The following is a proof which is independent of this characterisation. First assume that ‖ ‖ is non-archimedean. Let x, y ∈ K. Using that ‖ ‖ extends | | we then obtain |x + y| = ‖x + y‖ ≤ max{‖x‖, ‖y‖} = max{|x|, |y|} which shows that | | is non-archimedean. Now assume that | | is non-archimedean. Let x, y ∈ K̂. Let ε > 0. Since K is dense in K̂ there exist u, v ∈ K such that ‖x − u‖ < ε and ‖y...

A temporal constraint language Γ is a set of relations with first-order definitions in (Q;<). Let CSP(Γ) denote the set of constraint satisfaction problem instances with relations from Γ. CSP(Γ) admits robust approximation if, for any ε ≥ 0, given a (1 − ε)-satisfiable instance of CSP(Γ), we can compute an assignment that satisfies at least a (1−f(ε))-fraction of constraints in polynomial time....

A quantile summary is a data structure that approximates to ε-relative error the order statistics of a much larger underlying dataset. In this paper we develop a randomized online quantile summary for the cash register data input model and comparison data domain model that uses O( ε log 1 ε ) words of memory. This improves upon the previous best upper bound of O( ε log 3/2 1 ε ) by Agarwal et a...

Let Xm,ε be the distribution over m bits (X1, . . . , Xm) where the Xi are independent and each Xi equals 1 with probability (1 + ε)/2 and 0 with probability (1 − ε)/2. We consider the smallest value ε∗ of ε such that the distributions Xm,ε and Xm,0 can be distinguished with constant advantage by a function f : {0, 1}m → S which is the product of k functions f1, f2, . . . , fk on disjoint input...

A quantile summary is a data structure that approximates to ε error the order statistics of a much larger underlying dataset. In this paper we develop a randomized online quantile summary for the cash register data input model and comparison data domain model that uses O((1/ε) log(1/ε)) words of memory. This improves upon the previous best upper bound of O((1/ε) log3/2(1/ε)) by Agarwal et al. (...

We explore a number of problems related to learning and covering structured distributions. Hypothesis Selection: We provide an improved and generalized algorithm for selecting a good candidate distribution from among competing hypotheses. Namely, given a collection of N hypotheses containing at least one candidate that is ε-close to an unknown distribution, our algorithm outputs a candidate whi...

Inhibitor of IκB kinases (IKK) are key regulators of NF-κB signaling. Three IKK isoforms-α, β, and ε-have been linked to oncogenesis, yet the precise components of NF-κB signaling in ovarian cancer have not yet been dissected. We surveyed 120 ovarian cancer specimens for IKK-ε expression. Notably, cytoplasmic expression was elevated in metastatic lesions relative to primary tumors (P = 0.03). T...

1 Optimality theorems for JL Yesterday we saw for MJL that we could achieve target dimension m = O(ε −2 log N), and for DJL we could achieve m = O(ε −2 log(1/δ)). The following theorems tell us that not much improvement is possible for MJL, and for DJL we have the optimal bound. Theorem 1 ([Alo03]). For any N > 1 and ε < 1/2, there exist N + 1 points in R N such that achieving the MJL guarantee...

In the polytope membership problem, a convex polytope K in R is given, and the objective is to preprocess K into a data structure so that, given a query point q ∈ R, it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting and assume that d is a constant. Given an approximation parameter ε > 0, the query can be answered either way if the distance...