The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming, SDP, completion problem, by using the linear mapping between Euclidean distance matrices, EDM, and semidefinite matrices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances, we are able to efficiently solve many huge instances of this NP-hard problem to high accuracy, by finding a representation of the minimal face of the SDP cone that contains the SDP matrix representation of the EDM. The main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the SDP cone that correspond to cliques of the SNL problem.
@cite_27 @cite_6 propose the SpaseLoc heuristic. It is limited to @math and uses an solver for small localized subproblems. They then sew these subproblems together. So & Ye @cite_7 show that the problem of solving a noiseless with a unique solution can be phrased as an and thus can be solved in polynomial time. They also give an efficient criterion for checking whether a given instance has a unique solution for @math .
@cite_27 @cite_7 @cite_6
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The authors consider the problem of learning a linear threshold function (a halfspace in n dimensions, also called a "perceptron"). Methods for solving this problem generally fall into two categories. In the absence of noise, this problem can be formulated as a linear program and solved in polynomial time with the ellipsoid algorithm (or interior point methods). On the other hand, simple greedy algorithms such as the perceptron algorithm seem to work well in practice and can be made noise tolerant; but, their running time depends on a separation parameter (which quantifies the amount of "wiggle room" available) and can be exponential in the description length of the input. They show how simple greedy methods can be used to find weak hypotheses (hypotheses that classify noticeably more than half of the examples) in polynomial time, without dependence on any separation parameter. This results in a polynomial-time algorithm for learning linear threshold functions in the PAC model in the presence of random classification noise. The algorithm is based on a new method for removing outliers in data. Specifically, for any set S of points in R sup n , each given to b bits of precision, they show that one can remove only a small fraction of S so that in the remaining set T, for every vector v, max sub x spl epsiv T (v spl middot x) sup 2 spl les poly(n,b)|T| sup -1 spl Sigma sub x spl epsiv T (v spl middot x) sup 2 . After removing these outliers, they are able to show that a modified version of the perceptron learning algorithm works in polynomial time, even in the presence of random classification noise. An adaptive rule-based algorithm, SpaseLoc, is described to solve localization problems for ad hoc wireless sensor networks. A large problem is solved as a sequence of very small subproblems, each of which is solved by semidefinite programming relaxation of a geometric optimization model. The subproblems are generated according to a set of sensor anchor selection rules. Computational results compared with existing approaches show that the SpaseLoc algorithm scales well and provides excellent localization accuracy. We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order- @math tensor @math of the form @math , where @math is a signal-to-noise ratio, @math is a unit vector, and @math is a random noise tensor, the goal is to recover the planted vector @math . For the case that @math has iid standard Gaussian entries, we give an efficient algorithm to recover @math whenever @math , and certify that the recovered vector is close to a maximum likelihood estimator, all with high probability over the random choice of @math . The previous best algorithms with provable guarantees required @math . In the regime @math , natural tensor-unfolding-based spectral relaxations for the underlying optimization problem break down (in the sense that their integrality gap is large). To go beyond this barrier, we use convex relaxations based on the sum-of-squares method. Our recovery algorithm proceeds by rounding a degree- @math sum-of-squares relaxations of the maximum-likelihood-estimation problem for the statistical model. To complement our algorithmic results, we show that degree- @math sum-of-squares relaxations break down for @math , which demonstrates that improving our current guarantees (by more than logarithmic factors) would require new techniques or might even be intractable. Finally, we show how to exploit additional problem structure in order to solve our sum-of-squares relaxations, up to some approximation, very efficiently. Our fastest algorithm runs in nearly-linear time using shifted (matrix) power iteration and has similar guarantees as above. The analysis of this algorithm also confirms a variant of a conjecture of Montanari and Richard about singular vectors of tensor unfoldings. Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (lscr 2) error term added to a sparsity-inducing (usually lscr1) regularizater. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex regularizer. We propose iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian (i.e., separable in the unknowns) plus the original sparsity-inducing regularizer; our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. Under mild conditions (namely convexity of the regularizer), we prove convergence of the proposed iterative algorithm to a minimum of the objective function. In addition to solving the standard lscr2-lscr1 case, our framework yields efficient solution techniques for other regularizers, such as an lscrinfin norm and group-separable regularizers. It also generalizes immediately to the case in which the data is complex rather than real. Experiments with CS problems show that our approach is competitive with the fastest known methods for the standard lscr2-lscr1 problem, as well as being efficient on problems with other separable regularization terms.