Dynamic approach to a stochastic domination: The FKG and Brascamp-Lieb inequalities

Funaki, Tadahisa; Toukairin, Kou

doi:10.1090/s0002-9939-07-08757-6

Cited by 5 publications

(1 citation statement)

References 17 publications

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“…The first two parts follow from Cafarelli's theorem, see e.g. Theorem 3.1 in [FT07] where one may set the function ψ to be a quadratic function within the interval (−T, ∞) and either 0 outside it when T < 0 or a linear function tangent to the graph of y = x 2 at the point…”

Section: B Additional Toolsmentioning

confidence: 99%

Improved Lower Bounds for Learning Intersections of Halfspaces

Klivans

Sherstov

2006

Learning Theory

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Recent work of Klivans, Stavropoulos, and Vasilyan initiated the study of testable learning with distribution shift (TDS learning), where a learner is given labeled samples from training distribution D, unlabeled samples from test distribution D ′ , and the goal is to output a classifier with low error on D ′ whenever the training samples pass a corresponding test. Their model deviates from all prior work in that no assumptions are made on D ′ . Instead, the test must accept (with high probability) when the marginals of the training and test distributions are equal.Here we focus on the fundamental case of intersections of halfspaces with respect to Gaussian training distributions and prove a variety of new upper bounds including a 2 (k/ǫ) O( 1) poly(d)-time algorithm for TDS learning intersections of k homogeneous halfspaces to accuracy ǫ (prior work achieved d (k/ǫ) O(1) ). We work under the mild assumption that the Gaussian training distribution contains at least an ǫ fraction of both positive and negative examples (ǫ-balanced). We also prove the first set of SQ lower-bounds for any TDS learning problem and show (1) the ǫ-balanced assumption is necessary for poly(d, 1/ǫ)-time TDS learning for a single halfspace and (2) a d Ω(log 1/ǫ) lower bound for the intersection of two general halfspaces, even with the ǫ-balanced assumption.Our techniques significantly expand the toolkit for TDS learning. We use dimension reduction and coverings to give efficient algorithms for computing a localized version of discrepancy distance, a key metric from the domain adaptation literature.

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Section: B Additional Toolsmentioning

confidence: 99%

Improved Lower Bounds for Learning Intersections of Halfspaces

Klivans

Sherstov

2006

Learning Theory

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Concentration under scaling limits for weakly pinned Gaussian random walks

Bolthausen

Funaki

Otobe

2008

Probab. Theory Relat. Fields

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We study scaling limits for d-dimensional Gaussian random walks perturbed by an attractive force toward a certain subspace of R d , especially under the critical situation that the rate functional of the corresponding large deviation principle admits two minimizers. We obtain different type of limits, in a positive recurrent regime, depending on the co-dimension of the subspace and the conditions imposed at the final time under the presence or absence of a wall. The motivation comes from the study of polymers or (1 + 1)-dimensional interfaces with δ-pinning.

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