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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)
×ØÖÖØº The best constant and the extreme cases in an inequality of H.P. Rosen-thal, relating the p moment of a sum of independent symmetric random variables to that of the p and 2 moments of the individual variables, are computed in the range 2 < p ≤ 4. This complements the work of Utev who has done the same for p > 4. The qualitative nature of the extreme… (More)