Robust smoothed analysis of a condition number for linear programming

Bürgisser, Peter; Amelunxen, Dennis

doi:10.1007/s10107-010-0346-x

Cited by 7 publications

(8 citation statements)

References 34 publications

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“…Possible choices for the measures' family are the Gaussians N (f , σ 2 I) (used, for instance, in [14,19,32,33]) and the uniform measure on disks B(f , r) (used in [2,11,12]). Other families may also be used and an emerging impression is that smoothed analysis is robust in the sense that its dependence on the chosen family of measures is low.…”

Section: Smoothed Analysis Of LVmentioning

confidence: 99%

“…The state of the art at the end of the Bézout series, i.e., in [26], showed an incomplete picture. For (2), the rule consisted of taking a regular subdivision of E f,g for a given k, executing the path-following procedure, and repeating with k replaced by 2k if the final point could not be shown to be an approximate zero of f (Shub and Smale provided criteria for checking this). Concerning (1), Shub and Smale proved that good initial pairs (g, ζ) (in the sense that the average number of iterations for the rule above was polynomial in the size of f ) existed for each degree pattern d, but they could not exhibit a procedure to generate one such pair.…”

Section: Introductionmentioning

confidence: 99%

“…In an oversight of this non-constructibility, Beltrán and Pardo [6] provided a new version of their randomized algorithm 2 with an improved complexity of O(D 3/2 nN ).…”

Section: Introductionmentioning

confidence: 99%

“…2 The algorithm in [6] explicitly calls as a subroutine "the homotopy algorithm of [21]" without noticing that the partition in [21] is non-algorithmic. Actually, the word 'algorithm' is never used in [21].…”

Section: Introductionmentioning

confidence: 99%

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On a problem posed by Steve Smale

2011

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Abstract. The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltrán and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a non-constructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it LV,à la Beltrán-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and σ −1 , where σ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system f , of the expected running time of LV with input f . In addition to its dependence on N this bound also depends on the condition of f . Fourthly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is N O(log log N) . This is nearly a solution to Smale's 17th problem.

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Section: Smoothed Analysis Of LVmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In an oversight of this non-constructibility, Beltrán and Pardo [6] provided a new version of their randomized algorithm 2 with an improved complexity of O(D 3/2 nN ).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a problem posed by Steve Smale

2011

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“…The bound 2 ln(m + 1) + 3.31 is not only sharper (in that it is independent of n) but also more precise (in that it does not rely on the O notation). 2 1.3. Coverage processes versus condition numbers.…”

Section: Computation Of Points In Polyhedral Cones (Cppc) Given a Matrixmentioning

confidence: 99%

Coverage processes on spheres and condition numbers for linear programming

Bürgisser¹,

Cucker²,

Lotz³

2010

Ann. Probab.

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This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let p(n, m, α) be the probability that n spherical caps of angular radius α in S m do not cover the whole sphere S m . We give an exact formula for p(n, m, α) in the case α ∈ [π/2, π] and an upper bound for p(n, m, α) in the case α ∈ [0, π/2] which tends to p(n, m, π/2) when α → π/2. In the case α ∈ [0, π/2] this yields upper bounds for the expected number of spherical caps of radius α that are needed to cover S m .Secondly, we study the condition number C (A) of the linear programming feasibility problem ∃x ∈ R m+1 Ax ≤ 0, x = 0 where A ∈ R n×(m+1) is randomly chosen according to the standard normal distribution. We exactly determine the distribution of C (A) conditioned to A being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that E(ln C (A)) ≤ 2 ln(m + 1) + 3.31 for all n > m, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.[31] Zähle, M. (1990). A kinematic formula and moment measures of random sets. Math.

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Sharp Recovery Bounds for Convex Demixing, with Applications

2014

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Demixing refers to the challenge of identifying two structured signals given only the sum of the two signals and prior information about their structures. Examples include the problem of separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis, and the problem of decomposing an observed matrix into a low-rank matrix plus a sparse matrix. This paper describes and analyzes a framework, based on convex optimization, for solving these demixing problems, and many others. This work introduces a randomized signal model which ensures that the two structures are incoherent, i.e., generically oriented. For an observation from this model, this approach identifies a summary statistic that reflects the complexity of a particular signal. The difficulty of separating two structured, incoherent signals depends only on the total complexity of the two structures. Some applications include (i) demixing two signals that are sparse in mutually incoherent bases; (ii) decoding spread-spectrum transmissions in the presence of impulsive errors; and (iii) removing sparse corruptions from a low-rank matrix. In each case, the theoretical analysis of the convex demixing method closely matches its empirical behavior.Communicated by Emmanuel Candès.

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Robust smoothed analysis of a condition number for linear programming

Cited by 7 publications

References 34 publications

On a problem posed by Steve Smale

On a problem posed by Steve Smale

Coverage processes on spheres and condition numbers for linear programming

Sharp Recovery Bounds for Convex Demixing, with Applications

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