Abstract Let λ 1 , … , λ n denote the eigenvalues of a n × n matrix, ordered by nonincreasing absolute value, and let γ 1 ≥ ⋯ ≥ γ n denote the tropical eigenvalues of an associated n × n matrix, obtained by replacing every entry of the original matrix by its absolute value. We show that for all 1 ≤ k ≤ n , | λ 1 ⋯ λ k | ≤ C n , k γ 1 ⋯ γ k , where C n , k is a combinatorial constant depending only on k and on the pattern of the matrix. This generalizes an inequality by Friedland (1986), corresponding to the special case k = 1 .
When considering a field with an archimedean absolute value, like the field of complex numbers equipped with its usual absolute value, the amoeba of a hypersurface does not coincide any longer with a tropical hypersurface, however, it can be approximated by such a hypersurface, called spine, in particular, Passare and Rullg rd @cite_20 showed that the latter is a deformation retract of the former. In a recent work, Avenda n o, Kogan, Nisse and Rojas @cite_23 gave estimates of the distance between a tropical hypersurface which is a more easily computable variant of the spine, and the amoeba of a original hypersurface. However, it does not seem that the present bounds could be derived by the same method.
@cite_23 @cite_20
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Abstract Given any complex Laurent polynomial f , Amoeba ( f ) is the image of its complex zero set under the coordinate-wise log absolute value map. We discuss an efficiently constructible polyhedral approximation, ArchTrop ( f ) , of Amoeba ( f ) , and derive explicit upper and lower bounds, solely as a function of the number of monomial terms of f , for the Hausdorff distance between these two sets. We also show that deciding whether a given point lies in ArchTrop ( f ) is doable in polynomial-time, for any fixed dimension, unlike the corresponding problem for Amoeba ( f ) , which is NP -hard already in one variable. ArchTrop ( f ) can thus serve as a canonical low order approximation to start a higher order iterative polynomial system solving algorithm, e.g., homotopy continuation. The amoeba of a holomorphic function @math is, by definition, the image in @math of the zero locus of @math under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function @math with the property that its Monge-Ampere mass is concentrated to the amoeba of @math We obtain results of two kinds; by approximating @math with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of @math ; by computing the Monge-Ampere measure, we find sharp bounds for the area of amoebas in @math . We also consider systems of functions @math and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations. We present a framework for robustly estimating registration between a 3D volume image and a 2D projection image and evaluate its precision and robustness in spine interventions for vertebral localization in the presence of anatomical deformation. The framework employs a normalized gradient information similarity metric and multi-start covariance matrix adaptation evolution strategy optimization with local-restarts, which provided improved robustness against deformation and content mismatch. The parallelized implementation allowed orders-of-magnitude acceleration in computation time and improved the robustness of registration via multi-start global optimization. Experiments involved a cadaver specimen and two CT datasets (supine and prone) and 36 C-arm fluoroscopy images acquired with the specimen in four positions (supine, prone, supine with lordosis, prone with kyphosis), three regions (thoracic, abdominal, and lumbar), and three levels of geometric magnification (1.7, 2.0, 2.4). Registration accuracy was evaluated in terms of projection distance error (PDE) between the estimated and true target points in the projection image, including 14 400 random trials (200 trials on the 72 registration scenarios) with initialization error up to ±200 mm and ±10°. The resulting median PDE was better than 0.1 mm in all cases, depending somewhat on the resolution of input CT and fluoroscopy images. The cadaver experiments illustrated the tradeoff between robustness and computation time, yielding a success rate of 99.993 in vertebral labeling (with 'success' defined as PDE <5 mm) using 1,718 664 ± 96 582 function evaluations computed in 54.0 ± 3.5 s on a mid-range GPU (nVidia, GeForce GTX690). Parameters yielding a faster search (e.g., fewer multi-starts) reduced robustness under conditions of large deformation and poor initialization (99.535 success for the same data registered in 13.1 s), but given good initialization (e.g., ±5 mm, assuming a robust initial run) the same registration could be solved with 99.993 success in 6.3 s. The ability to register CT to fluoroscopy in a manner robust to patient deformation could be valuable in applications such as radiation therapy, interventional radiology, and an assistant to target localization (e.g., vertebral labeling) in image-guided spine surgery. The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50).