement matrix @math and the noise vector @math have zero-mean normal distributions with variances @math and @math respectively. For the LASSO estimator @math , we attempt to calculate the Normalized Square Error (NSE) defined as @math as a function of the noise level @math , the number of observations @math and the structure of the signal. We show that, the structure of the signal @math and choice of the function @math enter the error formulae through the summary parameters @math and @math , which are defined as the Gaussian squared-distances to the subdifferential cone and to the @math -scaled subdifferential, respectively. The first LASSO estimator assumes a-priori knowledge of @math and is given by @math . We prove that its worst case NSE is achieved when @math and concentrates around @math . Secondly, we consider @math , for some @math . This time the NSE formula depends on the choice of @math and is given by @math . We then establish a mapping between this and the third estimator @math . Finally, for a number of important structured signal classes, we translate our abstract formulae to closed-form upper bounds on the NSE. This paper proves that an "old dog", namely -- the classical Johnson-Lindenstrauss transform, "performs new tricks" -- it gives a novel way of preserving differential privacy. We show that if we take two databases, @math and @math , such that (i) @math is a rank-1 matrix of bounded norm and (ii) all singular values of @math and @math are sufficiently large, then multiplying either @math or @math with a vector of iid normal Gaussians yields two statistically close distributions in the sense of differential privacy. Furthermore, a small, deterministic and alteration of the input is enough to assert that all singular values of @math are large. We apply the Johnson-Lindenstrauss transform to the task of approximating cut-queries: the number of edges crossing a @math -cut in a graph. We show that the JL transform allows us to that preserves edge differential privacy (where two graphs are neighbors if they differ on a single edge) while adding only @math random noise to any given query (w.h.p). Comparing the additive noise of our algorithm to existing algorithms for answering cut-queries in a differentially private manner, we outperform all others on small cuts ( @math ). We also apply our technique to the task of estimating the variance of a given matrix in any given direction. The JL transform allows us to that preserves differential privacy w.r.t bounded changes (each row in the matrix can change by at most a norm-1 vector) while adding random noise of magnitude independent of the size of the matrix (w.h.p). In contrast, existing algorithms introduce an error which depends on the matrix dimensions. In statistics and machine learning, people are often interested in the eigenvectors (or singular vectors) of certain matrices (e.g. covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. One usually employs Davis-Kahan @math theorem to bound the difference between the eigenvectors of a matrix @math and those of a perturbed matrix @math , in terms of @math norm. In this paper, we prove that when @math is a low-rank and incoherent matrix, the @math norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of @math or @math for left and right vectors, where @math and @math are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments. Given @math noisy samples with @math dimensions, where @math , we show that the multi-step thresholding procedure based on the Lasso -- we call it the Thresholded Lasso , can accurately estimate a sparse vector @math in a linear model @math , where @math is a design matrix normalized to have column @math norm @math , and @math . We show that under the restricted eigenvalue (RE) condition (Bickel-Ritov-Tsybakov 09), it is possible to achieve the @math loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle while selecting a sufficiently sparse model -- hence achieving sparse oracle inequalities ; the oracle would supply perfect information about which coordinates are non-zero and which are above the noise level. In some sense, the Thresholded Lasso recovers the choices that would have been made by the @math penalized least squares estimators, in that it selects a sufficiently sparse model without sacrificing the accuracy in estimating @math and in predicting @math . We also show for the Gauss-Dantzig selector (Cand e s-Tao 07), if @math obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, one will achieve the sparse oracle inequalities as above, while allowing at most @math irrelevant variables in the model in the worst case, where @math is the smallest integer such that for @math , @math . Our simulation results on the Thresholded Lasso match our theoretical analysis excellently.