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Abstract Given a compact doubling metric measure spaceXthat supports a 2-Poincaré inequality, we construct a Dirichlet form on$$N^{1,2}(X)$$ that is comparable to the upper gradient energy form on$$N^{1,2}(X)$$ . Our approach is based on the approximation ofXby a family of graphs that is doubling and supports a 2-Poincaré inequality (see [20]). We construct a bilinear form on$$N^{1,2}(X)$$ using the Dirichlet form on the graph. We show that the$$\Gamma $$ -limit$$\mathcal {E}$$ of this family of bilinear forms (by taking a subsequence) exists and that$$\mathcal {E}$$ is a Dirichlet form onX. Properties of$$\mathcal {E}$$ are established. Moreover, we prove that$$\mathcal {E}$$ has the property of matching boundary values on a domain$$\Omega \subseteq X$$ . This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form$$\mathcal {E}$$ ) on a domain inXwith a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.
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Private measures, random walks, and synthetic data
Abstract Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex—but very common—machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates aprivate measurefrom a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data in general compact metric spaces, for any fixed privacy budget$$\varepsilon $$ bounded away from zero. A key ingredient in our construction is a newsuperregular random walk, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmically slowly.
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- PAR ID:
- 10501637
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Probability Theory and Related Fields
- Volume:
- 189
- Issue:
- 1-2
- ISSN:
- 0178-8051
- Format(s):
- Medium: X Size: p. 569-611
- Size(s):
- p. 569-611
- Sponsoring Org:
- National Science Foundation
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