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The paper is devoted to the comparison between global properties and local properties of symmetric convex bodies of high dimension By global properties we refer to properties of the original body in question and its images under linear transformations while the local properties pertain to the structure of lower dimensional sections and projections of the… (More)
New concepts related to approximating a Lipschitz function between Banach spaces by aane functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the… (More)
MAX CUT is the problem of partitioning the vertices of a graph into two sets max imizing the number of edges joining these sets This problem is NP hard Goemans and Williamson proposed an algorithm that rst uses a semide nite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n dimensional sphere and then uses a random… (More)
In this paper we consider four previously known parameters of sign matrices from a complexity-theoretic perspective. The main technical contributions are tight (or nearly tight) inequalities that we establish among these parameters. Several new open problems are raised as well.
We prove a concentration inequality for functions, Lipschitz with respect to the Euclidean metric, on the ball of`n p , 1 p < 2 equipped with the normalized Lebesgue measure.
We compute the number of summands in q-averages of norms needed to approximate an Euclidean norm. It turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the Euclidean norm. Particular attention is given to the case q = 1 (in which the average is replaced with the maxima). This is closely… (More)
The main result is that a Banach space X is not super-reflexive if and only if the diamond graphs D n Lipschitz embed into X with distortions independent of n. One of the consequences of that and previously known results is that dimension reduction a-la Johnson–Lindenstrauss fails in any non super reflexive space with non trivial type. We also introduce the… (More)
It has been suggested in the past that adaptation effects may serve a useful role in perception. This paper shows that if the adaptation process follows a simple scheme, called proportional gain adjustment, then it can fulfil two useful functions: correction of errors and recalibration. The proposed scheme controls the gain of the system. Although it is… (More)
Concentration inequalities are estimates for the degree of approximation of functions on metric probability spaces around their mean. It turns out that in many natural situations one can give very good such estimates, and that these are extremely useful. We survey here some of the main methods for proving such inequalities and give a few examples to the way… (More)