Converse implication

Abstract: The paper considers spin systems on the d-dimensional integer lattice Zd with nearest-neighbor interactions. A sharp equivalence is proved between decay with distance of spin correlations (a spatial property of the equilibrium state) and rapid mixing of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). Specifically, we show that if the mixing time of the Glauber dynamics is O(n log n) then spin correlations decay exponentially fast with distance. We also prove the converse implication for monotone systems, and for general systems we prove that exponential decay of correlations implies O(n log n) mixing time of a dynamics that updates sufficiently large blocks (rather than single sites). While the above equivalence was already known to hold in various forms, we give proofs that are purely combinatorial and avoid the functional analysis machinery employed in previous proofs. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004Supported by EPSRC grant “Sharper Analysis of Randomised Algorithms: a Computational Approach” and by EC IST Project RAND-APX.Supported in part by NSF grants CCR-9820951 and CCR-0121555, and by DARPA Cooperative Agreement F30602-00-2-060.

Abstract: We consider the question of whether P ne NP implies that there exists some concept class that is efficientlyrepresentable but is still hard to learn in the PAC model of Valiant (CACM '84), where the learner is allowed to output any efficient hypothesis approximating the concept, including an "improper" hypothesis that is not itself in the concept class. We show that unless the polynomial hierarchy collapses, such a statement cannot be proven via a large class of reductions including Karp reductions, truth-table reductions, and a restricted form of non-adaptive Turing reductions. Also, a proof that uses a Turing reduction of constant levels of adaptivity would imply an important consequence in cryptography as it yields a transformation from any average-case hard problem in NP to a one-way function. Our results hold even in the stronger model of agnostic learning. These results are obtained by showing that lower bounds for improper learning are intimately related to the complexity of zero-knowledge arguments and to the existence of weak cryptographic primitives. In particular, we prove that if alanguage L reduces to the task of improper learning of circuits, then, depending on the type of the reduction in use, either (1) L has a statistical zero-knowledge argument system, or (2) the worst-case hardness of L implies the existence of a weak variant of one-way functions defined by Ostrovsky-Wigderson (ISTCS '93). Interestingly, we observe that the converse implication also holds. Namely, if (1) or (2) hold then the intractability of L implies that improper learning is hard.

Abstract: In Euclidean space a set of constant width has the property that it is not a proper subset of any set of the same diameter. The converse implication is also true. Here we show that if Euclidean is replaced byn-dimensional Banach space the direct statement is true, but the converse statement is false. Attention is drawn to the problem of characterising those Banach spaces of finite dimension for which the converse is true.

Abstract: can be approximated by finite rank operators uniformly on compact sets. It is clear that Xhasabasis ~ XhasBAP =~ XhasAP The fact that the converse implication to the second one does not hold in general was discovered by Figiel and Johnson [8] soon after Enflo's example [7] of a space without AP. The main purpose of this paper is to show that also the implication "BAP=~basis" does not hold in general; this answers problems asked by a number of authors (e.g. [14], [18], [27]). Before stating the result, we recall more notation. For a given basis (x n) of X one denotes by bc (x n) (the basis constant of ix~)) the smallest K such that